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Anjan Biswas Daniela Milovic Matthew Edwards
Mathematical Theory of Dispersion-Managed Optical Solitons With 23 figures
• -
~~~1iftB1\&U HIGHER EDUCATION PRESS
til Springer
Authorsl IAnjan Biswas lIJept or ApplIed Mathematics 1& Theoretical Physics IDelaware State University 1200 N DUpOllt Highway pover, DE 19901-2277, USA IE-mail: blswas.al1)
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Daniela Miloviq Faculty or Electromc Engmeenngl Department of Te1ecommunicationN University of Nis Scrbia E-mail:
[email protected]
Matthew EdWards
ISchool of Arts and Scicnccs lIJepartmel1t or PhYSlcsl (Alabama A & M UmversIly INormal, AL-35762, USAI IE-mail· matthew edwardsCiiJaam!! ed!!1
IISSN 18678440
e ISSN 18678459
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IPreface
rIhe concept of dispersion-managed (Dr",!) optical solitons was introduced il~ Whe early 1990s. The advent of sllch Dl'1il solit.ons has changed t.he world of oppeal solitons. In fad. the\' arc governed by the dispersion-managed nonlinearl IS('hri>ciinger's eqnat.ions (DI'vI-NLSE), unlike in the ('ase of claHHical solit.OIlHI Iwhich are governed by t,he pure nonlinear Schri)dinger's eqnat.ions (NLSE). lIt is to be noted that the pure N LSh is integrable by the clai:li:lical method of Ilnverse Scattering Transform, while the cai:le of Dl\/l-NLSE is not integrable. [This leads to a lot of challenges and hindrances in studying the DlVI-NLSE. ~1<-lIlY methods have been intTodnced in order to Hhuiy the D1'vI-NLSE. Theyl ~lrc the varIatIOnal pnw'lplc, solIton perturbatIOn theory, moment, method aHI livell as the ai:lymptotic analYi:lis. These methods of i:ltudying the ))1\/1 solitonsl ~lave introduced a \vider picture in this areaj IThis book introduces and exposes the concept of Dl\I solitons from scratch. rLater the Hohton pert.nrbatIOn theory and t,he 'vanatIOnal pnnciple are mt.ro~inced t,o Hhuiy the dynamics of pnlseH t,hat, propagat,e throngh opt.i('al fiberH. rrhe types of optical fi bers that are studied in this book are the polarizationl !preserving fibers. birefringent fibers and finallv the case of multiple channelsl as also taken into consideration. The asymptotic analvsis is used to studv thel hmlHi-linear pulHes in opt.i('al fiberH ,\There along '\Tit,h the diHpersion, t,he nonIlinearity is ali:lo managed. Later, the Gabitov-Turitsyn equationi:l are derivedl ~or these three types of optical fibers using the ai:lymptotic analysii:l. SubsekiuenUv. hIgher order asymptotIC analysIs IS earned out to denve the lllgherl brder Gabitov-Turitsvn equations [or these types o[ optical fibers. Finallv, Whe issne of opt.i('cLl ('rosstalk iH t,onched upon to complete the dis('nHHionj lIt needs to be noted that there are quite a few technical aspects that ar~ ~kipped in this text. Those issues are the collision induced !requellcy andl ~'IIrung jItter along ,vIth the amphtude jItter. I he other Important Issue that ihas been deliberately skipped is the issne of four-wave mixing. F inaIIy, onel lOt, her import.ant if)f)ue of Dlvl f)olitons that has not been touched upon if) thel ~1,spect of soliton radiation. These issues are not yeL exhaustively studied inl Ithe context of DIVf solitons and therefore requires further development bcforel
Ivi
Preface!
[being incorporated in thii:l text. Although there are a few paperi:l that hav~ been published in this context. a substantial amount o[ vmrk is yet to be donel Wo complete theHe chapt,erH. IThis book is organized aH folhn'ls: Chapt,er 1 introdu('eH t,he ne('eHHity awll ~mportance of i:ltudying the dispersion-managed optical i:lolitons ai:l opposed to ~,he classical or conventional optical solitoni:l. Chapter 2 introduces the techbicalities o[ dispersion-managed optical solitons. the conserved quantities asl Iwell aH t,he solit,on perturbation theory. Finally thiH ('hapter ends ''lith a brief [ntrociuction t,o the variational principle. Chapt,er 3 fo('uHes on t,he polarizaILion preserving libel's. Three types o! pulses are studied in this chapter. TheYI larc Gaul:ll:lian, supcr-Gaussian and l:Iuper-l:Icch puli:lCI:I. The l:Ioliton parameterl kivnamicl:I are derived. Finallv the stochastic perturbation of optical solitonsl ~s studied ,,,it,h t,he aid of soliton pert.llrbation theory. Chapters 4 and 5 deall Rvith the birefringent fibers and multiple channels. Chapter 6 details out thel laspect or optical crosstalk in both the linear as well as the nonlinear regime. IChapter 7 derives the Gabitov-Turitsvn equation for polarization-preserving lfibers. birefringent fibers as ''lell aH in the case of mult.iple channels by us~ng the technique of multiple-scale perturbation expansion. In Chapter 8, thel ~ssue of quasi-linear pulses are studied \vhich another form of optical pulses Ithat are studied where the nonlinearitv is also managed in addition to thel [group velocity disperHion. Finally, in Chapter g, the higher order asyrnptot.i('1 ~1Ilalysis iH carried out t,o derive t,he higher-order Gabitov-Turit,syn equationHI ~,hat serves ai:l an opening to the future research in this direction1 IThis book is primarily intended [or graduate students at the l\Iasters andl lDocLoral levels in Applied l\Iathematics. Applied Physics and Engineering. !Also undergraduate students, ,,,it,h senior standing, in PhYHicH and Engineer~ng will benefit out of thii:l book. The pre-requii:lite of this book ii:l a knowledg~ pf Partial and Stochastic Differential Equations, Perturbation Theory andl IQuant urn Mechanics.1 Anjan I3is''laHI II )anjela IVljlovjd
Ivlatthew Edward"
IAcknow ledgementsl
rIhe first author of the book, namely Anjall Biswas, is extremely grateful t.o Ihis parent.s for all t.heir unconditional love in his upbringing, blessings, edukation, support, encouragement and sacrifices throughout his life, till Lodav. OCIe also want,s to thank his spOllse for all the help. lIe iH most grat,eflll t,o biH onlY 5-vear-old Hon Sonml for pruviding hiln ''lith all the fUll, laughter, Ilight and unforgettable momenb, throughout the period of writing thb: book. ~~lithout such intimate moments with his son, this book would have not beenl possible. IThe work by the first author of t,he book, Anjan I3iS"\V<-1H, ,vas fully snp-
[JOrled by NSF-CREST Grant. No: IIRD-0630388 and Army Research Officel I(ARO) along with the Air Force Office of Scientific Research (AFOSR) unkler the award number: W54428- RT- ISP and these supports arc genuine! v andl ~inccrcl v appreciated. This author is extremely thankful to Prof. Fcngshad [Liu, the Chair of the Depart,rnent of Applied l'viat,hernaticH and Theoretica] fhysics for his constant enconragement. and all the help of many kinds, indnd~ng release time, he received since he joined Delaware State Oniven:lity. He is !llso thanklul and gratclul to the grant 01 Pro!. Liu, numbered DAAD 19-0;111-0375, [rom which he received financial support in numerous occasions.Thq !first author iH alHo extremely thankful to Dr. NOllreddine I'vfelikechi,the Dear~ pf College of l\iJathematics, Natural Sciences and Technology and Dr. Harr!l II.,. Williams, Provost and Vice Prei:lident of Delaware State Oniven:lity who IliIso helped hU11 on several occaSIOns WIth Ius research activItlCS at Dclawarq IState University. IThe second author of the book namely Daniela J'vlilovic, first and forefIlost. thanks God tor all the blessings throughout her life and studies. Shel ~,lso expresses prolound thanks to her parents l\Tilorad and Zora l\Tilovic: lor ~,hcir conl:itant encouragcmcnt. unconditional lovc, ::;clflcl:)l:) ::;acrifice::;. provid~ng a warm, comfort.able atmoRphere in ,vhich Rhe could think, write ann live. IShe offerR a special worn of t,hankR to her only lO-year-old son VllkaRin ,vho !brought unforgettable moments of jov and happiness into her life and inspiredl her to write this bookJ
Acknowlcdgcmcnb
IThe work of the second author Daniela lVlilovic was supported by thel k\rmy grant numbered DAAD 19-0;1-1-0375 and this support is genuinely andl ~incerely appreciat,ed. The He('ond ant,hor iH also thankful t,o Prof. FengHhar~ [Liu, the Chair of the Depart,rnent of Applied l'viat,hernaticH and Theoretica] IPhYi:lici:l tor all his i:lupport during her visit to this Department in Spring 2008.
Contents
11
12
Introduct.jon
1
References .................................................
3
Nonhnear Schrodulger's EquatIon ........................
5
ru
Dcrival ion 2 1 -I
p.2 ~.;1
T jmjlo1.ions or conycn! jOllO] soljtons
2.1.2 Dispenlion-rnanagcrncnt............................ 2.1.3 ]'vIathernatieal formulation. . . . . . . . . . . . . . . . . . . . . . . .. Integrals of motion..................................... Soliton perturbaUon theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 23-1
13
or NLSE .
Perl Jlrl)}lj.jon terms
.5 9
10 11
15 18 19
Q.4
VariatIOnal princIple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2.4.1 Perturbation termi:> . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. IReferences .................................................
20 21 22
Polarization Preserving Fibers. . . . . . . . . . . . . . . . . . . . . . . . . . .. B.l Introduction........................................... 0.2 Integrals of moUon ..................................... 3.2.1 GanHHian pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.2.2 Super-Gaussian pulses ............................ 3.2.3 Super-Sech pubes ................................ p.;l Variational principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ~L1.1 Gaussian pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.3.2 Super-Gaussian pulses ............................ B.4 Perturbation t,ernlH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.4.1 Gaussian pubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. :1.4.2 Super-Guuooiuu pubes ............................ ~1.5 Stochal:ltic perturbation ................................. if{eterenceR .................................................
27] 211 28 29 30 32 ;15 ;15 36 36 371 :171 :18 42
CQntents
~
Ii
Birefringent Fibers 4 -I Tn1.rocillc1 jon 4.2 IntegralH of IIlohon ..................................... 4.2.1 GanHHian pubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.2.2 Super-Gaw:li:lian pulses ............................ 4.3 Variational principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.:1.1 Gaussian pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.3.2 Super-Gaussian pnIseH ............................ 4.4 Perturbation t,ernlH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.1 Gaussian pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4.4.2 Supcr-Gaul:ll:lian pul::;CI:I ............................ IR <'feIn J (,('a
45 45 47] 49 49 50 52 53 55
571
58 60
Multiple Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63 fS 1
IntIodllctjon Integrals of mohOll ......... 5.2.1 Gaussian pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5.2.2 Super-Gaussian pulses ............................ VarIatIOnal prIncIple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
63
641 66 67] t5.3 tit) 5.3.1 Gaussian pubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59 5.3.2 Super-Gaui:li:lian pulses ............................ 70 f5 4 per1.llrha1 jon 1DrIlls 70 5.4.1 Gaussian pulses ................................. . 7;1 5.4.2 Super-Gaussian pnbeH ........................... . 741 IReferences 75 ~.2
16
OptIcal Crosstalk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 771 6.1 Tn-band crosstalk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 78 80 ~.2 Gaussian optical pulse h:J. I Kj1. error ro1.e 841
p.3
Seeh optical pulse ...................................... Super-Sech opheal pnbe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
85 88
lReferences .................................................
93
p.4
17
Gabitov-Turitsyn Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95 17.1 Tntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95 17.2 Polarization-preserving libel'S ............................ 96 7.2.1 Special solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99 17.3 13irelringent fibers ...................................... 100 17.4 J)WlliVl system ........................................ 1041 17.5 Properties of the kernel ................................. 1071 751T
oaakaa
em)('
1 08
7.5.2 Lossy Case ...................................... lOt) lReferenC'es 109
!CQntents
18
xjl
Quasi-linear Pulses ....................................... III l3 -I Tn1.rocillcj jon 1-1-1 ~.2
~.3
S.4
Polarization-preserving fibers ............................ 8.2.1 Losskss system .................................. 8.2.2 LOi:li:lY sYi:ltem .................................... Hiretringent tibers ...................................... t(~1.1 Losslcss system .................................. 8.3.2 lossy syst.ern ..................................... !vl11It.iple channels ....................................... 8.4.1 Lossless system .................................. H.4.2 LOl:ll:lv ::;vl:ltcrn ....................................
IR ('fern J (,('S
19
112 112 115 121 122 1241 129 129
l:n
1:141
Higher Order Gabitov-Turitsyn Equations ................ 1:171 91
Introdllctjon
~.;l
Polarization preserving fibers .......... 1371 Birefringent fibers ...................................... 142 DWDM systems ........................................ 149
p.2 ~J.4
IR eferen ('es
1371
1541
[ndex ......................................................... 1571
IChapter
~
lInt rod uction
rtJispersion-l\lanaged optical solitons or UK/1 solitons was first intronuced il~ Ithe carly -19908 and since then it bccarnc 1:1. very attractivC' topic ror op~1('aI COIIlIIlllIllcahons. DlspcrslOIl- ]\'lanagcIIlcnt. IS very Important 1Il dense! Iwavelength-diviHion IIlultiplexed (D\VDJ'vI) Hyst.enlH. DJ'vI Holitons allow thel ~ormation of ultra-long-haul Tera bit level optical networks working in alii pptical mode and maintaining optical tram;parency over vast geographicall ~·cgions. The more wavelengths in usc. the greater the need [or dispcrsion~nanagcrncnt. In I11gh channd COllnt syst.enlH, channel spacmg IS 'very dos(~ ~1Ild requires ('ontiIllloml broadband management. of dispersion and diHper~ion slope [1 101. IThe discovery of dispersion-managed optical soliton has introduced plentyl pf new methods [or high rate data transmission. By proper choice of pa~'arneters both Gordon-H;-llls effect and fonr-,vave-mixing can be significant.1}l ~llppressed t.lms pruviding nearly error-free transmission. It ,vas shmvn ir~ 11997 that bright solitons can exist in 1)1'\/) fibers and they can exist at normall !path average dispersion. Dark solitons can also exist in such fibers, even at !anomalous path averaged dispersion. This implies that the range of existenccj ~)f bright and dark solitons uverlap and therefore it becomes possible, for thel lfiri:>t time, to analYl';e interaction between bright and dark i:>olitons. IThe propagation of 1)1'\/) solitoni:> ii:> governed by the dispersion-managedl ~lOnlinear Schrodinger's equation (DJ'vI-NLSE). Dispersion management forcesl ~a('h soht,on to propagate m the normal (iISperSlOn regIme of a fiber dUrIngl ~'very map period. \\/hen t,he map period is a traction of the nonlinear lengt,h, ~,he nonlinear effects become insignificant, leading to linear puJi:>e evolutionl pver the map period. If the ,elf-phase modulation (SPM) effects are balancedl Ibv the average Ull:>pCrl:)lOn. I:>ohtonl:> can I:>UrVIVe m an average I:>cnl:>e evcn on al ~ongcr length scalel IThe N LSB that governs the classical or convent.ional solit,on propagatioIi1 ~s integrable by the classical method of Inverse Scattering Transforrn (TST), ~nlike DJ'vi-NLSE which ii:l not integrable by 1ST. I:)inee the Painleve tei:lt of ~ntegrability [2] will fail. So, i:leveral metho(li:\ have been introduced in orderl
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
1 Introduction ~o study l)1'\il-NLSh that includes variational approach~ collective variablesl !approach. soliton pcrturbation theory. moment mcthod as wcll as the asvmp~,otic c-lIlc-LlYHis. These methods really bronght wider pict.nre in the area of DJ'vI Holitons IThe main feature of Divi soliton ii, that it does not maintain ib, i:lhape, lividth or peak power. unlike a fundamental soliton. However, Ill\/l solitonl !parameters repeat through dispersion map [rom period to period. This makesl
rtJl\l Holitons appli('able in ('ommuni('ations in spit,e of changes in Hhape, "\vidthl ~n' peak power. From a sYHt,enlH standpoint, theHe DJ'vl solitons perform better. IBy a proper choice 01 initial pulse energy, width and chirp will periodicailyl !propagate m the t)ame dIt)per::;lOn map. I he pube energlCt) much t)maIIer thar~ g;nhcal energv should be aVOIded m desIgmng Dl\l t)ohton ::;vt)tem. whereasl ~f t,he puhm energy iR t,he same as the critical energy, it is the mORt, suitablel ~ituation. An inappropriate choice of initial pulse energy may cause pulsel ~ntcraction
and thus lead to detrimental pulse distortion. The required mapl
!penod becomes shorter as the bIt rate mcrea::;es. IThe main difference bet"\veen the average group velo(,ity dispersion (GVD) ~olitoni:l and Dlvl solitons liei:l in iti:l higher peak power requirements for i:lUS-
Itaining DJ'vI solitons. The larger energv of Dl\I solitons bencfits a soliton sysItem by improving the signal-to-noise ratio (SNTI) and dccreasing the timingl Ijitter. The use of periodi(, disperHion map enableH nltra high data t.ranHmis~ion Clver large diHt.c-1Ilces "\Vit.llOut using any in-line opt.i('al filters sin('e thel
!periodic use of dispersion-compeni:lating fibers (DCF) reducei:l timing jitter a large factor. IAn important application 01 the dispersion-managcment is in upgrad-
!w
~ng
the existing terreHt.rial nehvorkH employing Ht.c-1Ildanl fibers. Recent, ex-
lPeriments show that the use of 1)1'\/1 solitons hai:l the potential of reali~ingj ~.ransoceanic light-wave i:lystems capable of operating with a capacity of 1 [b/s or more. !Optical amplifiers compensatc fiber losses but on thc other hand inducel ~.iming
jitter. This phenomena iH mainly caused by the change of
solit,or~
which affects the group velocity or the i:lpeed at which the puli:lel IPropagates through the hber. The timing jitter can become an appreciabl~ [raction o[ a bit slot [or long-haul svstcms as bit slots becomc less thaI~ 1100 ps. If left nw'ontrolled, Hu('h jitter can cause large pmver penaltieH, Fori ~requency
rtJl\l sohtons tnnmg JItter IS consHlerably smaller t,han t,hat, for fundamentall ~olitoni:l and the physical reason for jitter reduction is related to the enhancedl fnergy of the D1\l i:lolitons. From a practical point of view~ reduced timingl Ijitter o[ DJ\..f solitons permits much longer transmission distances. lAB the bit rate increases. solit,on-soliton int,eraction becomes a crit.i('al is~ue.
CoIliHion length dependH on the details of dispersion map. The
Hysten~
lPerlorrnance can be optirni:;;ed b.'i an appropriate choice 01 rnap strength1 IDispert)ion- management can be efficientlv u::;cd in several situationt) at) fol~
IHdcrcnccs
31
11. For optimum pulHe generation in a mode-locked laser operating at a wave~ength around 1 pm or shorter. The normal chromatic dispersion has to Ibe uver ('ompensatcd m order t,o utIhze the anomalous (hSperSlOn rcglInc, ~vhere soliton effects can help to obtain shorter pulses. It is nsually alsol Inecessary to compenHate carefully the higher-order diHpersion, i.e., to con~rol the group delay disperHion over a significant optical bandwidth1 2, In a mode-locked fiber laser, dispersive and nonlinear cIIects can becomel ~o strong that the pulse paramet,ers (including the pulse duration andl I('hirp) 'vary signifi(,antly dnring each resonat,or ronnd-t.rip. \Vith a snitlable combination o! hbers exhibiting normal and anomalous dispersion, ill ~tretched-pube fiber latler can bc realized, which can generate pubctl (Dl\J ~oliton:::l) "\vith tlignificantly higher pube energy than with c,g. :::loliton model ~ocking,
3, Similar effect can be llsed in optical fiber comnlllnications: a fiber-optic link! Iconsisting of a periodic arrangement of fibers wiLh nonnal and anomalous! idisper:::lion can help to :::luppress nonlinear effects :::luch as channel crosstalkj ~ria four-wave mixing. It is possible t,o suppress the Gordon-IIans timingl jitter at the same time, if the average chromatic disperHion is zero.
IReferences F. Abdullaev. S. Darmanyan &: P. Khabibullaev. Optical SoMons. Springer- New York. fNY. LSA. (1993). ~. 1'v1. ,J. Ahluwi('h &: H. Stcglll". /3oh/'o'fl8 (Jm] /Ju: TfliOt-;'f"8t-: SC(J/Je'f"iny TnJ'fl8j'()nn. STAT\:I. iPhiladelphia, PA. CSA, (1981), p. G. P. AgrawaL Nonlinear Fiber Optics. Academic Press, San Diego, CA. US.A. (1995). ft. N. N. Akhmediev & A. i\nkicwi('7;. ,'30litons, Nonlinear Pulse.'! and Rcanu;. Chapmanl land Hall, London. UK. (1997).1 ff). A, 13iswas & g, Konar. introduction to Non-Kerr Law Optical S'oliton.s, eRe Press, IBoca Raton, PL USA. (2006). n. A. Hasegawa x,-, Y. Kodama. 8olttons in Optical Communications. Clarendon Press. !Oxford, UK. (1995), 17. Y. Kivshar & G. P. Agrmo..'iJ.l. Optical Soliton.'!: From Fibe'r.'! to Photonic Cry.'!tals. IAcademic Press, San Diego. CA. CSA. (200:3). fS. B. A. l\1alomed. Soliton Manaqement in Periodic Systems. Springer, Heidelberg. DE. (2006). P. L. Mollenaller & J. P. Gordon. Soliton zn Optical Fibers: Hmdamenlals and Applzcakions. Academic Press, San Diego, CA. LSA, (2006).1 ~O. V. E. Zaklmrov & S. \iVabnitz. Optical Solitons: Theoretical Challenges and Iruln.~tT'ia~ IPer8pedi've.~. Springer, Heidelberg. DE. (1999).1 OC.
IChapter 2
rNonlinear Schrodinger's Equation
this chapter the nonlinear Schrodinger's equation (NLSE) will be derived the basic principles of Electromagnetic Theory. This equation will be ~hen modified in presence of dispersion-management. The conserved quanti~ies of this dispersion-managed NLSE (DM-NLSE) will be derived. The varilational principle used for solving the DM-NLSE will be introduced. Finally, ~his chapter will end with a brief introduction to the soliton perturbation ~n
~rom
~heory.
~.1
Derivation of NLSE
IPulse propagatIOn through nonlmear and dIsperSIve medmm IS governed by ~he NLSE that is derived from Maxwell's equation. Maxwell's equations arise ~n Electromagnetic Waves and are described in a medium that is assumed to !be isotropic with no free charges (i.e., no plasma). NLSE can be modified for ~he case of plasma generation by adding terms to the NLSE that account for lffiultI-photon absorptIOn and plasma mdex change. IThe starting point is the Maxwell's equations which takes the following Worms, assuming no free charges:
(2.1) '\l·B=O
aB
'\lxE=-
at
'\l x H =
aD at
(2.2)
(2.3) (2.4)
~here E, H are electrical and magnetic fields while D, B are the respective !flux densities and all vectors depend on three spatial coordinates and time
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
16
2 Nonlinear Schrodinger's Equation
I( t). The spatial and time dependence of vector fields is not shown explicitly keep the notations simple. First of all, separate D into linear and nonlinear parts as (2.5) D=EoE+P
~o
~here EO is the permittivity of vacuum and P is the induced polarization. IF'or optical fibers, the magnetic flux (B), that is written as
+M
B=/1o H
(2.6)
~here /10 is the vacuum permeability, can be simplified by taking M = 0 [because silica glass has no magnetic effect. The second term in the rightIhand side of Eq.(2.5) represents nonlinear polarization that accounts for all !higher order effects, frequency generation, nonlinear absorption, nonlinear ~ndex changes and others. The first term in the right-hand side of Eq.(2.5) ~m the other hand accounts for linear response. IThe wave equation can now be derived by taking the curl of Eq. (2.3) and IUsing Eqs.(2.4)-(2.6) as
\7 x \7 x E = -
I
1
a2 E
c2 at 2
-
/10
ap at2
(2.7)
!Where the speed of light in a vacuum is defined as c = y'EOiiO. By virtue pf Eq.(2.5) and using the identity from Vector Calculus, Eq.(2.7) can be
(2.8) IThis is the nonlinear wave equation in the time domain. Assume that the 1P0larIzatIOn does not change dUrIng the propagatIOn, so scalars can be used ~nstead of vectors. The coupling between orthogonal polarization states is, in general, very small and can be neglected. Applying Fourier transform (FT) ~o E(r, t) and per, t):
r(r,
I: I: I:
IE(r,w)
=
E(r,t)eiwtdt
(2.9)
Ip(r,w)
=
P(r,t)eiwtdt
(2.10)
t)
= EO
x(p, t - t')E(r, t - t')dt'
(2.11)
IThe linear susceptibility X is scalar for an isotropic medium such as silica glass. Equation (2.11) represents the delayed nature of temporal response ~hich is a feature that has important implications for optical fiber commuinications through chromatic dispersion. By virtue of Eq.(2.11), Eq.(2.7) can [be written in the frequency domain as
12] Derivation of NLSE
7
hIV -X \7 X E
=
W2 -
-E(r,w)--;?E
(2.12)
~ow, the dielectric constant is frequency dependent and is, in general, complex. It is written as (2.13) E(r,w) = 1 + x(r,w)
land x( r, w) is the Fourier transform of x( r, t). Its real and imaginary parts lare related as
. >.)2
E
= ( n+ ~w
(2.14)
lHere the refractive index n and the absorption coefficient a are both freIquency dependent and related to X as
In = {I + R(X)}2
(2.15)
!where Rand 8' represents the real and imaginary parts respectively. Fiber Idispersion (chromatic and material dispersion), related to the frequency delPendence of n, is one of the most important factors that limit performance pf the fiber-optic communication systems. !For fibers with low optical losses, Eq.(2.12) can be simplified by taking E ~o be real and replacing it by n 2 . Since n( r, w) does not depend on spatial Icoordinate in both the core and in the cladding of a step-index fiber, the ~oIIowing identity can be used:
It'
X
\7 x E
== \7 (\7 . E) - \7 2 E
(2.16)
!Equation (2.16) holds approximately as long as the index change occur rver a length scale much longer than the wavelength. By using Eq.(2.16) ~n Eq.(2.12), it is possible to obtain (2.17) !where the wave number ko is defined as k _ w _ 21f 0- ~ A
(2.18)
land >. is the vacuum wavelength of the optical field oscillating at the freIquency w. A specific solution of the wave equation (2.17) is called optical Imode and it must satisfy certain boundary conditions and its spatial dis~ribution remains while propagating along optical fiber. Single-mode fibers ~upport only the HEn mode, also known as the fundamental mode of the !fiber. All higher order modes are cut off at the operating wavelength. Each [requency component of the optical field propagates through a single mode fiber as @(r,w) = xF(x,y)iJ(O,w)e i (3z (2.19)
18
2 Nonlinear Schrodinger's Equation
Iwhere X is the polarization unit vector, B(O, w) is the Fourier transform of the ~nitial amplitude, and f3 is the propagation constant. The field distribution pf the fundamental fiber mode F(x, y) can be approximated by the Gaussian Idistribution (2.20) Iwhere w is the field radius and is referred to as the spot size. It is determined [by fitting the exact distribution to the Gaussian function. To force a fiber ~o support only fundamental mode, the V-number of a fiber is an indicative Imeasure of how many higher-order modes can propagate through the fiber [19,74]. It is defined as
V
=
27f
>:aNA
(2.21)
Iwhere k = 27f/A is the wavelength of the radiation or the wave number, a ~s the fiber core radius and N A is the numerical aperture relating to the ~efractive index-step between the core and the cladding. To force a fiber to pnly support the propagatIOn of the lowest-order fundamental mode, the VIllumber must be such that V :s; 2.405 [19, 74]. Fiber becomes multi-mode ~hen the core radIUs IS mcreased, and thus the V -number becomes hIgher land the step-index remains unchanged. The spot size is for 1.2 < V < 2.4 [74]. (2.22) IThe effective core area Aeff = 7fW2 determines the light confinement in the Icore. Nonlinear effects become stronger in fibers with smaller values of the effective core area. IFor the laser pulse contammg many optIcal cycles, the optIcal sIgnal trav~ling in the z direction can be expressed as the product of a complex envelope IA(z, t) and a carrier wave:
iE(z, t)
= A(z, t)ei{wot-,B(wo)z}
(2.23)
IBy taking the Fourier transform of the signal A(z, t)e'wot into its frequency ~pectrum A(z,w) at any plane z and propagate each frequency component [orward a small distance dz, and then inverse Fourier-transform these components to temporal domain, it is possible to obtain signal envelope A(z, t) lat the plane z + dz:
r (z + dz, t) IA
=
~ 27f
1 1 00
-00
dtlw
00
A(z, t')e i{ b.w(t-t')} e-ib.,BdZdt'
(2.24)
-00
Iwhere tlw = w - wo, tlf3 = f3(w) - f3(wo). The propagation constant f3(w) has ~he weak nonlinear dispersive form as (2.25)
12] Derivation of NLSE
9
Iwhere {3m = (dm{3/dwmt=wo' with (31 = 1/vg and Vg being the group ve~ocity. The group velocity dispersion (GVD) coefficient {32 is related to the Idispersion parameter D by
(~) = _ 27fC (32 rIn = !£ d)" Vg
)..2
(2.26)
IThe coefficient (33 is related to the dispersion slope S as
r10
=
(27fC) 2
--:\2
{33
+
47fC )..3 {32
(2.27)
IBy replacing ~w by i (8A(z, t)) /8t it is possible to calculate (8A(z, t)) /8z ~rom Eq.(2.24) as (2.28) IThis is the basic propagation equation that governs the evolution of pulses ~nside a single-mode fiber. In absence of dispersion, namely when (32 = (33 = 0, pptical pulse propagates without change in shape such that A(z, t) = A(O, t[B1Z). Equation (2.28) can be extended to include self-phase modulation and Icross-phase modulation effects by adding a nonlinear term. The resulting ~quation is known as the nonlinear Schrodinger's equation (NLSE) and has ~he form:
(2.29)
f?1.1 Limitations
oJ conventional
solitons
[n fibers with constant GVD solitons can propagate over unlimited distances !without any distortion. However, in reality, optical fibers have small but fiInite losses thus requiring amplification for long distance unregenerated trans!mission. Laser amplifiers such as erbium-doped fiber amplifiers (EDFA) are !usually used where the amplified spontaneous emission (ASE) noise is added ~o the signal at each amplification. The noise modulates the soliton frequency !randomly that leads to the random timing jitter through GVD of the fiber. IThis timing jitter is called the Gordon-Haus (GH) timing jitter [19]. The !magnitude of the GH timing jitter (variance of the fluctuation of the arrival ~ime of a pulse) is proportional to the GVD ofthe fiber when the pulse width ~s kept constant. Thus one can expect smaller timing jitter when the GVD is !reduced. However, the GVD cannot be reduced arbitrarily because the soli-
110 ~,on
2 Nonlinear Schrodil1ger's Equationl
energy and consequently the signal-to-noise ratio (SIR) at the receive~
Iwill reduce as SIR is proportional Lo the fiber GVD. Thus. Lhe GH Limingl Ijitter HetH a fundamental limit. to Holiton transrnissionj IThe t.ranHlation from frequency flnctllatioml to t.irning jitter through fiber ICV I) is a major i:lource of hindrance to performance enhancement of solitonl ~ystems. Hei:lides the GH effect the interaction ot i:lolitol1s ai:l well as the inter!action of solitons v.lith acoustic ",raves arc reduced due to the smaller valucl ~)f the G\iD, but "\vit,h a limit placed by the SIR requirement. IThe nonlinear interact-ion bet-ween adjaceIlt pulses t,hrongh exponentiall Itails 01 sech pulses leads to packing solitons with small separation. Separabon "\vith larger than fivc timcl:l the pube "\vidth arc m;uallv neccl:ll:larv. Thul:l, ~,hc :::;pcctral "\vidth of thc I:lignal i:::; con:::;idcrably larger than that of modula~.ion formats nsed in linear transmission, like t,he non-retnrn t,o zero (NRZ) ~onnat, that leads to inefficiency in bandwidth utilization. This is an inherent klrawback or the soliton transmissionJ
2.1.2 Dispersion-managementl Wor fiber t.ramllnission sYHt,enlH operating in linear regimes, t,he tot'eLl diHper~lOn acC'nnmlated over the sYHtem Hhonld be made aH dOHe t,o zero aH possIblel ~,o avoid dii:lpersive pulse broadening. \IV-hen a dii:lpersion-shifted fiber is ui:led, Iwhere the fiber GVD is almost constanLlv egual to zero along the svstem, however. fiber nonlinearitv which is small but non-negligible. severclv de[grades the performance of D\VDJ'vf syHtemH mainly due to t,he F\V]'vI-indllC'ed froi:li:ltalk between channels. The effect of F \i\i 1\/1 is also effective in singlel fhannel systems if the signal wavelength is almoi:lt coincident with the I';erol klispersion vmvclength of the fiber. In this case the noise in the vicinity of the! ~ignal spectrum is amplified and the signal spectral width is broadened. ITo reduce the nonlinearity indnced performanC'e degradation, nonzero diHlPeri:lion fiberi:l needs to be used as transmission fibers that requires some i:lort pf dispersion compeni:lation. Dispersion compensation can be achieved in 31 bumber of v·mys. One wav is the pre- and post-compensation in lumped fash~on by nHing diHperHion-cornpensating fibers or one can llHe the C'hirped fiber [gratings. A third way iH by using t,he span-by-span C'ompensation "\vit,h amIPlifier spani:l coni:ltructed by pOi:litive- and negative-dii:lpersion trani:lmissionl lfibers. l\Tidpoint inveri:lion of signal i:lpectrum can also be used as dispersionl kOlnpensaLionj !Nonlinear ret,nrn-to zero (RZ) pulse transmission in dispersionrompenHat,ed Hystems t,hat, conHist of alternating normal- and anomalollH§ispersion fibers was first reported in 1992 1441. An experirnental dernonstra~.ion of :::;uch I:\yl:\tcrn wal:\ firl:\t rcportcd in 1995 [44]. A nurncricall:\tudy I:\hmv[ng thc pCrIodIcally I:\tatlOnarv I:\ohton-hke l:\olutlOn CXIl:\t 1Il I:\uch I:\vl:\tcrn "\val:\l ~eported in 1995 [44]. Since then it has been widely recognized that soliton-
12] Derivation of NLSE
]]
pke RZ pulses which are periodically stationary propagating through optical !fibers with alternating sign of GVD is much more advantageous over conven~ional solitons in fibers with constant GVD. Such a periodically stationary ~olution is called a dispersion-managed (DM) soliton. [When there exists a nonzero residual averaged dispersion and the fiber Inonlinearity is insignificant, the averaged pulse width is broadened with a ~ate determined by the averaged dispersion. The pulse is eventually dispersed put. Fiber nonlinearity can compensate for the pulse broadening if the chirp ~nduced by the nonlinearity can counter balance the chirp induced by the laccumulated dispersion. When the initial pulse amplitude is properly chosen, ~o that the balance between the nonlinearity and dispersion is achieved, the ~volution of the pulse becomes periodically stationary and a DM soliton is !thus created. ~ DM soliton is a periodically stationary isolated pulse propagating in a !fiber with alternating positive and negative dispersion. There is no explicit Iclosed-form expression available for the DM soliton solution even for the simIPlest dIsperSIOn arrangement. ThIS IS contrary to the case of conventIOnal ~olitons in fibers with constant dispersion where soliton solutions are explic~tly given. FilII numerical simulations of the NLSE with varying dispersion, is Iffiost accurate but time consuming and difficult to extract physical insights. ~nalytical approaches by means of which one can get physical insights of the ~undamental nature of the DM solitons are strongly desired. In this chapter, ~he variational principle, that is the primary and most widely studied techpique of attacking the analytical aspects of DM soliton, will be introduced. ~lso, the soliton perturbation theory that is used to study the other technical laspects of DM solitons like the coIlision-induced frequency and timing jitter, lamplItude JItter, wIll be taken up later III thIS chapter. The other analytIcal ~echniques namely the multiple-scale perturbation method and the asymp~otic analysis will be seen in the context of quasi-linear pulses in the later Ichapters.
f2.1.3 Mathematical formulation IThe relevant equation is the nonlinear Schrodinger's equation (NLSE) with Idamping and periodic amplification [3-6]: (2.30) !Here, r is the normalized loss coefficient, Za is the normalized characteristic lamplifier spacing, and Z and t represent the normalized propagation distance land the normalized time, respectively, while q represents the wave profile ~xpressed in the non-dimensional units.
2 Nonlinear Schrodinger's Equation
IAlso, D(z) is used to model strong dispersion management. Now, decompose the fiber dispersion D(z) into two components namely a path-averaged Iconstant value ba and a term representing the large rapid variation due to ~arge local values of the dispersion [6]. Thus, 1
+ -~(()
D(z) = ba Iwhere ( = z/za. The function lamplification period, namely
~ ~o
Za
(2.31 )
is taken to have average zero over an
~(()
~~) = ~
Za
dz = 0 Jto ~ (3...) Za a
(2.32)
that, the path-averaged dispersion D will have an average ba , namely,
rD) = - Za1
l
za
0
D(z)dz = ba
(2.33)
IThe proportionality factor in front of ~((), in Eq. (2.31), is chosen so that !both ba and ~(() are quantities of order one. In practical situations, disper~ion management is often performed by concatenating together two or more ~ections of given length with different values of fiber dispersion. In the speIcial case of a two-step map, it is convenient to write the dispersion map as a lPeriodic extension of [6]
~(() ~here
={
~~:
0::::: 1(1 < ~ ~
: : : 1(1
(2.34)
LSI and LS 2 are given by
~I = 28 ()
28 1 - ()
~2=---
~ith
<~
the map strength
8
(2.35) (2.36)
defined as (2.37)
IConversely, (2.38) land
() =
~2
~2-~1
(2.39)
]3
12] Derivation of NLSE
IA typical two-step dispersion map is shown in Figure 2.1.
IFig. 2.1 Schematic diagram of a two-step dispersion map.
ITaking into account the loss and amplification cycles by looking for a of Eq.(2.30) of the form q(z, t) = P(z)u(z, t) for real P and taking IP to satisfy ~olution
IPz
+ r P - [e rZa -
N
1]
L 8 (z -
nza)p
=
0
(2.40)
n=I
1(1) transforms to I.
zU z
D(z) + -2-Utt + g(z)lul 2 u =
~(z) = ~or
z
E
p 2 (z)
0
= a6e-2r(z-nzaJ
(2.41) (2.42)
[nza, (n + l)za) and n > 0 and also (2.43)
~o
that over each amplification period
~ g(z)) = - 1
Za
l
0
za
g(z)dz
=
1
(2.44)
[Equation (2.41) is commonly known as the DM-NLSE and it governs the propagation of a dispersion-managed soliton through a polarization preserv~ng optical fiber with damping and periodic amplification [3-6]. Figure 2.2 is profiles of DM solitons in the linear and logarithmic scales respectively. [n a polarization preserved optical fiber, it was seen in the last section ~hat the propagation of solitons is governed by the scalar DM-NLSE given
(b)
j)(z)~. ,q(Z)~I,NLSF;
1) if t in E g , (2,4 e .ruetl,od.orTnve"e ,a tl te o N L .t b h r E g ,(2,4.tJ o.iotegrate.NLSFJ.by le b ,i s. ii, Po
r~9T).
d. i t i, 'c<;ove.:e n..n C/}a.o,fo .seatte'iOK
15
12.2 Integrals of motion
~.2
Integrals of motion
lOne of the intrinsic properties of DM-NLSE is that it has conserved quan~ities that is also known as Integrals of Motion. Conservation quantities are la common feature in Mathematical Physics, where they describe the con~ervation of fundamental physical quantities. In this section, the conserved Iquantities of DM-NLSE, given by Eq.(2.41), will be derived. Rewriting the rLSE in the form [21,37] aT aX _ 0 (2.45) +
az
at -
represents the conservation law. Here T is known as the density while X known as the flux. Also neither the density nor the flux involve derivatives Iwith respect to t. Thus, T and X may depend upon t, z, q, qt, qtt, ... , but not Iqz. Now, if both T and X t are integrable on (-00,00), so that X ---+ constant las Itl---+ 00, then Eq.(2.41) can be integrated to yield ~t
~s
:Z (I:Tdt) = 0
I:
Tdt
=
constant
(2.46)
(2.47)
IThe integral of T, over all t, is therefore called the constant oj motion or the lintegral of motion. For a dynamical system, with a finite number of degrees pf freedom to be integrable, the system needs to have as many conserved Iquantities as the degrees of freedom. The first conserved quantity for the INLSE will now be derived. IPerforming the operation (2.41) xq* yields (2.48) IThe complex conjugate of Eq.(2.48) is (2.49) !Operating Eqs.(2.48)-(2.49) gives (2.50) which can be rewritten as
(2.51 )
2 Nonlinear Schrodinger's Equation ~o
that the flux is u*Ut - uuj;. Integrating Eq.(2.51) with respect to t yields (2.52)
~ince for a soliton u, Ut, Utt, ... approach zero as It I --+ 00, as mentioned [before. Thus, the first conserved quantity for the NLSE is given by
(2.53) IThis conserved quantity is known as the energy or wave power while math~matically, it is the L2 norm. Now, the second conserved quantity will be Iderived. The complex conjugate of Eq.(2.41) is given by (2.54) IPerforming the operation u;x (2.41) i (uzUt - UtU;)
+ D 2z
(UtUtt
+ UtX
+ UtUtt ) + g(z)
row, performing the operation q* x eft (2.41) '( IZ
*) U *Utz - uU tz
(2.54) gives (UUt
+U
+ u*Ut) luI 2 =
0 (2.55)
x eft (2.54) yields
D(z) (U *Uttt + UU ttt *) + -2-
IThen, Eqs.(2.55)-(2.56) leads to
(2.57) ~hich
simplifies to
liD(z)fz
(UUt - u*Ut)
1+ ~~ [21ut 1
2 -
(u*Utt
+UUtt ) -
[ntegrating Eq.(2.58) with respect to t gives
41 u 12 ]
=
0
(2.58)
12.2 Integrals of motion
17
(2.59) SO that
M
=
iD(z)
1
00
(U*Ut -
uu;)dt
=
constant
(2.60)
Iwhich is the second conserved quantity that is also known as the linear moImentum. These are the only two conserved quantities that are known so far ~or the DM-NLSE. The third important quantity, namely the Hamiltonian I(H), that is given by (2.61 ) not a conserved quantity. It can however be proved that H is an integral pf motion of DM-NLSE if D(z) and g(z) are constants, in which case, it Ican actually be proved that there exists infinitely many conserved quantities ~or the NLSE given by Eq.(2.41). This is because when D(z) and g(z) are Iconstants, it reduces to the case of classical or conventional solitons. IThe conserved quantItIes wIll now be explIcItly evaluated by assummg that ~he soliton pulses are given in the form [56] ~s
lu(z, t) = A(z)J [B(z) {t - t(z) }] f exp[iC(z)
{t - t(z)} 2
-
iK,(z) {t - t(z)}
+ iB(z)]
(2.62)
!Where j represents the shape of the pulse. It could be a Gaussian type or Ian SG type pulse. Sometimes super-sech pulses are also considered. Here ~he parameters A(z), B(z), C(z), K,(z), t(z) and B(z) respectively represent ~he soliton amplitude, the inverse width of the pulse, chirp, frequency, the Icenter of the pulse and the phase of the pulse. It needs to be noted that, ~his approach is only approximate and does not account for characteristics ~uch as energy loss due to continuum radiation, damping of the amplitude pscillations and changing of the pulse shape. For such a pulse form given by [Eq.(2.62), the integrals of motion are (2.63)
lu' in = ~D(z)
1
00
-00
(U*Ut -
Iwhile the Hamiltonian is given by
uu;)dt
=
A2 o,2,o -K,D(z)Jjl
(2.64)
2 Nonlinear Schrodinger's Equation
D(z) ( rL -2-
A2 BIo,0,2
+ 4~ I 2 ,2,0 + ----s- I O,2,0 A 2C 2
ti 2 A2
)
-
g(z) A4 -2- IiI O,4,0
(2.65)
Iwhere the following notation was introduced:
(2.66) ~or
nonnegative integers a, band c.
~.3
Soliton perturbation theory
IThe soliton parameters that were introduced in the previous subsection are pow defined as [56]
() Z
z. J()
=
2D z
-00
(uu*t - u*u t ) dt
I~oo lul 2 dt
(2.67)
(2.68) (2.69) (2.70) !Where B is defined as the root-mean-square (RMS) width of the soliton. In lease of DM solitons, the alternately varying dispersion as seen in Eqs.(2.31) land (2.34), makes the soliton breathe periodically. Thus, the soliton width Idoes not stay constant although the energy of the soliton does. Hence, the IRMS width is used in the analytical study of DM solitons. From these defini~ions, one can derive the variations of the soliton frequency, chirp, RMS-width dti = 0 dz
(2.71) (2.72)
dB dz
=
-4D(z) C I 2 ,2,0 B I O,2,0
!Also, the velocity of the soliton is given by
(2.73)
19
12.3 Soliton perturbation theory V
dt
=- =
(2.74)
-K,D(z)
dz
[2.8.1 Perturbation terms ~n
presence of perturbation terms, the perturbed NLSE is given by (2.75)
~here R represents the spatio-differential operator, although sometimes it Icould, very well, represent an integral operator. Also, the perturbation pa~ameter E is the relative width of the spectrum and 0 < E « 1 by virtue pf quasi-monochromaticity. In presence of these perturbation terms given by IEq.(2.75), the adiabatic variation of soliton parameters are given by [37, 56]
~
Idz =
E
1
00
00
(2.76)
(u*R+uR*)dt
i: i:
lFi"om Eqs.(2.67)-(2.70), it is possible to obtain [56]
I~: = E;(Z) [i 1-
2 2 10 4 0 -g(z)AB -'-' 1220
1
00
K,D(z)
iE B +--
h
2 0
A2
Iv = ~:
=
2
1
00
(2.77)
** t(utR-utR)dt
B31°O t 2 (u* R + uR*) dx
-00
4E
C
A2
2 2 0 - 00
h,2,0 10 2 0
-K,D(z)
(U*R+UR*)dX]
+ -1-
(u* R - uR*) dx
-00
ddB = -4D(z) C z B
(u;R - Ut R *) dt
+~
1
i:
(2.78)
00
E - oo
+~
t 2 (u* R
t (u* R
+ uR*) dx
+ uR*) dt
(2.79) (2.80)
2 Nonlinear Schrodinger's Equation
120
~.4
Variational principle
lFor a finite dimensional problem of a single particle, the temporal developIment of its position is given by the Hamilton's principle of least action [22, 137]. It states that the action given by the time integral of the Lagrangian is Ian extremum, namely,
l
o
t2
L(x,x)dt=O
(2.81)
h
!where x is the position of the particle and x = dx / dt. The variational problem 1(2.81) then leads to the familiar Euler-Lagrange's (EL) equation [37]:
8L _ ~ (8L) 8x dt 8x
=0
(2.82)
iHere, for Eq.(2.41), the Lagrangian is given by (2.83) row, using Eq.(2.62), the Lagrangian given by Eq.(2.83), reduces to =-
D(z)A ( B 4 fo 0 2 3 2B "
+ 4C 2 f2 "2 0 + Ii2 B2 )
fo 2 0
"
(2.84) IBy the principle of least action, namely Eq.(2.81), the EL equation is [16]
8L _ 8p
!i ( 8L ) dz
8pz
- 0
k\There p is one of the six soliton parameters. Substituting A, B, C, [or pin Eq.(2.85) the following set of equations are obtained: dA dz
-=
dB
-
dz
-ACD(z)
= -2BCD(z)
(2.85) Ii, t
and ()
(2.86) (2.87) (2.88)
dli = 0 dz
(2.89)
21
12.4 YanatlOnal prmcIple
dt
= -liD(z)
dz IdO
=
Ich
(li2 _ 10,0,2 B2) D(z) 2 10,2,0
(2.90)
+ 5g(z)A2 10,4,0 10,2,0
4
(2.91)
IThis Dynamical System of soliton parameters can be further analyzed based ~m the particular type of pulses that will be studied in the subsequent chap~ers. Furthermore, perturbation terms will be added and the adiabatic pa~ameter dynamics of soliton parameters can also be obtained.
19.4.1 Perturbation terms IThe perturbed DM-NLSE is going to be studied in this subsection. The adia[batic parameter dynamics of solitons is going to be studied in this subsection Iby the aId of vanatIOnal prmclple. The perturbed DM-NLSE that IS gIven by (2.92) iHere, once again, R represents the perturbation terms and E is the perturIbation parameter. In presence of the perturbation terms the EL equation Iillodify to [27] (2.93) IWhere p represents the six soliton parameters. Once again, substituting A, B, in Eq.(2.93) the following adiabatic evolution equations lare obtamed:
10, Ii, t and 0 for p dd
z
ddB z
=
=
-ACD -
2
LEI
0 2 0 2 2 0
-2BCD - AL EBI
1 ~[Re-i>](Io,2,oT2 1 ~[Re-i>](Io,2,oT2
- 3I2,2,0)f(T)dT (2.94)
-00
00
020 220
-00
- I 2,2,0)f(T)dT (2.95)
gA2B2I040
D------'-'
-
2
dli 2E dz - ABIo 2
~2 AI
1
220 00
0
-00
100 -00
~[Re-i>]
4
h
f(T)
# + 2Td
20
T
(2.96) (2.97)
2 Nonlinear Schrodinger's Equation
122
(2.98)
dB = dz
I
(",2 _I ,0,2 B2) D + 5gA2 I ,4,0 O
2
O
I O,2,0
~ 2AB~0'2'0
I:{B~[Re-iq)l 4
1+ 4"'~[Re-iq)lTf(T) !where the notations land
~
=
(3f(T)
+ 2T ~~) (2.99)
}dT
T = B(z) C(z) {t - t(z)} 2
I O,2,0
-
(t - t(z)) ",(z) {t - t(z)}
(2.100)
+ B(z)
(2.101)
!Was used. Also, Rand <.S represent the real and imaginary parts, respectively. rote that Eqs.(2.86)-(2.91) are special cases of Eqs.(2.94)-(2.99) respectively ~or E = O. These equations can be modified based on a particular type of !pulse that is considered in optical fibers. This will be studied in detail in the ~oIIowing chapter.
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1
~()n
169.
170. 171.
172.
17:1.
174.
transmissi(HI qualit.y wit.h coherent. int.crtcren(:e
Pr()gnc,~.'1 in
F,la:tn)'{rUJ,gneti,c,~j
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2 Nonlinear Schrodinger's Equationl
17,'), l'vI. Stcfanovic, D, Draca, i\, Panajotovic &. D. Tvlilovic. "Tndividual and joint influcncc bf second and t.hird order dispersion on transmission quality in the presence of coherent ~nterference". Optik. \/01120, No 13, 6:16-6H.(2009). 176. B. St.ojanovic, D. 1\1. 1\-1ilovic & A. Bii:n".'iJ,s. "Timing shift of optical pulses due t.o interthannel crosstalk". Progre.~8 in F:lerJromagrJdic.~ Re.~mrch A!. Vol L 21-30. (2008).1 177. S. K Turitsyn, 1. Gabitov, E. \.\1. Laedke, V. Kl\Jezentse\', S. L. 1\'111she1'. E. G. Shapiro. Schafer, & K. H. Spatschek. "Variational approach t.o optical pulse propagation inl rIispersion compensated transmission syst.em". OpUr8 CommunJ,caJwnK Vol 151, Nol 11-:1,117-135, (199,), 178. S. VValmit7., Y. Kodama. &. i\. R. i\ce"Ves "COTitrol of optical solitoTi interactions". Oplical Fiber Technology. Vol 1, 187-217. (1995).1 179. P. K. A. \Vai &. C. R. T\:Ienyuk. "PolarLmtion mode dispersion, de-correlation, and kliffusion in optical fibers ·with randomly varying birefringence" . .Jou.rnat of Lightwavd Technology. Vol 14, No 2, 148-157. (1996). ISO. V. E. Zakharov &. S. VVabniL-I. Opt1.ml Sohlon.~: Theoret1.ml Challenge.~ and rndu8lrw~ I!-'erspecfives. Springer, New York, CA. USA. (1999)J
rr.
IChapter 3
[polarization Preserving Fibers
13.1
Introductjonl
IBirefringence in conventional single-mode fibers changei:l randomly due to Ivariations in the core shape and anisotropic stress acting on the corc. Linearlvl [lOlarized light. lallIlched into t,he fiber rea('hes very quickly into a state of ~lrbit.nlry polarizat.ion. Pulse broadening iH a result of the fact. that different. ~requency components of an optical pulse acquire different polaril';ation Htates. iThis is known as polarizalion mode dispersion (PJ'vID) and is a limiting [actorl [or terrestrial high biL rate optical communication SVSLClllS. Nmvadavs it is bOHHible t,o manufacture fibers for '\Thieh random fluctuations in the ('ore Hhapel ~1Ild size are not t,he guverning factor in determining the state of polarization. ISuch tiberi, are called polarization pre8er1 1ing (PP) fibers. A large amount of birefringence is introduced intentionall v in these fibers bv the aid of desigl~ ~nodifications so that small random birefringence fluctuations do not aiIecL ~,he light polarizat.ion significantly.1 IPolarization prei:lerving optical fibers preserves the plane of polarizationl pf the light launched into it. This type of fiber is a single-mode fiber and is ~lso called the polarizalion rnainlainmq (PM) .fiber. The polari"alion is pre~erved bv introducing asymmcLrv in the fiber structure. This asvmmcLrv mavl ~)e eit,her in fiber internal Ht.reHHeH (Ht.reHH-incinced birefringence) or in fiber ~hape (geometrical birefringence). Ai:lymmetry causei:l different propagationl ronstants for two perpendicular polarized modes that are transmitted by thel pteI'. Cross-coupling bcLween these modes are reduced as compared to thel ~:onvent.ional Hingle-mode fiber. IPolarization mlllt.iplexing Hhonld not, ,york nnless polarizat.ion-prcscrving III bers are not used. It turns out that even though polari~a,t.ion states 01 the bit Itrain does change in an unpredictable rnanner, the orthogonal nature 01 an.\-.'I bvo neighboring bits is nearlv preserved. Because of this orthogonality. the! ~nteraction among Holitons iH mllch v{eaker as compared to the ('o-polarize(~
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
3 PolarIzatlOn Preservmg FIbers
128
case. The reduced interaction lowers the bit-error rate (BER) and the transmission distance of a Gb/s soliton system. IThe PP fiber has a higher attenuation than conventional single-mode fiber. IThese are the main reasons why this type of fiber is rarely used for longIdistance transmission. They are instead commonly used for telecommunica~ion applications, fiber-optic sensing and interferometry. Besides that PM !fibers are not optically symmetrical and have strong internal birefringence Icaused by stress applying members. The internal birefringence is significantly Ihigher than normal levels of bend-induced birefringence. In this way, it pre~erves the state of polarization when the laser beam is correctly aligned to either of the two axes. IThe two most critical measures of a terminated PP fiber are extinction ~atio and key alignment accuracy. Extinction ratio can be easily degraded !by inappropriate adhesives or untested curing procedures. A PM fiber can imaintain a 25-35 dB extinction ratio but after concatenation, this figure can !be as low as 10-15 dB if not processed with great care. The fiber orientation Iffiisalignment can cause a crosstalk between TE and TM modes even when the ~xtinction ratio is good. For example, a perfectly terminated PM fiber with 127 dB extinction ratio can generate 15 dB crosstalk due to key misalignment. IThe fiber length is also a potential source of extinction ratio degradation and ~t is extremely important to keep the entire fiber free from twists, bends and ~emperature vanatIOns. IThe dimensionless form of DM-NLSE, in polarization preserving fibers, is given by ~olitons
~ncreases
I.
ZU Z
D(z) + -2-Utt + g(z)lul 2 u = 0
(3.1)
IThis equation governs the propagation of optical solitons in 1 + 1 dimensions. ~t is derived in the previous chapter. The mathematical structure of the pptical pulse for (3.1) is taken to be
r(z, t)
=
A(z)J [B(z) {t - t(z) }] I· exp[iC(z) { t
- l(z)} 2
-
ili(z) {t - l(z)}
+ iO(z)]
(3.2)
f represents the functional form of the pulse while the interpretation pf other soliton parameters are already discussed in the previous introductory Ichapter. ~here
13.2 Integrals of motion IRecall that the two integrals of motion of the polarization preserving fibers lare given by [25]
29
13.2 Integrals of motion
(3.3)
. 1=
land
= ~D(z) _
(U*Uf - uu;)dt
~ o,2,o = -K,D(z)]jl
(3.4)
IThese are respectively the energy and the linear momentum of the soliton. ~lso, it was mentioned in the previous chapter that the third important Iquantity, namely the Hamiltonian, is given by
(3.5) not an integral ofthe motion, unless D(z) and g(z) are constants, in which actually exists infinitely many conserved quantities. ~n the following three subsections, these conserved quantities and the iHamiltonian of solitons are going to be computed for the different kind of !pulses that are going to be studied in this book. Although there are var~ous kinds of pulses that are studied in the literature of DM solitons, the !Gaussian, super-Gaussian and the super-sech type pulses are only considered ~n this book. The other kinds of pulses that are not touched upon in this ~ext are Gauss-Hermite pulses [40], cosh-Gaussian pulses and sinh-Gaussian !pulses [25] and many more. ~s
~here
13.2.1 Gaussian pulses Wor a pulse of Gaussian type, f(T) = e- T22 • So, the pulse format is given by ~ (z,
t)
=
1 2 )2 A(z)e-2B (z) ( f-f(z)
I· exp[iC(z) { t
- l(z)} 2
-
iK,(z) {t - l(z)}
+ iB(z)]
(3.6)
!Figure.3.1 shows the plot of a Gaussian profile. IThus, the conserved quantities respectively reduce to
(3.7) (3.8) !while the Hamiltonian is
3 PolarIzatlOn Preservmg FIbers
130
IFig. 3.1 Breathing Gaussian soliton.
IH = ~
i:
~ ~~2
[D(z)I U tI 2
-
g(z)luI 4 ]dt
{D(z) (B4 _ 2K? B2
+ 4C2) _
J2g(z)A 2B2}
(3.9)
lFigure 3.2 shows the root-mean square variation of the width of the Gaussian sohton.
13.2.2 Super- Gaussian pulses Wor a super-Gaussian (SG) pulse, f(T) e-T~'" with m ~ 1 where the parameter m controls the degree of edge sharpness. With m = 1, the case of la chirped Gaussian pulse is recovered while for larger values of m the pulse gradually becomes square shaped with sharper leading and trailing edges [25]. IThe pulse profile is given by
I· exp[iC(z) { t -
l(z)} 2
-
ili(z) {t - l(z)}
+ i8(z)]
(3.10)
[n Figure 3.3, one can see the shapes of the pulses as the parameter m varies. lFor an SG pulse, the integrals of motion respectively are
(3.11)
13.2 Integrals of motion
31
B
10
50
100 Z{lQiiJ
150
200
IFig. 3.2 RMS width variation of a Gaussian pulse.
t·o .8
--m=l
----m=2 ...... m=3 -·-·-m=4 -.- ·-m=5
0.6 0.4
0.2
o
4
x IFig. 3.3 SG pulses for various values of m.
. 1=_=
= ~D(z) 2
!while the Hamiltonian
(U*Ut -
IS
~ uu;) dt = -liD(z)-f mB
(3.12)
3 PolarIzatlOn Preservmg FIbers
132
1=
~ [D(Z) {!~22 (2~) + m:2 r (2~) - ;~r (2~)} ~ g(z)2~2~r (2~)]
(3.13)
!Figures 3.4 and 3.5 display the breathing 8G pulse and the root-mean square Ivariation of the soliton.
IFig. 3.4 Breating SG pulse for m = 2.
13.2.3 Super-Sech pulses row, it is assumed that the solution ofEq.(3.1) is given by a super-sech (88) Ichirped pulse J(T) = 1/ coshn T. Thus, the pulse profile is given by
l,(z t)
['
=
A(z)
coshn [B(z)
(t - t)]
I· exp[iC(z) { t
- l(z)}
2- i"'(z) {t - l(z)} + iB(z)]
(3.14)
iHere, the parameter n > O. The 88 pulses for various values of the parameter [L is given in Figure 3.6.
!For 88 pulses, the integrals of motion are given as
13.2 Integrals of motion 16
33
.............................................................................. .
14
12 B
10
o
100 Z(Kiii)
50
200
150
IFig. 3.5 RMS width variation of an SG pulse for m = 2 .
.0
- - n=1 ........ n=2
0.8
n=3 ........ n=4 ............ ---- n=5
0.6 .~ ..
<> U)
'"
0.4
0.2
:: ;{f
0.0
-2
f-4
0
2
4
oX
IFig. 3.6 SS pulses for various values of n.
M
=
. 1
!..D(z) 2
(3.15)
00
(U*Ut -
-00
!while the Hamiltonian is
uu;) dt
=
A2 -KD(z)-B B
(3.16)
134
3 PolarIzatlOn Preservmg FIbers
(3.17) Iwhere the Gauss' generalized hyper-geometric function is defined as
(3.18) !where B(l, m) is the beta function and the Poehammer symbol or the rising ~actorial notation (m)j is defined as
I(m)n
=
m(m
+ 1)(m + 2)··· (m + n -
= (m+n-1)! = r(m+n) 1
(m -1)!
r(m)
1) (3.19)
IFigures 3.7 and 3.8 show the profile of a super-sech pulse and the root-mean ~quare variation of the width of an SS pulse.
o IFig. 3.7 Breathing SS pulses for n = 2.
13.3 YanatlOnal prmcIple
10
35
50
100
150
200
z(km) IFig. 3.8 RMS width variation for SS pulse for n = 2.
13.3 Variational principle IThe variational principle that was derived in the previous chapter will now Ibe used to obtain the soliton parameter dynamics for Gaussian and super!Gaussian pulses. It needs to be noted that it is now left to the reader, as Ian exercise, to obtain these parameter dynamics for super-sech pulses, in a ~ery similar fashion. The Lagrangian given by Eq.(2.84) is computed for the !Gaussian and super-Gaussian pulses and then substituting the six soliton !parameters for pis Eq.(2.85) yields the following parameter dynamics in the lriext two subsectIOns.
13.3.1 Gaussian pulses dA dz dB dz
-
=
-ACD(z)
= -2BCD(z)
(3.20) (3.21) (3.22)
dr;,
dz
=0
(3.23)
3 PolarIzatlOn Preservmg FIbers
136
(3.24) (3.25) IEquations (3.20)-(3.25) represent the evolution equations of the parameters pf a Gaussian soliton propagating through an optical fiber. These evolution ~quations can be used to study various issues including the pulse interaction.
13.3.2 Super- Gaussian pulses dA dz dB dz
=
(3.26)
-ACD(z)
= -2BCD(z)
(3.27)
=0
(3.29)
dC dz d",
dz
dt
-=
dz
dB dz
2
!5:..- _ m22~ 2
-"'D(z)
r (4m-l) ----zm B2 r (...L) 2m
D(z)
(3.30) 1
+ 5g(z)A2- 2 4 =+1 2m
(3.31 )
rote, here, that for m = 1, Eqs.(3.26)-(3.31) all reduce to Eqs.(3.20)-(3.25) ~espectively for Gaussian solitons.
13.4 Perturbation terms [n this section, similarly the adiabatic parameter dynamics of Gaussian and ISG solitons will be obtained. This is also based on the variational principle. IThe starting point is the EL equation given by Eq.(2.93). Subsequently, re~ations (2.94)-(2.99) respectively reduce to the following set of relations of ~he soliton parameters that are described in the following two subsections for !Gaussian and SG pulses, respectively.
18 4 Perturbatjon terms
37
13.4.1 Gaussian pulses
-dA = -ACD(z) - dz
E
1
00
2n _
~[Re-'"> ](4T 2
- 3)e- T 2 dT
(3.32) (3.33)
(3.34) (3.35) (3.36)
(3.37) IThese equations now represent the evolution equations for the parameters pf a Gaussian pulse propagating through an optical fiber in presence of the !perturbation terms.
13.4.2 Super-Gaussian pulses
3 PolarIzatlOn Preservmg FIbers
138
(1) -1- r (-
. -T2- r --
I
{
m2 2;'
3 )} e- 7 2 '" dT 2m
m2 2;"
2m
2 r (4m-l)
dC dz
B4~ ~ -2C2 2"',;;:2 r(2~)
(3.39)
D() z
(3.40) d",
dz
1 m2 2,;,,;;:1 AB r....L
=-
E-
-
dt _ dz - -",D(z)
d()
dz
00
2
!5:..- _ m22~ 2
+ E1- m2 r
.
{2mT 2m - 1 B28'[Re- z>]
-00
(2!rJ 1 '"
1 m22in AB r
+E
AB
1
00
-00
r (4m-l) ~ B2 r (~)
,;,,,, 1 (2m)
1
00
(3.41)
~[Re
-i>
D()
z
]Te
_7 2 ",
(3.42)
dT 1
+ 59 (Z)A2 24';'';;1
.>
{B8'[Re- Z ](3 - 4mT2m)
-00
(3.43) ISo, now, these are the adiabatic evolution of the soliton parameters for an ISG pulse in presence of the perturbation terms.
13.5 Stochastic perturbation IBesides these issues, one also needs to take into account, from practical con~iderations, the stochastic aspects. These effects can be classified into three [basic types [1, 14]: ~. ~.
Stochasticity associated with the chaotic nature of the initial pulse due to partial coherence of the laser generated radiation. Stochasticity due to random non-uniformities in the optical fibers like the !flllctuations in the values of dielectric constant the random variations of !the fiber diameter and more
p.5
;19
Stochastic perturbation
3. The chaotic field caused by a dynamic stochai:lticity might arise from 31 Iperiodic modulation of the system parameters or when a periodic array of ~mlseH propagate in a fiber optic reHonator. [hus, stochasLicitv is inevitable in optical soliton communicationsJ IStochaHt.icity are baHically of hvo types, narnely, homogeneous and non~lOmogeneous [1, 14]. In inhomogeneouH case, t,he Htochasticit.y iH present ir~ ~,he input pulse of the fiber. So, the parameter dynamics are deterministicl [but however the initial values are random. \i\tThile in the homogeneous cas~ Ithe stochasticitv originates due to the random perturbation of the Ilber like! Whe denHity tlllct.nat.ion of the fiber material or t,he random variations in thel
Ifi her
djameter etc
IThe perturbed NLSE given by Eq.(2.75) is going to be studied in this sec~1orl. ,vhere the perturbatIOn term R WIll now mdude a random }JerturbatIOr~ ~,erm. rIms, m tIns sectIOnJ
R
~ Oil.
+ (lUll + o-(z, t)
(:1.44)
pere in (3.44), n < 0 (> 0) is the attenuation (amplification) coefficient, {l the coefficient of bandpass filtering. Also, a(z, t) representH the randon~ ~tochai:ltic perturbation term. IThe use of optical ampHfieri:l affects the evolution of i:loHtons considerably. [he reason is that amplifiers. although needed to restore soliton energv, in~.rodnces noiHe originating from amplified spontaneous emission (ASE) [14]. ITo study the impact of noise on soliton evolnt.ion, the evolnt.ion of the mear~ fnergy of the soliton due to ASE will be i:ltudied in this section. Tn cai:le of [unwed amplilicaLion. solitons are perturbed bv AgE in a discrete lashion at Ithe 10caLion of the amplifiers. It can be assumed that noise is distributed alII ~llong the fiber lengt,h Hince t,he amplifier Hpacing satiHfies Za « 1. In (3.44), r-(z, t) representH the noise t,erm ''lith GanHHian statistics and iH assumed t,hat, p-(z. t) [14[ is a function of z only so that ,,(z. t) ~ ,,(z). Now, the complexl ~tochastic term a(z) can be decomposed into real and imaginary parts as ~s
(3.45) [t is further assumed that
"I (z) and "Az) are independently delta correlated; (3.46)
(0-1 (Z)O-l (ZI))
~ 2D,/i(z -
ZI)
(3.47) (3.48)
Iwhcre Dl and D2 are related to the ASE l:ipectral dcnl:iitv. In thil:i taRRumed t,hat, Dl - D2 - D. ThnRl
I("(z))
~
()
GltlC.
it is
(:1.49)
810
3 PolarIzatlOn Preservmg FIbers
land
(a(z)a(z')) ~n
=
2o(z - z')
(3.50)
soliton units, one gets (3.51 )
~here
Fn is the amplifier noise figure, while Po _ (G - 1)2 GGlnG
(3.52)
related to the amplifier gain G and finally Nph is the average number of photons in the pulse propagating as a fundamental soliton [14]. ~n presence of the perturbation terms, given by Eq.(3.44), the adiabatic Idynamics of the soliton energy (E) and the frequency (K,) are given by ~s
(3.53)
dK, dz
(3.54) ~hich are obtained by the aid of Eqs.(2.76) and (2.77), respectively. Here in IEqs.(3.53) and (3.54), T and ¢ are as in Eqs.(2.100) and (2.101), respectively. !Equations (3.53) and (3.54), as it appears, are difficult to analyze. If the ~erms with a1 and a2 are suppressed, one recovers a dynamical system, for a !fixed chirp, which has a stable fixed point, namely a sink, of Eqs.(3.53) and 1(3.54) given by (B, R) where
B=
ar (~) + va 2 r (~) -
8(J2C2m(2m -l)r (~)
(3m(2m -
l)r e;~l )
r (~)
1/2
(3.55)
R=O
(3.56)
41
13.5 Stochastic perturbation
!Now, linearizing about this fixed point, Eqs.(3.53) and (3.54) respectively Ireduce to dE dz
=10
2aE - (3B 2 m(2m - 1)
r (2m-I) 2m r (2;")
-
8(3C 2E B2
r (.JL) r (2!rJ
~
(3.57)
(3.58) !Equations (3.57) and (3.58) are known as Langevin equations which will now Ibe analyzed further to obtain the long term behavior of the SG pulses in !presence of the stochastic perturbations. IAfter rescaling, Eq.(3.57) reduces to dE dz
dE - ((1 - E)]
-=
F=
I:
e-!r2'"
(0"1
(3.59)
cos¢ + 0"2 sin¢) dT
(3.60)
IThe stochastic phase factor is, now, defined by
¢(z, y)
=
l
z
(3.61 )
((s)ds
!where z > y. Assuming that ( is a Gaussian stochastic variable one gets (3.62) (e[4>(z,Y)H(z',y')])
Ie =
2(z + z' - y - y') -
eO()
=
(3.63)
Iz - z'l - Iy - y'l
(3.64)
land
(((y)e-4>(z,y))
=
(((y)((y')e[-4>(z,y)-4>(z',y')])
f(e-4>(Z'Y))
= 2Do(y -
=
oeD(z-y)
y')eD(}
+ ";;'e O(}
(3.65) (3.66)
812
3 PolarIzatlOn Preservmg FIbers
!Now, solving Eq.(3.59) gives
I(E(Z))
=
1D_E; { 1 - e-EZ(l-D)}
(3.67)
Iwhere, the initial energy Eo is given by
mA (1) 1'Iv0 = E(O) = --s-f 2m 2
(3.68)
IF'rom Eq.(3.67), it follows that
lim (E(z))
z->oo
=
DEo
------=
1- D
(3.69)
IThus, it shows that in presence of the stochastic perturbation term of the ~orm that is considered in this section, the SG pulses will propagate down ~he fiber with a fixed mean energy that is given by (3.69) as long as D < l. lHowever, if D > 1, (E(z)) becomes unbounded for large z.
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/\. Riswm; &. S. KonaT". Tntr()r/nction to Non Kerr T,aw Optical ,'30lit()ns. eRe Press, [Boca Raton, FL. USA. (2000) D. Breuer. F. Kuppers, A. l-lattheus. E. G. Shapiro. L Gabitoy & S. K. Thritsvn. (·SYIlllllet.ricaI dil:ipersion compermation for standard monomode fiber based comIllUniration svsj.ems with large amplifier spacing". Opl.1,c.~ f-.r:I.I.r:rs. Vol 22. No 1:3.982-984. K1997). D. Breuer, K. Ji.'trgensen, P. Kupperl:i, A. 1\.fattheus, I. Gabitov &. S. K. Turibyn. "Optimal schemes for dispersion compensation of standard monomode Ilber based ~inks". Optics Communications. Vol --10, No 1-3, 15-18. (1997). A. b:hrhardt, M. I-:,iselt, G. GoGkopl, L. Ki.illcr, v\. Picpcr, R. Schnabel, G. H. v\c [bel'. "Semiconductor laser ampliller as opt.ical switching gate". JOlLrnal oj f-.1,ghJ11!a1l~ Technology. Vol 11, No 8, 1287-1295. (199:1). A. D. Pil:ihman, G. D. Duff & A. J. Nagel. "MeClsurement and simulation of multi[path interference for 1.7 Gbjs light.wave transmission systems using single- and mult.i[requeney laser". Journal of Lightwave Technology. Vol 8, No 6, 89-'1-905. (1990). e. 1\-1. Khalique &. A. Bil:iwas. "A Lie symmetry approach to nonlinear Schri)dinger'l:il rquCltJon \\lth non-Kur Im\ nonhmanty Cmnmunications in Nonlinear Science and INumr:ru;o] SmwJoJion. Vol 14, No 12,40:3:3-4040. (2009).1 H. Kohl, A. 13iswas. D. l\1ilovie & E. Zerrad. "Perturbation ot Gaussian optical solitonsl an dispersion-managed fibers". Applied Mathernatic8 and Computation. Vol 199, No 1, 12GO-2G8. (2009).1
@2. H. Kohl. A. 13i"va" D. Milovic & Eo Zenad. "Pe,tmbation of ,upeHech colito,," i,) Illisperl:iion-nmnaged optical fibers" international Jo'Urnal oj 1 heoretical Physics. Voll ~7. No 7, 20:18-2064. (2008)1
12:1. H. flit.
12r.;. I2G.
127.
,28.
129.
po.
pI. Pl. Kn.
Kohl. A. 13iswas, D. l\1ilovie & Eo Zerrad. "Adiabatic dynamics ot Ciaussian and 8uper-Gaussian solitons in dispersion-mmmged optical fibel"1:i . Frvgn!88 in b'lectmmag Vuo:lirs Rr:searrh. Vol 84, 27 53. (:2008). R. KohL D. I\Iilovic. E. Zen'ad I( A. Biswas. "Pert.mbation 01 super Gaussian optical 8olitons in llil:ipersion-managell fibers". Mathematical and Cmnp'Uter Modelling. Voll ~9, No 7 8, 4H3, 427. (2009). R. Kohl. D. l\.lilovic, E. Zenad &. A. Diswas. "Soliton perturbation theory lorl ~ispcrl:iion-managcd optical fibers". Journal oj Nonlinear Optical f'hY8ic8 and Ala I ~erialH. Vol 18, No 2. (200U)1 P. 11. Lushnikov. 'Dlspenaon managed sohl.ons 1Il opt.lcal libel's \\lth ~elO a\erage klispersion". Optics Letters. Vol 25, No 16, 11--1-'1-11'16. (lOOO)J l'vI. P. l'vlahrnood & S. B. Qru:lri. "l'vlodcling propagation of chirped solitons in ani ~llipt..ically low birefringent.. single-mode optical fiber" . .lournal of Nonlinear OpLical IJ-'hysics and Materials. Vol 8, No -'1, -'169-'175. (1999)J A-I. 1'I,'Taj,SLltnol,O. "Analysis of interaction bc/.\vccli stretched pulses pmpagaUng illl kiispersion-managed llbers". TF.:F.:F.: Phol.orJic.~ Tr:chrJOloQ1! LellerK Vol 10, No :3, :37:3p75. (1998)1 H. MichineL "Pubel:i nonlinear surtw:.:e ,,,mves andl:ioliton emission at nonlinear graded lindex waveguides". OpI.1,ca.l and Qv.anl.v.m FJledronir,s. Vol 30, No 2, 79-97. (1998)j A. Panajotoyic, D. l\1ilovie & A ..Mittie. "Boundary case of pulse propagation analytic! 80lution in the presence of interference and higher order dispersion 1 ELS1RS 2005 Confr:renre Prorer:ding8. !)47-!)50. Nis-Serbia. (2005)j A. Panajotovie. D. l\.1ilovic, A. Biswas & E. ZelTad. "lnfluence of even order dispersiOl~ r)n super-l:iech soliton t.ransmil:ision qualit.y under coherent. crosstalk" . Research Letters Op£irs. Vol 2008, 01:3986, !) pages. (2008).1 A. Panajotoyic, D. l\.1ilovic & A. 13iswas. "lnfluence of even order dispersion on soli~on trmmmisl:iion qualit.y wit.h coherent. int.erference Prvgre88 in Electrvmagnet"ic6j IRe8mrch R. Vol :3, 0:3-72. (2008). V. N. Serkin & A. Hasegawa. "Soliton management in the nonlinear Schrodinger equa~ion model ,,,lith varying dispersion, nonlinearity and gain". JETP Letters. Vol 72, N(~
Vn
12, 89-92. (2000).
3 Polarization Preserving Fibersl
111
O. V. Sinkin, V. S. Grigoryan &: C. R. TVlcnyllk. "Accurate probahilistic treatment be hit-paU,ern-dependent. nonlinear dist.ort.ions in REB calculations for \VDT\,T HZ sys~ems". Journal of Lightwave Technology. Vol 25, No 10, 2959-2967. (2007)) ,15. 1\'1. Stchmovic & D. 1\.Jilovic. "TllC impact of out-of-baTid crosstalk UTI optical connTlU ~4.
Inication link preferences". Journal of Ophml CommuniraliorJ8.
Vol 26. No 2, 69-72.
(2005).
& D. T\Hlovic. "Individual and joint influence second and t.hird order dispersion on transmission quality in the presence of coherent ~nterference". Optik. \/01120, No 13, 6:16-6H.(2009). B. Stojanovic, D. 1\1. I\Hlovic & A. Bii:n".'iJ,s. "Timing shift of optical pulses due t.o interr,hannel crosstalk". Progre.~8 in F:lerJromagrJdic.~ Re.~earch A!. Vol L 21-30. (2008).1 S. K. Tnritsyn, I. Gabij,ov, E. \V. Laedke, V. K. I'vTe;.\enj,sev, S. L. T\:Illsher, E. G. Shapiro, [I. Schafer, & K. 11. Spatschek. "Variational approach to optical pulse propagation il~ Illisper::;ion compensat.ed t.ransmi::;sion ::;ystelll Optics C:ornnmnicutiorts. Vol 1.51, N(~ 11-:3. 117-I:J5. (1998). P. K. A. Wai & C. H. l\'lenvuk. "Polarization mode dispersion, de-correlation, and Illiffmion in optical fiGer::; ,,,lith randomly varying birefringence Journal oj LighhJJuwj Technology. Vol 14, No 2, 148-157. (1990). V. E. za.kharov & S. \Vabnitz. Optical Solitons: Theoretical Challenges and Industria4 !perspecfh'es. Springer, Heidelberg. DE. (1999).1
~6. 11. Stcfanovic, D. Draca, A. Panajotovic ~)f
110.
IChapter 4
!Birefringent Fibers
kl.l Introductjonl IAn ideal ii,otropic fiber propagates undisturbed in any Htate of polarizationl ~aunchcd into the fiber. Under ideal conditions of perfect cylindrical geometrvl ~tnd IsotropIC' materIal, a mode excIted "\vlt,h ItH polarIzatIOIl 1Il OIle (hrechor~ iwonld not conple ''lith t,he mode in t,he orthogonal direction. Real fibers lPossei:li:l Home amount of anisotropy because of an accidental lOi':;i:\ of circularl ~YllllllcLrv. This loss is due to either a non-circular geometry of the fiber or al bon-sYllllllcLrical stress field in the fiber cross section. Thus, small deviationsl ~rorn the C'ylindrical geometry or HIIlall fluctuations in mat,erial anisotrOp}l ~'eslllt in a mixing of the t,vo polarization states and t,he mode degeneracy is Ibroken. Thm,~ the mode propagation COTIi,tant becomes i:dightly different tor Ithe modes polarized in orthogonal directions. This propertv is referred to asl
Imodal birefringence [;151.1 IThe intrimlic birefringen(,e is int.roduced in the manufacturing process~which a permanent feature of the fiber. It comprii,es any effect that cam,es a de~riation from the perfect rotational symmetry of the ideal fiber. An elliptikal core gives rise to the geomcLrical birefringence. while a non-svll1ll1cLrica] ~tress field in the fiber cross-section induces the stress birefringence which is ~ntrodll('ed by the photo-elastic effect. during the fiber manufacturing pro('eHHI ~s
[351·
"pC,·lcle"'b"iccre"'.f"r"inCCg"'e",cclc=eC-=c=aCCn-aCTls"o-cb=e-c".r"e".a"t=edTCw=h=e=n=e=v"eccr=a'"fi"b=e=r=u=n"d=eCCrg='o=e=s='=eTla=.s"·t",ic~1 due to external perturbatIOns lIke the hydrostatIc pressure. longltudl-
~tresses
~lal Ht.rain, Hqueezing, t;\viHting, bending and ot,her Huch external tact.ors. Thel [)ertllrbation induced in t,he permittivity tensor t,hrollgh the phot,o-daHti(' efII eeL Jilts the degeneracy 01 the linearly polari:;;ed modes and induces extrinsicl
!iirefringence
1351.
IThe birefringence can abo significantly affect the I:)oliton propagation ir~ pptical fibers. In a highly birefringent, polarization maintaining fiber, t,he dif~eren('e of phase velocities between the two orthogonally polarized modes is
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
816
4 Birefringent Fibers
high to avoid coupling between these two modes. In highly birefibers, the beat length (inverse of intrinsic birefringence) is much ~ower than the nonlinear length and solitons remain stable whether launched Iclose to the slow or fast axis. On the contrary, for weakly birefringent fibers ~olitons remain stable along the slow axis but become unstable along the ~ast axis. Under certain conditions the two orthogonally polarized solitons Ican trap one another and move at a common group velocity, despite the !polarization dispersion. This phenomenon is known as soliton trapping and Ican be applied in soliton dragging logic gates. Other applications of nonlin~ar birefringence include high-resolution distributed fiber sensing and passive Imode-Iocking of fiber lasers [35]. IThe propagation of solitons in birefringent nonlinear fibers has attracted !much attention in recent years. The equations that describe the pulse proplagation through these fibers was originally derived by Menyuk [11]. They Ican be solved approximately in certain special cases only. The localized pulse ~volution in a birefringent fiber has been studied analytically, numerically land experimentally [11] on the basis of a simplified chirp-free model without pscillating terms under the assumptions that the two polarizations exhibit Idifferent group velocities. In this chapter, the equations that describe the lPulse propagation in birefringent fibers are of the following dimensionless ~ignificantly
~ringent
~orm:
!Equations (4.1) and (4.2) are known as the Dispersion Managed Vector Non~inear Schrodinger's Equation (DM-VNLSE). Here, u and v are slowly varying ~nvelopes of the two linearly polarized components of the field along the x land y axis. Also, 6 is the group velocity mismatch between the two polariza~ion components and is called the birefringence parameter, (3 corresponds to ~he difference between the propagation constants, 0: is the cross-phase modtulation coefficient and y is the coefficient of the coherent energy coupling ~erm. These equations are, in general, not integrable. However, they can be ~olved analytically only for certain specific cases [5, 11]. ~n this analysis, the terms with 0 will be neglected as 0 < 10- 3 [5]. Also, !neglecting [J and the coherent energy coupling given by the coefficient of the [y term, one arrives at the special case of Eqs.(4.1) and (4.2) as
(4.3) (4.4)
47
81.2 Integrals of motion
14.2 Integrals of motion IThe two integrals of the motion of Eqs.(4.1) and (4.2) are the energy (E) and ~he momentum (M) of the pulse that are respectively given by [11]
(4.5) M
= ~D(Z)
[00
(u*Ut - uu;
+ v*Vt -
vv;)dt
(4.6)
IBy Noether's theorem [5], each of these two conserved quantities corresponds ~o a symmetry of the system. The conservation of energy is a result of the ~ranslational invariance of Eqs.(4.1) and (4.2) relative to phase shifts, while ~he conservation of the momentum is a consequence of the translational in~ariance in t [5]. The Hamiltonian (H) given by
9 Z (I U 14 - -2-
+ 1v 14) - 2'.8 2 (U*Ut* *-
UU t
+v
Vt - VV t*)
las in the case of polarization preserving fibers, is not a constant of motion, IUnless D(z) and g(z) are constants in which it is a consequence of the trans~ational invariance in z. For Eqs.(4.3) and (4.4), the Hamiltonian is given
!bY
(4.8) [Now, it is assumed that the solutions of Eqs.(4.3) and (4.4) are given by Ichirped pulses of the form [5, 11] ~(z, t) = A 1 (z)! [Bl(Z) {t - h(z)}] exp[iC1 (z) {t - h(z)}2
f-land
Iv(z, t)
ilil(Z) {t - h(z)}
+ iBl(Z)]
(4.9)
= A 2(z)! [B2(z) {t - t 2(z)}] exp[iC2 (z) {t - t 2(z)}2
f--
iIi2(Z) {t - t2(Z)}
+ iB2(Z)]
(4.10)
!where! represents the shape of the pulse. Also, here the parameters Aj(z), IBj(z), Cj(z), Iij(Z), tj(z) and Bj(z) (for j = 1,2) represent the pulse ampli-
818
4 Birefringent Fibers
the inverse width of the pulses, chirps, frequencies, the centers of the pulses and the phases of the pulses, respectively. Once again, this approach ~s only approximate and does not account for characteristics such as energy ~oss due to continuum radiation, damping of the amplitude oscillations and Ichanging of the pulse shape. Also, for such pulse forms, the integrals of motion are ~udes,
~ -D
Ai (1) A§ (2) ) () Z ( K1 -B fo 2 0 + K2 -B fo 2 0 1 " 2' ,
r I:[¥
(4.12)
IThe Hamiltonian is now given by =
(lutl 2 + Ivtl 2 )
-
~ (lul + Iv1 4
4
)
~ alul 2 1vl 2 - ~(1- a) (u 2v*2 + v 2u*2)] dt
+ D(z) 2
_
1
~here
g(z) 2
(A2 B 2I(2) 2
0,0,2
+ 4 A2C2 I(2) + K B~
2,2,0
A2 I(2) ) B2 0,2,0
A~ I(2) - aA 21 A22 f + (1- a)A 1 A 2 J 040
B2
(4.13)
"
I: I:
the following notations are introduced:
~=
~=
j2 [B1(Z) (t - h(z))] j2 [B2(Z) (t - t2(Z))] dt
(4.14)
j [B1(Z) (t - h(z))] j [B2(Z) (t - t2(Z))] cos[C1 (t - td 2
t-- C 2 (t -
t2)2
+ K2(t -
t2) - K1(t - td
+ ((h -
(/2)]dt
(4.15)
land
(4.16) [or j
=
1, 2.
81.2 Integrals of motion
49
14.2.1 Gaussian pulses IThe conserved quantities, for Gaussian pulses in a birefringent fiber, reduce to
f
=
I:
1M = ~D(Z) = 1
~hile
1:
(lul 2 + Ivl2)dt
-D(z)
(U*Ut -
=
(~~ + ~~) ~
uu;
+ V*Vt -
("1 AiBl + "2 B2A§) VE2
(4.17)
vv;)dt (4.18)
the Hamiltonian is given by
(4.19) !Where the quantities I and J, for the case of Gaussian pulses, respectively Ifeduced to
(4.20) land
(4.21)
14. 2. 2 Super- Gaussian pulses IThe conserved quantities for SG pulses are
4 Birefringent Fibers
150
D(z} (1);1 Ai + 1);2 A§) r (~) rL _m22'" B1 B2 2m
(4.23)
Mrhile the Hamiltonian here is
D(z)A~ [B2~r B 22
2
2",-1
2~
(4m - 1) + ciB2 2';',; : r (.2.-) 2m
2
mm
2
IWhere the quantities 1 and J, for SG pulses, respectively reduce to (4.25)
(4.26)
14.3 Variational principle ISince Eqs.(4.3) and (4.4) can not be solved by the aid ofthe Inverse Scattering ITransform, they will be studied by the aid of the variational principle. This ~s based on the observation that these equations support the chirped soliton ~olution whose shape is close to that of a Gaussian or sometimes a Super-
51
f'I..3 YanatlOnal prmcIple
i:
lGaussian(SG) pulse. For DM-VNLSE, the Lagrangian is given by [5]:
f
=
~
[i (u*u z
-
uu;)
+ i (v*v z
-
vv;)
H- is (v*Ut - uv;) + is (u*Vt - vu;) - D(z)(lutI2 + IVtI2) 1+ g(z)(luI 4 + Iv1 4 ) + 2ag(z)luI 2IvI 2 1+ 2(3 (u*v + uv*) + ')'(u 2v*2 + v 2u*2)]dt
(4.27)
lFor the reduced DM-VNLSE, the Lagrangian given by (4.3)-(4.4) is given by
L
I
=-
1 [i ( 00
-00
2
u *U z
-
uU z*) +
i(
-
2
v *V z
-
D(z) vV z*) - - (I Ut 12 2
+ 1Vt 12) (4.28)
row, using the pulses given by Eqs.(4.9) and (4.10) the Lagrangian reduces to
Fi= ag(z)AiA~I
(4.29)
IThe EL equation given by Eq.(2.82) will be utilized to obtain the dynamics pf pulse parameters for birefringent fibers. In EL equation, p now represents ~me of the twelve soliton parameters. Substituting A j , B j , G j , K,j, tj and ()j I(j = 1,2) for p in (2.82), the following set of equations are obtained: (4.30) (4.31)
(4.32)
152
4 Birefringent Fibers
d"'l
dz
dtl
=0
(4.33) (4.34)
-D(Z)"'l
-=
dz
(4.35) (4.36) (4.37)
(4.38)
d"'2
dz
=
0
(4.39) (4.40)
d()2 dz
=
D( ) z
(",~ 2
_
B2IO~0'2) 2 1(2) 0,2,0
+
5g(z)A~ IO~4,0 4
[(2) 0,2,0
(4.41) IThe explicit form of the parameter dynamics for the Gaussian and SG pulses ~ill now be obtained in the following two subsections.
14.3.1 Gaussian pulses IThe evolution equations (4.30)-(4.41) respectively reduce to (4.42) (4.43)
f'I..3 YanatlOnal prmcIple
53
(4.44)
dK,l dz dtl dz
-
=
0
= -D(Z)K,l
(4.45) (4.46)
(4.47) (4.48) (4.49)
(4.50) (4.51) (4.52)
(4.53) IThese equations are useful in studying the various physical aspects of the ~olitons in birefringent optical fibers, namely, the timing, amplitude or the [requency jitter, the evolution of the coherent energy and much more.
14.3.2 Super- Gaussian pulses [n this case, the evolution equations (4.30)-(4.41) respectively reduce to
154
4 Birefringent Fibers
(4.54) (4.55)
dCl dz
D(z)
=
4
1
- g(z)Al B 124 >7>+1
I
2>7>
r (2!rJ r( 2m 3) d"'l dz
=
m
2 2 >7>-3 >7>
ag(z)A~ B 2
0
D(z)
=
(4.58)
{",22 _m22~ r r(~) B2} (2;") 1
5 g(z)Ai 2 4 >7>+1 - B1 2>7>
+
I
(4.56) (4.57)
dtl dz = -D(Z)"'l IdOl ~
I
r( 2m 3)
+2
3m ag(z)A~ 2 >7>-1 B2 2>7>
I
r (....L) 2m
aA2
dz = -D(z)A2C 2
(4.59) (4.60) (4.61)
2 r(4m-l)
dC2 = D(z) dz
~
2"',:;;:2
~ B4_2C 2 r(2~)
a"'2 dz
=
0
(4.63) (4.64)
~ = D( ~
+
I
z
)
{",22 _ 22~ r r(~) B2} (2;") 2
5 2
4>7>+1 2>7>
m
3m ag(z)AfB2
2
g(z)A2 +
2
2>7>-1 2",
B
1
I
r ( 2m 1 )
(4.65)
5.5
Q Q Perturbatjon terms
14.4 Perturbation terms ~n
this section, DM-VNLSE in presence of the perturbation terms will be The perturbed DM-VNLSE that are going to be analyzed are given
~tudied.
[by
liU z
+ D~Z) Utt + g(z)(luI 2 + alvl 2 )u = iER1[u, u*; v, v*]
~Vz + D~Z) Vtt + g(z)(lvI 2 + alul 2 )v =
iER 2 [v, v*; u, u*]
(4.66) (4.67)
iHere, Rl and R2 represent the perturbation terms and the perturbation pa~ameter E, as before, is the relative width of the spectrum. In presence of the perturbation terms, the EL equations modify to [27]
8L 8p
d (8L) dz 8pz
--1
.
=ZE
1
00
-00
(8U* R l - - R l*-8U) dt 8p 8p
(4.68)
represents twelve soliton parameters. Once again, substituting A j , IBj, G j , lij, tj and OJ, where j = 1,2, for pin (4.68) and (4.69), the following ladIabatic evolutIOn equatIOns are obtamed: ~here p
) . 10(1) ,2,oTl2 - 312(1)) ,2,0 f ( Tl dTl
(4.70)
I (
f (1 (1) ,2,0 f ( Tl ) dTl 0,2,oTl2 - 12(1))
(4.71)
(4.72) dlil dz
=
2E
00
A1Bl1~ld 0
1- 2G R[R 1
1
-00
1 e- i 1> l
{B 2<S[Rl e - i 1>1] df 1 dTl
hfh)
}d
Tl
(4.73)
4 Birefringent Fibers
156
(4.74)
(4.76)
r((2) 10,2,OT22 - 12(2)) ,2,0 f ( T2 ) dT2
(4.77)
1
dC2 dz
= D( ) (Bi 1o~0,2 2 1(2)
z
2
2,2,0
f (f(T2)
g(z)A§B~ 10~4,0 4
1(2) 2,2,0
+ 2T21!;) dT2
d"'2 _ 2E () dz A2B210 22 0
--
_ 2C2) _
1
00
-00
(4.78) B20<[R -iq,l] df 2'S 2e dT2
1- 2C2~[R2e-iq,2hfh) } dT2
(4.79) (4.80)
57
Q Q Perturbatjon terms
1+ 41i2~[R2e-iq,2]T2!(T2) } dT2
(4.81)
!where the notations land j = 1,2 were used. Once again, ~ and "..5 represent the real and imaginary parts, respectively. This dynamics will now be simplified for the Gaussian land SG solitons in the following subsections. ~or
14.14.1 Gaussian pulses ere, again, using f (Tj) = e -rJ where j = 1, 2 and using the integrals fiji c or j = 1, 2 in Eqs. 4.70 - 4.81 , the adiabatic parameter dynamics of per~urbed Gaussian pulses are obtained as foIIows: (4.82)
(4.83)
I~l = - A~~l
f£ I:
(4.84)
{Bi"..5[R 1 e- i q,1] 2Tl
t+- 2Cl~[Rle-iq,lh}e-r~dTl
(4.85) (4.86)
d(h
dz
= D(z)
+ 5;ng(z)Ai 4v2
4 Birefringent Fibers
158
(4.87) (4.88) (4.89)
(4.90)
(4.91) (4.92)
+ ~ A IB
1
V
27f
2
2
1
00
{B 2SS[R2e- i
-00
t+- 4"2~[R2e-i
14.14.2 Super-Gaussian pulses !For the SG pulses, the adiabatic parameter dynamics reduces to
(4.93)
59
Q Q Perturbatjon terms
(4.94)
Tf
f { ~r m22=
(1) 1 (-3)} - - -------ar 2m
m22=
2m
e- T 12 = dTI
(4.95)
del = D(z)
dz
1 r (2~) ~ g(z)A2I BI224=+1 r( 2m
3 ) 2m
m
3 2
2 2 =-3 A2BI r 2m
I
( 2m 3 ) (4.96)
(4.97) (4.98)
~
to AlIBI
;~:
I:
{BI <S[R l e- i (h](3 - 4mTrm)
1+ 4h;1~[Rle-iq,1h}e-T?= dTI
(4.99)
(4.100)
4 Birefringent Fibers
160
(4.101)
dC2 = D(z) dz
(4.102)
-d"'2 = dz
1
-E--
A2B2
2rn-l
m2 --:;m;-1°O
r
..1....
-00
{2m~2m-l B 22s[R 2
2
2
.'" e-"'f'2] (4.103) (4.104)
dB 2 dz
=
D(z)
2
1
+ 5g(z)A2 2 --:;m;4=+"
(4.105) !Here, once again, Eqs.(4.94)-(4.105) all respectively reduce to Eqs.(4.82)1(4.93) for the special case m = 1. These are the parameter dynamics of Ichirped Gaussian and SG solitons can be very useful to study vector solitons an a birefringent media.
lB.eferen ces ~. ~.
fJ.
ft.
F. Abdullaev, S. Darmanyan & P. Khabibullaev. Optical Solitons. Springer-Verlag, INew York, NY. USA. (1993). M. J. Ablowitz & H. Segur. Solitons and the Inverse Scattering Transform. SIAM. IPhiladelphia, PA. USA. (1981). M. J. Ablowitz, G. Biondini, S. Chakravarty, R. B. Jenkins & J. R. Sauer. "Four-wave ~ixing in wavelength-division multiplexed soliton systems-Ideal fibers. Journal of pptical Society of America B. Vol 14, 1788-1794. (1997). G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San Deigo, CA. USA. (1995).
IHdcrcnccs
61
f5. N. N. Akhmediev & A. /\nkicwi("7;. ,'30litons, Nonlinear Palses and Rcanu;. Chapmanl find Hall, London. UK. (1997).1 !3. D. Andcnmn. "Variational approach to nonlinear pulse propagation in optical fibers Iphysical Review 11. Vol 27, No 6, 3135-:3145. (19tl3). 17. A. 13iswas. "Dvnamics of Gaussian and super-Gaussian solitons in birefringent optical lfibers". Pmgress in Hlectmmagnetics Nesmrch. Vol :n, 119-1:19. (2001). S. A. Bis. .vas. "Dispersion-managed vector solitons in optical fibers'. Fiber' and integrated Oplics. Vol 20, No 5, S03-S15. (2001)j P. A. 13is\vas. "Gaussian solitons in birefringent optical fibers" international .Journal of Ip'ure and l1pplied Mathematics. Vol 2, No 1, tl7-104. (2002).1 110. A. Biswa..s. "Super-Gaussian solil.ons in optical fibers". Fifwr and Tnlcgrnl.rod Oplics. r;0121, No 2, 115-124, (2002),
~~~~~~~~ 112. A. Biswas. "Tnt.{'gro-difrcl"{'nl.ial pertul"baUon 01" disp{,l"sion-manag{xi solil.ons". J01Lrna~ lof £lectmmagnetic Waves and Applications. Vol 17, No "1, 6H-665. (20m). ~3. A. Biswas. "Dispersion-managed solitons in optical couplers' Journal oj Nonlinear' Oplical Physics and Maleria1s. Vol 12, No 1. 4S-74. (200:3). 111. A. 13iswas. "Perturbations of dispersion-managed optical solitons". Progress in Elec-I ~fVmagnetics Resean:h. Vol 4tl, tl5-123. (2004)1 II!). A. Biswas. "Dispersion-T\Tanaged solit.ons in bil"{'fl"ingent fibers and mult.ipl{' challlll'ls'·. ~6.
117.
118.
119.
pO.
t21.
122,
,2:3.
,24.
,2.5.
,26.
linternational Mathematical .Journal. Vol 5, No 5, . 177-515. (200/1). /\. Biswm; &, S. KonaT". Tntrodnction to Non-Kerr T,aw Optical ,'30litons. CRe Press, lRoca Raton, FL. USA. (2006). R. KohL A. Biswas. D. Milovic &; E. Zerrad. "Pert.urbation 01 Gallssian optical solitonsl an dispersion-managed fiGers". Applied Mathernatic8 and Compntation. Vol 199, No 1, poO 208, (2009),1 R. Kohl. A. Bisv,ras. D. Milovic &; E. Zerrad. "Perturbation of super sech solitons inl Illispcrlolion-Ilwuagcd optical fibers". international Jonrnal oj Theoretical f'ftysics. Voll ~7. No 7, 20:38 2064. (2008)1 R. Kohl. A. Diswas, D. Milovic &. Eo Zenad. "Adiabatic dynamics 01 Ciaussian and ~upcr-Gaussian solitons in dispcrsion-managcd optical fibers. Progre8S in l:Jectmmag Indics Research. Vol 84, 27 53. (2008). R. Kohl. D. Milovic. E. Zen'ad &: A. Biswas. "Pert.mbation of super-Gaussian optical ~o1itons in dispersion-managed fibers". Mathematical and Compnter Modelling. Voll ~9. No 7-8. 4"18, 427. (2009). R. Kohl. D. Milovic. E. Zenad x,-, A. Biswas. "Optical solitons bv He's variational !Principle in a non-Kerr law media". Jonrnal of infrared. Millimeter and Terahertz IWaves. Vol 30, No 5. 526-5:37. (2009)1 R Kohl, D, Milo,k, K Zen"" & A, Hi","" "Soliton pe,tlH'hation theo",' [0,1 klispersion-managed optical fibers" . .Jonrnal of Nonlinear Optical Physics and Ma-I ~erials. Vol 18, No 2. (2009)1 l'vI. F. I'vTahmood &. S. B. Qadri. "1'vTodeling propagat.ion or chirped solitons in ani ~l1iptically low birefringent single-mode optical fiber" . .Jonrnal of Nonlinear Optical Iphysics and Materials. Vol 8, No 4, 469-475. (1999)1 D. Marcuse. C. R. T\.Tenyuk &. P. K. A. \Vai "Application or t.he Manakov-PMD equation ~o studies of signal propagation in optical fibers with randomlv varving birefringence" . IJonrnal of Lightw(l"('e Technology. Vol 15, No 9, 17:35-1746. (1997). 1\.I. 1\lal.sumol.o. "Analysis or inl.eraction bct.wcell stretched pulses pmpagal.ing inl klispersion-managed fibers". i£££ Photonics Technology Letters. Vol 10, No :1. 37:1~75, (1998)1 A. Panajotovic. D. Milovic &. A. Mi1.1.ic. "Boundary ca..se or pulse propagat.ion analyl.icl ~olution in the presence of interference and higher order dispersion". T£L81K82005 Conference PfVcn~dings. 547-550. Nis-Serbia. (2005)1
/1 13irefringent Fibersl
~2
127. A. Panajot.ovic, D. rvlilovic, A. Biswas & E. Zcrrru:l. "Influcnce of even ordcr dispersionl ~)n slIper sech sollton transmlSSlOn qllallty under coherent crosst.alk ReM~arch. Leller.~ ~n Optics. Vol 200~, 613986, 5 pages. (2008)) 128. A. Panajotovic. D. I\·Iilovic & A. Biswas. "Influence of even order dispersion on soli~,on t.ransmission qualitY" with coherent interference". Proqre88 w. FJledromaqnelicd lHesearch lJ. Vol :1, 6:1-72. (200~). 129. O. V. Sinkin, V. S. Grigoryan & C. IL Menyuk. "Accurate probabilistic treat.ment ~)f bit-pattern-dependent. nonlinear dist.ort.ions in HER calculations for \VDT\·T RZ sys~ems". Journal of Lightwave Technology. Vol 25, No 10, 2959-2967. (2007)) ~O. C. D. Stacey, IL M. Jenkins, J. Banerji & A. IL Davis. "Demonstration of fundamental Imode only pmpaga1.ion in highly m1l1Lirnode fibr{' for high powcr EDFAs". Opl.ic.1 Comnw.1aml.wn.~. Vol 269, :~10-:~14. (2007)j pI. 1d. Stetanovic & D. 1dilovic. "The impact of out-ot-band crosstalk on optical commu~licat.ion link prcten.'ncelol Jonrnal oj Optical Cmnrnnnication8. \'0126, No 2, 69-72.
(2005). p2. AI. Stefanovic, D. Dmca. A. Panajo!ovic Xi D. Milovic. "Individual and join! influence second and third order dilolpersion on transmission quality in the prelolence of coherent linterference". Oplik. Vol 120, No 1:~, 636-641.(2009). Kn. 13. Stojanovic. D. 1\1. l\.1ilovic & A. 13is\vas. ''Timing shitt. of optical pulses due to inter~hannel croslolt.alk". PmYTe88 in Elecimmayneiic8 Re8emdt M. Vol 1, 21-30. (2008)1 ~4. S. K. Tudt,yn, I. Gabilnv, E. W. Laedke. V. K. Meheul,ev. S. L. M,,,be,', E. G. Shapim, [I. Schater, & K. 11. Spatschek. "Variational approach to optical pulse propagation il~ Illisperlolion compensat.ed t.ransmilolsion Iolystem Optics C:ornnmnicatiOrt8. Vol 1.51, N(~ 11-:3.117-1:)5. (1998). p5. T. H. Wolinski. "Polarimetric optical fibers and sensors". Progress in Optics. Vol XL. ~-75. (2000)1
rA
IChapter 5
rMultiple Channels
5.1 Introduction IThe successful design of low-loss dispersion-shifted and dispersion-flattened pptical fibers with the low dispersion over a relatively large wavelength range Ican be used to reduce or completely eliminate the group velocity mismatch ~or the multi-channel WDM systems resulting in the desirable simultaneous larrival of time aligned bit pulses, thus creating a new class of bit-paraIIel ~avelength links that is used in high-speed single fiber computer buses. In ~pite of the intrinsically small value of the nonlinearity-induced change in the ~efractive index of the fused silica, nonlinear effects in optical fibers cannot be ~gnored even at relatIvely low powers. In partIcular, m WDM systems wIth the ~imultaneous transmission of pulses of different wavelengths, the cross-phase iffiodulation (XPM) effects needs to be taken into account. Although the XPM ~ill not cause the energy to be exchanged among the different wavelengths, ~t will lead to the interaction of pulses and thus the pulse positions and ~hapes gets altered significantly. The multi-channel WDM transmission of copropagating wave envelopes in a nonlinear optical fiber, including the XPM ~ffect, can be modeled bY the foIIowing N-coupled NLSE in the dimensionless Worm [5]:
fq£') + D;Z) qi? + g(z) { I.e') I' +
t,
""m l.e
m )
I' }.e') ~ 0
(5.1)
!where 1 < I < N. Equation (5.1) is the model for the bit-parallel WDM ~oliton transmission. Here aim are known as the XPM coefficients. It is well [known [32] that the straightforward use of this system for the description pf WDM transmission could potentially give incorrect results. However, this !model can be applied to describe the WDM transmission for dispersion flat~ened fibers, the dispersion of which weakly depends on the operating wave~ength.
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
,') ,Multiple Channels
lIt was first pointed out by Yeh and Bergman [51 in model (5.1) that whenl Itwo or more optical pulses co-propagate simultaneouslv and aITed each other ~,hrough t,he intemlity dependen('e of the refractive index, the XPI'vI term car~ ~)e mled t,o produce an interesting pnl.'w shepherding effect. In particular, Yehl land Bergman i:ltudied (5.1) numerically the evolution of two pulses whos~ pperating wavelengths are separated by 4 nm ()q - 1.55 p,m and ).2 - 1.5461 lum). For this case, the F'\.T]'vI dIed is negligible. '\.Then these pulses are ini~.i<-llly
oHset by a half pulse ''lidth, the pulses tend to att.ract. each other. when the tV{() co-propagating pulses, on t,wo separate ,vavdengths,
~lowever,
~,re
separated by a suJilciently large distance, these two pulses will not inter-
!act with each othcr. Tv{O widclv separated pubes can be brought tmfficientlyl pose to each other by launching anothcr pube on a I:)cparate vl'avclcngth ().3 F 1.542 pm) with the proper magnitude and at t,he proper hme. Accordingj ~o Yeh and Bergman this pnlse is called the shepherd pulse becanse of its ~hepherding
behavior \\'ith other pulses. Computer experiments shm\'ed that
lmv- magnitude shepherd pulse does not posses sufficient attractive strengtl~ ~,o pull the shepherd pulses together. Additionally. it was noh('ed t,hat, at <-~ ~l,
Rrery large amplitude of the shepherd pulse, the shepherded pulsei:l tend to break up. and [or the broadened shepherd pulse the eITed is getting weaker. !Therefore, these experiments suggest that, for Eq.(5.1), there exists an opti~IlllIn pulse ''lIth a (Trt,am magruhuic, pulse WIdth and pulse shape that car~ [)rovide t,he best, alignment for these pulses. As a matter of fact, this observa~,ion meani:l that the mOi:lt effective shepherding effect should be observed for pulses that are close to the so called "nonlinear modes'~ o[ the model, that is ~olitons of the nonlinear model (5.1)1 ~!\nother important medium in which t,he model given by Eq.(5.1) arises is ~,he photo-refractive medium [5]. In the case of incoherent beam propagationl ~n a biased photo-refractive crYi:ltaL which is a non-instantaneous non Iin~ar media, the diITradion behavior o[ that incoherent beam is to be treatedl ~ome\\'hat diITerenLlv. The diITracLion behavior of an incoherent beam can be ~fl'ect.ively
described by the sum of t,he intensity contributions from all its
('0-
Iherent components. Then the governing equation of i\ i:lelf-trapped mutuallYI ~ncoherent wave packeti:l in i:luch a media is given by Eq.(5.1 )J lE'quation (5.1) is, in general, not integrable. However, it can be solveJ1 ~1Ilalytically for certain very specifi(' cases, namely, ,vhen D(z) = g(z) = 1 ~llong with OIm = 1, V'm, n. In Eq.(5.1), the special case, ,vhere J.V = 2, feduces to the case of birefringent Hberi:l. This case is studied in the previousl ~hapter.
p,2 ~t
Integrals of motion
needs to be noted that Eq. (5.1) does not have infinitely many cormervatior~ In fact, it has at least two integrals of motion ann they are the energ)l
~aws.
15.2 Integrals of motion
65
I( E) and the linear momentum (M) that are respectively given by [5]
(5.2) land
M = iD(z) 2
t 1 (q(l)*q~l) 00
-
_-00
q(l)q~l)*) dt
(5.3)
IThe Hamiltonian (H) given by
(5.4) however, not a conserved quantity unless, in addition to D(z) and g(z) constants, the matrix of XPM coefficients A = ((Xij) N x N is a symmetric Iillatrix, namely, (Xij - (Xji for 1 :::; i,j :::; N. Thus, for a birefringent fiber, the liriatrix should be of the form ~s,
~eing
A=
(5.5)
IWhile for a triple channeled fiber,
(5.6) land so on. Now, it is assumed that the solution of Eq.(5.1) is given by a Ichirped pulse, in the lth core, of the form [5] Iq(l) (z,
t) = Az(z)f [Bz(z) {t - tz(z)} 1exp[iCz(z){ t - tz(z)} 2
1- ilbz(z){ t -
tz(z)}
+ iBz(z)]
(5.7)
~here f represents the shape of the pulse. It could be a Gaussian type or Ian SG type pulse. Also, here the parameters Az(z), Bz(z), Cz(z), Ibz(z), tz(z) land Bz(z) respectively represent the soliton amplitude, the inverse width of ~he pulse, chirp, frequency, the center of the pulse and the phase of the pulse an the lth channel. Using the variational principle, a set of evolution equa~ions for the pulse parameters will be derived. Once again, this approach is kmly approximate and does not account for characteristics such as the energy ~oss due to the continuum radiation, damping of the amplitude oscillations land changing of the pulse shape. For convenience, the following integrals are
Idefined'
5 Multiple Channels
166
(5.8) (5.9) Iwhere a, band c are integers and 1 ::; l ::; N. For such a pulse form given by 1(5.7), integrals of motion are
100 I _ -00
2
N
= '""" ~
q(l)
I
N
A2
= '""" ~ _ B I l(l) 0,2,0
_
dt
(5.10)
I
(5.11) !While the Hamiltonian is
=
D(z) '""" (A2 Bll(l) 2
~
1=1
I
0,0,2
+ 4 ArClI(I) + KrBAr 1(1) ) B3 2,2,0 0,2,0 I
I
(5.12)
15.2.1 Gaussian pulses Wor a pulse of Gaussian type, substitute f( T) = e-!T2. Thus, the conserved Iquantities respectively reduce to
(5.13)
15.2 Integrals of motion
67
(5.14) Mrhile the Hamiltonian is
= D(z) 2
(A2B ArC? "'rAr) vE'2 ~ ~ + Br + BI II
N
N
g(z) " "a A2 A2 2 ~~ 1m I m 1=1 m
. exp
I
~here,
{
27r
I
Bl+B;,B;,) (-tl - -) 2} tm
2 (Bl
(5.15)
for Gaussian pulses, the integral JI,m, from Eq.(5.9) reduces to
JI,m
I
=
{Bl B;, (- -) 2} Bl 27r + B;, exp 2 (Bl + B;,) tl - tm
(5.16)
15.2.2 Super-Gaussian pulses
Wor
SG pulse, choose 1(T) = e-!T 2P with P 2: 1, where the parameter p Icontrols the degree of edge sharpness. For an SG pulse, the integral JI,m, ~rom Eq.(5.9) reduces to (5.17) !which cannot be obtained in a closed form, unlike in the case of Gaussian ~oliton, and unless a particular value of p is considered. So, the integrals of !motion respectively are (5.18)
168
=
D(2 r (~) 2p
. p2 2P 1
t
5 Multiple Channels
r.z
l-1
Ar
(5.19)
Bz
Mrhile the Hamiltonian is
= ~ [D(Z) {ArBZr (2 P -1) + 4 A rCl r ~
2p
2
pBr
(~) 2p
r.rBAr r (~) } _ g(z) ~ a A2 A2 J Z,m ] 2 2 L.J Zm Z
+
~.3
2
p
P
Z
m
(5.20)
m
Z
Variational principle
IThe soliton parameter dynamics will be obtained by the aid of variational !principle. In order to study this principle, the Lagrangian of Eq.(5.1) is reeded. This is given by [5]
+ g(z)
Iq(l) 14
+ 2g(z) ~1 aZm Iq(l) 12 Iq (m) 12] dt
(5.21)
row, using Eqs.(5.7)-(5.9), the Lagrangian given by Eq.(5.21), reduces to L
= "L.J [_ D(z) {Ar2Bz 1(Z) + 2ArCl 1(Z) + r.r Ar 1(Z) } 0,0,2 B3 2,2,0 2B 0,2,0 Z=l
I I
Z
Ar
Z
Ar (t zdr.z _ dez) l(l)
g(z) At l(l) _ dCz I(Z) + 2 Bz 0,4,0 Br dz 2,2,0 Bz
dz
dz
0,2,0
N
+ g(z)Ar L azmA~Jz,m] m-l-Z
(5.22)
IBy the EL equation, seen in Chapter 2, the soliton parameter dynamics are
dAz
dz = -D(z)AzCz
(5.23)
69
15.3 YanatlOnal prmcIple
dBI dz
dCl dz
=
D(z) 2
{
=
(5.24)
-2D(z)B1CI
(l) B41o,0,2 _ 4C2 } I 1(!) I 2,2,0
(5.25)
dl l
-
dz
dBI ~ dz
=
D(z) 2
{
2 _
"'I
d"'l = 0 dz
(5.26)
=
(5.27)
-D(Z)"'1 (I)
2B2 10,0,2 I r(l)
}
+~ (
0,2,0
(I)
)A2 10,4,0 4 g z I r(l) 0,2,0
3 g(z) " B I L...J almAmJl,m 2JW 2
0,2,0
(5.28)
moll
IThis dynamical system of soliton parameters will now be modified to the ~wo special cases where two types of pulses will be considered. They are the !Gaussian and super-Gaussian pulses that are studied in the following two subsectIOns.
15.3.1 Gaussian pulses lFor Gaussian pulses, the pulse parameter dynamics reduces to dAI dz = -D(z)A1CI
(5.29)
dBI dz = -2D(z)B1CI
(5.30)
dCl (4 J2 4 ~ 3 -=2D(z) Bl -Cl2) --g(Z)A1Bl-v2g(z)Bl
.moll LB N
I
m
aim A4m {2 Bl 2 Bm - 2} JB2 +B2 exp 2(B2 +B2) (tl-t m ) I m I m d"'l = 0 dz
dl l
-dz =
-D(Z)"'1
(5.31 ) (5.32)
(5.33)
5 Multiple Channels
170
(5.34)
15.3.2 Super-Gaussian pulses lFor SG pulses, the dynamical system simplifies to
dAz dz dBz dz
=
=
-D(z)AzCz
(5.35)
-2D(z)BzCz
(5.36)
dKz =0 dz
(5.38)
dz = -D(Z)KZ
(5.39)
dtz
-----''--c----'---!---} + 3 P ()B ~ aZmA;, JZ,m - - - g z zL 2r ...l. Bm 2p m.,...-"-Z
4~+1 g(1 At
22P
z
(5.40)
5.4 Perturbation terms [n this section, the NLSE for multiple channels in presence of perturbation ~erms will be studied. The perturbed DWDM system given by
71
1.5 4 Perturbatjon terms
(5.41) once again, 1 :::; l :::; N while m =I- l. In presence of perturbation terms, EL equation modify to [5]
~here,
~he
loL _
raP
~
(OL) = iE dz opz
1
(R oq(l)* _ R* oq(l) ) dt op op
00
-00
(5.42)
~here p represents the 6N soliton parameters. Once again, substituting A z, IBz, G z, tiz, tz and Bz for pin Eq.(5.42), it is possible to arrive at the following ladiabatic evolution equations:
IdAz
~ = -D(z)AzGz
+EBzl°O 4
(5.43)
-00
IdBI dz
=
-2D(z)Bz Gz
2
EB21°O + _z 2Az
TZ
1
JCilJCil 2,2,0 0,2,0
-00
g(z) B3"
-(-Z)- z ~ aZm 21 2 ,2,0 m#-Z
~ ~ 4 A
Z
_1_1 (Z)
1220
00
A2
+ q(Z) R*)dt
(5.44)
T
mJZ,m
[B (
-00
(q(l)*R
z q
(I) R*
-
q
(1)* R)
(5.45) dtiz = ~_1_1°O ['B ( (Z) R* _ (Z)* R) dz A (I) Z z qt qt z 1020 - 0 0
dtz dz = - D (z ) tiZ
+ AzE
1
1
00
-I(I) _ 020
00
(5.46) TZ ((1)* qR
+ q(Z)R
*) dt
(5.47)
5 Multiple Channels
172
+ ~~ 1
2 Az
_1_1 1~~~,o
00
[3iBz(q(l) R* - q(l)*R)
-00
1+ 2iTZ(q~Z) R* - q?)* R)
+ 4/iZTZ(q(l)* R + q(l) R*)]dt
(5.48)
!Now, relations (5.43)-(5.48) can be rewritten in the following alternative Iconvenient forms. These are also known as the phase-amplitude forms. dA Z = -D(z)AzCz - ~ dz 2
I
-dBz = dz
-2D(z)BzCz -
1 ~[Re-iq,l] (l __ 1 1)
Bz Az
00
1(l)
0,2,0
-00
00
(0-
~[Re- iq, I]
( -TZ1(l)
3_)!(TZ) dTZ 1(Z)
(5.49)
!(TZ)dTz
(5.50)
2,2,0
-
2,2,0
-00
--
1(Z)
0,2,0
g(z) B3" A2 T -(-Z)- z ~ aZ m mJZ,m 212,2,0 m=l-Z
(5.51)
(5.53)
3 g(z) B Z ~ A2 J +"2----W~ aZ m m Z,m + 10,2,0
m=l-Z
3!(TZ)+2TZ dd! TZ
(0
(l) 2A zBz1 o,2,0
+4/i~[Re-iq,I]T!(TZ)
1 {B 00
O<[Re -iq,l] z'S
-00
dTZ
(5.54)
1.5 4 Perturbatjon terms
!where
TZ land ~or
=
B(z) (t - tZ(z))
(5.55)
~z = CZ(Z) {t - tZ(z)} 2- K;Z(Z) {t - tZ(z)} + (}z(z)
1 :::; I :::; N. Also,
~
(5.56)
and 8' represent the real and imaginary parts, respec-
~ively.
15.4.1 Gaussian pulses ow, substituting the integrals I(Z)b for the indicated values of a, band c in ~qs.(5.49)-(5.54), the adiabatic parameter dynamics for Gaussian pulse are a, ,c
(5.57)
(5.58)
dCz dz
I
=
4
v'2
2
4
2D(z)(Bz - Cz ) - Tg(z)A z Bz - v'2g(z)Br
L
aZmA;, m#Z Bmy'Bl + B;'
(5.59)
(5.61 )
(}z -_ ~ dz
D(z) - (K;Z2 - B2) Z 2
ILN
5v'2 3v'2 +g ()At z - +g (z )Bz 8
Bz
2
{22 aZmA4m B z Bm - - 2} B y' B2 + B2 exp 2 (B2 + B2 ) (tz - t m ) m#Z m Z m Z m
174
+ -EA 1B 27f
Z
1
5 Multiple Channels
00
Z
-
2 {BzSS[Re-""cp 1](3 - 4TZ)
(5.62) IThese equations now represent the evolution equations for the parameters pf a Gaussian pulse, for 1 :::; I :::; N, propagating through an optical fiber in !presence of the perturbation terms.
15.4.2 Super-Gaussian pulses perturbation terms of an SG pulse, substituting the integrals I(l)b ~oror 1the:::; I:::; N and the form of !(TZ) in Eqs.(5.49)-(5.54) leads to
a, ,c
IdAz p+21°O"cp dz = -D(z)AzCz - Ep2-:P -00 ~[Re-" I] (5.63)
~ dz
=
21
00 Bz -2D(z)BzCz - E-p2"P ~[Re-"'i"'l] Az_oo "A.
(5.64)
dCz = D(z) {B4 (2 _ l)r( -Tv-) dz 8 zp P r(~) -
g(z) 2
4
r(2p)
4p+l Az Bz r( 2p
3 ) 2p 2p
_ 8C2} z
pg(z) 3 " A;' r( 3 ) B z L...J aZ m BJz,m m#Z
m
(5.65)
(5.66) (5.67)
7.5
IReferences
dBz
D(z)
dz
2
+
5
2
4p+l
A4 g(z) BZ Z
2p
2p+l
+
EA 1B p22P (1) Z
Z
r
2
3
p
+"2 r(
1
00
-00
1 2p
)g(z)Bz
L N
m
aZm Z
A4 Bm
JZ,m
m
{B O<[R -icPl](3 4 2P) z'S e - pTZ
(5.68) ISo, now, these are the adiabatic evolution of the soliton parameters for an ISG pulse, for 1 :s; l :s; N, in presence of the perturbation terms.
IReferences M. J. Ablowitz & G. Biondini. "Multiscale pulse dynamics in communication systems strong dispersion management". Optics Letters. Vol 23, No 21, 1668-1670. (1998). ~. G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San Deigo, CA. USA. (1995). f3. N. N. Akhmediev & A. Ankiewicz. Solitons, Nonlinear Pulses and Beams. Chapman land Hall, London. UK. (1997). fl. D. Anderson. "Variational approach to nonlinear pulse propagation in optical fibers". IPhysical Review A. Vol 27, No 6, 3135-3145. (1983). f'). A. Biswas. "Dispersion-Managed solitons in multiple channels". Journal oj Nonlinear pptical Physics and Materials. Vol 13, No 1, 81-102. (2004). ~. T. Hirooka & A. Hasegawa. "Chirped soliton interaction in strongly dispersion~anaged wavelength-division-multiplexing systems". Optics Letters. Vol 23, No 10, 1768-770. (1998). 17. T. Hirooka & S. Wabnitz. "Stabilization of dispersion-managed soliton transmission [by nonlinear gain". Electronics Letters. Vol 35, No 8, 655-657. (1999). ~. T. Hirooka & S. Wabnitz. "Nonlinear gain control of dispersion-managed soliton amplitude and collisions". Optical Fiber Technology. Vol 6, No 2, 109-121. (2000). f.). P. M. Lushnikov. "Dispersion-managed solitons in optical fibers with zero average ~ispersion". Optics Letters. Vol 25, No 16, 1144-1146. (2000). ~O. T. R. Wolinski. "Polarimetric optical fibers and sensors". Progress in Optics. Vol XL, ~-75. (2000). ~.
~ith
IChapter 6
Optical Crosstalk!
~lodern all-optical networks that employ all-optical switches and optical addkIrop rnulLiplcxcrs (OA D1Vfs) provide a Lrcrncndous high spccd or several tens bf Tera-bits per second. Snch net"\vorks are challenging ''lith IIlany issneH t,hat. ~mve to be anhC'lpatcci aIIlong '\TIuch optIcal crosstalk IS an Important oneJ IAlmost every component in an optical communication system may introkJuce one of the two different types of croi:li:ltalki:\ which are known as in-bandl bI' out-oI-band crosstalks. In optical nctvmrks. using wavelength division lllul~.iplexing (\VDJ'vf), it is 'very common to mle the terrninologieH of intra-channe~ ~1Ild int,er-C'hannel respectively for iu-band and ont-of-band. The main differfnce between such croi:li:ltalk i:lignals lies in their wavelength valuei:l. Althoughl pptical components are gcLting bettcr bv rejecting more than 35 dB fron~ Itheir adjaccnt channels. there still exists residual signals. The fcature be~:ornes IIlore Hignificant, if channel powers are nneqnal. IThe cnrrent \VDJ'vi technology involveH IIlultiplexing He-veral hnncire(iH of Rvavelengths and transporting them in a single fiheL t.hus achieving This in 3J Ito tal capacitv. If a \VD1\I is used. in a fibcr radio ncLwork. [or example. thcnl ~ach base station (BS) can be assigned a single wavelength. A \VD1\I rcquircsl ~elective oph('al cOIIlponentH that can IIlnltiplex or deIIlulhplex channels (orl kirop and add channels). These components can ali:lo caui:le an optical croi:li:ltalk! Iby not fully removing unwanted channels. [\Vhen the crosstalk and input signals are on the same channel or wavc~ength, or have dOHc-valned "\v<-lVdcngt.!lH, it iH an in-band croHHtalk. ThiH kindl ~)f ('rosstalk accnIIlulat,eH very h-1Ht while paHHing t,hrough optical s"\vitcheH andl Ihave more serious effect than the same kind of crosstalk in the detection stag~ 171. The in-band croi:li:ltalk, however, can be either coherent (phai:le-correlated) pr incoherent (not phase-correlated) with the signal considered [41. If the sig~al croHHt,alk mixing takes place "\vithin the laser coherent lcngt,h, then inIband crosstalk is classified aH a coherent crosstalk. Otherwise the incoherent kTosstalk will appear. The cohercnt crosstalk impacts signal rnorc scvcrelvl
Uwn
the jI)('ol)('rent crosstalk The ollt-of-hand or jnkr-dJi-lIlIH'] (Tossta]k ()('-
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
6 Optical Crosstalk
178
Icurs when the crosstalk signal has wavelength that is different from that of luseful signals. IThe optical crosstalk even though originated from imperfections of small Icomponents mostly tends to generate large performance degradations and ~hus needs extra attentions. Experimental studies ofin-band and out-of-band Icrosstalks show that the out-of-band crosstalk can be efficiently removed by parrow-band filters and has no influence on signals. On the other hand,the in[band crosstalk results in closing of signal eye-diagram and increases bit-error ~ate (BER).
16.1 In-band crosstalk IThe in-band crosstalk arises from the signal leakage within optical crossIconnects (OXeS) mainly due to component imperfections. It can also arise from reflectIOns but thIS IS not a serIous source because these reflectIOns can Ibe controlled. The presence of crosstalk signals at the switch output is very ~mportant and needs improvements. ~ crosstalk signal may occur as a leakage of channel 1 which travels with ~he signal in channel 2 and remains present at the output of the multiplexer. IThe demultiplexer should be able to ideally separate all wavelengths to dif~erent optical fibers but due to the component imperfection, a small portion Iffiay leak into the adjacent channel. Throughout the stage of multiplexing all ~avelengths into a single fiber, the smaIl portion of optical signals that leaked ~nto the adjacent channel will leak back into the output of single fibers. Along ~ith different nonlinear effects and effects from higher order dispersion, the ~nfluence of the crosstalk on optical pulse propagation becomes more severe. IThere are two different approaches in studying the influence of crosstalks ~m the optical pulse propagation. The first one is the analytical method that Illses the linear model of a single mode fiber and the second one uses non[inear effects and direct NLSE solving. The analytical method analyzes the Ijoint influence of the higher order dispersion and crosstalk on optical pulse Idynamics in a closed form. IThe linear model of a single mode fiber with length L is given by its ~ransfer function as [1]
!H(w)
= exp { - (0; + ij3(w)) L}
(6.1)
kYhere 0; is the attenuation coefficient, j3(w) is the phase constant that can be ~xpressed in Taylor series about w = Wo as
(6.2)
79
16] In-baJJd crosstal k
IThe impulsive response is obtained by using the inverse Fourier transform of
IH(w):
~(t) = -!;
r
1:
I:
H(w)eiwtdw
2~e-aLei(Wot-(30L)
exp [i{W(t-,BIL)
1- ~,B2w2 L - ~,B3w3 L -
... } ] dw
(6.3)
IApplying Euler's lemma to Eq.(6.3) simplifies it to
h(t)
I
=
~e-aL exp [i {wot 27f
,BoL + O(t)}] JI';(t)
+ I';(t)
(6.4)
!where
IIs(t) =
(6.5)
I:
sin (wt -
O(t)
=
~ ~,BnLwn) dw
tan- 1 ls(t)
le(t)
(6.6) (6.7)
IThe characteristics of phase shifts, denoted by O( t), are induced by the higher prder dispersion terms. the impulsive response h(t) corresponds to the elec~rical field in physical sense and its envelope can be expressed as
(6.8) peneral forms of optical signals in medium with a dominant dispersion effect [s obtained by using Eqs.(6.4) and (6.7) with the integrals le(t) and ls(t) Idefined as
r1(t)
=
r2(t) =
I: I:
F(w) cos (wt-
~ ~!,BnLWn)dW
(6.9)
F(w) sin (wt -
~ ~!,BnLwn )dW
(6.10)
!where F(w) represents the Fourier transform of input signals.
6 Optical Crosstalk
180
16.2 Gaussian optical pulse IPulses from many lasers can be approximated by Gaussian pulses and the !most important feature of such pulses is that they maintain their shape during propagation. Therefore, the input optical signal with a Gaussian envelope is ~epresented by ~1(0, t) = JPoe-(t/T~)2+iwot
(6.11)
~here Po is the optical pulse peak power, T6 = To/V2 is pulse half-width (at 11/ e intensity point) and Wo is the optical carrier frequency.
IThe in-band interference is at the same frequency as an useful signal but time and phase shifted in regard to the useful signal, as
~he
(6.12) ~here Pi is the interference peak power, b (b = b'To) and ¢> are the time and IPhase shIft, respectIvely. The value of the tImmg ShIft, or the propagatIOn Idelay, depends on the nature of the in-band interference (coherent crosstalk pr multi-path reflection). The phase shift ¢> varies in a random manner due ~o temperature and wavelength variations in the range [0, ?fl. The envelope land phase of the resulting signal sTet) at the fiber input are [8]
(6.13) v'PiC(t/T~-b')
(t)
=
tan- 1
sin¢>
JPoe-(t/T~)2 + v'Pie-(t/T~-b')2 cos¢>
(6.14)
IThe in- band interference can be constructive or destructive, depending on ~he phase-shift value. If the worst case is considered, i.e., the destructive [nterference and if it is assumed that it appears at the beginning of an optical !fiber (e.g. a double reflection [3] or an in-band crosstalk resulting from WDM Icomponents used for routing and switching along optical networks [5])the keceiver pulse shape under the influence of n-th order dispersion is [9]
rn(t, L) VIJ:n =
e-aLei{wot-f3oL+O(Y)}
O(T) = tan- 1 12(T) h(T)
Jtt(T) + I?(T)
(6.15) (6.16)
81
16.2 Gaussian optical pulse
(6.17)
(6.18) Eqs.(6.17) and (6.18) are obtained by changing the following paramein Eqs.(6.9) and (6.10):
~here
~ers
bn
I
=
((3nL) -lin n.,
(6.19)
'
~t it is assumed that binary data sequences with values 0 and 1 are transirrutted along the optIcal fiber. For non-overlappmg of lIght pulses down an pptical fiber link, the digital bit rate (B) must be less than the reciprocal of ~he broadened (through certain order of dispersion) pulse duration. Depend~ng on the interference time shift value, the coherent interference can be left I(b < 0) or right (b > 0) shifted from the center of data in binary sequences. IThe propagation length can be expressed via dispersion length that is for r,-th order dIspersIOn IS gIven by
(6.20) IThe second order dIspersIOn mduces a symmetncal broadenmg. The greater ~ime-shift of interference induces the asymmetrical pulse deformation. Note ~hat both, the noisy nature of the input to a clock-recovery circuit and noise !produced by optical amplifiers, timing jitter can be induced. Then, as previkJusly mentioned,the asymmetrical pulse deformation can be dangerous. The [Worst case in the detection process is b = O. This situation is very often seen ~n switching systems [1, 2]. The eye diagram for the worst case is shown in !Figure 6.1(a). IThe crosstalk level is defined by the signal-to-noise ratio (SIR) i.e., the !ratio of useful signal optical power to crosstalk signal optical power. It is Idefined as
Po (6.21) Pi OCf one of the methods that compensate degrading influences of the second prder dispersion is employed, the third order dispersion remains and has a greater influence on the pulse shape. The third order dispersion distorts the pulse shape such that it becomes asymmetrical with an oscillatory structure pear one of its edges ((33 > 0 affects the trailing edge of the pulse while SIR
= 20 log -
6 Optical Crosstalk
182 0.5~-------------------------,
SIR=12dB b=O L=3 d 0.4 B=31 Gb/s: P3= Ops3/km. T,o=4ps
1
~0.3 ;:! 0)
~0.2
0.1
r.
1. O~-------------------------, SIR=12dB, b=-T04, L=3Lo 0.8
B=125Gb/s, P3=O.lps3/km,
=lps
~~0.6
4
8
12
(b)
SIR=12dB,b=O,L=3L d B=O.87Tb/s, P3=1.3ps3!km, P3=O.13ps3/km, To=O.141ps
0.2
O. O+-.--.-.,...,~~~~~~""";;::::;:::~...,.....j
-1. 0-0. 8-0. 6-0. 4-0. 20. 0 0.2 0.4 0.6 0.8 1. 0 1"
(c)
IFig.6.1 Eye diagram for (a) second order dispersion, (b) third order dispersion, (c) second land third order dispersion in the presence of coherent interference.
83
16.2 Gaussian optical pulse
183 < 0 affects the leading edge of the pulse). Because of the asymmetrical Ideformation of pulses induced by the third order dispersion (oscillation on the ~railing edge) the biggest error in the detection process will occur for small Inegative interference time shifts. The effect of a negative interference is more Idestructive than a positive interference at the receiving end of the fiber due ~o the timing shift of the resulting pulse. The opposite situation will happen ~or (33 < O. Great absolute values of time shifts can increase an inter-symbol ~nterference (lSI) if the transmission rate is high enough to induce a sizeable pverlapping of pulses. Figure 6.1 (b) shows the eye diagram for the case ofthe ~hird order dispersion, when the second order dispersion is suppressed in the presence of the most destructive interference. IThere is a case (equal dispersion length for the second and the third order Idispersions) when it is needed to investigate the joint influence of the second land the third order dispersions on pulse propagation along the linear optical lfiber. Then analytical expressions describing the pulse shape along the optical lfiber have the following form [8]:
r
2,3(t, L) =
~o e- aL ei {w ot-,6oL+li(r)} B(t) = tan- 1 I 2 (t) h(t)
JIr(t) + Ii(t)
(6.22) (6.23)
(6.24)
(6.25) land
(6.26) !Figure 6.1 (c) shows the joint influence of the second and the third order dispersions and in-band crosstalks.The Gaussian pulse at the receiver is broad~ned by the second order dispersion and it has a long trailing edge as a result pf the third order dispersion influence. Because of such pulse deformation, ~he position of in-band crosstalk signals in regard to center of a bit is very
6 Optical Crosstalk
184 ~mportant.
In a detection process,the bigger error is made in the following
Icases. If Ibl < To, for in-band crosstalk signal right shifted with respect to the Icenter of bit. ~. If Ibl ::::: To, for in-band crosstalk with respect to the center of bit. ~.
IThe results presented in Figure 6.2 below testify that the pulse becomes most Idistorted for the case when there is no time shift for in-band crosstalk signals. IThe eye diagram for this case is shown in Figure 6.l(c). 1.0~----------------------------~ L=3Ld ,P2=1.3ps2/km, P 3=0.13ps3/km
=0.141ps,SIR=10dB - i n absence of interference
0.6
~
0.5
~~
0.4
·3
1 .2 .1
. 0 +-r"""T"""1"""T"""l,....,-:!~"T"""T"-.=....;::;:~~~ H.0-0.8-0.6-0.4-0.2 0.0 0.2 0.40.6 0.81.0
IFig.6.2 The pulse shape at the end of the optical fiber (L = 3Ld) under the joint influence pf the second and the third order dispersions for SIR = 10 dB.
'6.2.1 Bit error rate !When the transmission rate (B) and the transmission distance (L) is fixed, la suitable measure of the line performance is the bit error rate (BER). Since ~he BER has to be extremely small, with numerical tools, it is very difficult land time consuming to perform full simulations of the system in order to Idetermine BER simply by counting the mistakes. Therefore, it is of great ~nterest to find a proper statistical approximation of the BER. IThe most commonly used technique to evaluate the Intensity Modulation!Direct Detection (IM-DD) system performance assumes a Gaussian white poise distribution on both the zero and the one levels [10]. It is difficult to Idetect interference time shifts especially when there are many connections land taps in the system because they may cause reflections too. Therefore, it ~s treated as a random variable with a uniform probability density function
85
16.3 Sech optical pulse ~(b).
The BER, in this case, is given by ER = -
11 1+ 2 '8
2
-
2'8
iD
1
00
-2
J27f( i N )1/2
. )2 (Z. - Zsig o
exp
22~
. )2 (Z. - Zsig 1
22~
di
di p(b)db
(6.27)
Iwhere i sigo and i Sigl are the mean value of the currents, in 0 and 1 states
~espectively, iD is the decision threshold and 2~ is the mean square noise Icurrent. The BER for joint influence of the second and the third order dispersions in the presence of coherent interference is shown in Figure 6.3.
~'~";~~~~~~:~-~~~-~~~'---~~-'
.01
...................
lE-4 lE-6 lE-8 ~ ~
._._._._._._._._._._._._._._._._._._._._._._._._._.
lE-lO
.-'~':~\\.-.-.
~ lE-12 lE-14
B=0.71 Tb/s,L=3L d
lE-16
in the absence of interference -··-·SIR=10dB ---- SIR=15 dB ....... SIR=20 dB
fJ,=1.3 ps'/km, fJ,=O.13 ps'/km,
IE-18 lE-20
lE-22+-~--.-~---.--~-r--~-.--~-.~
5
10
20
15
25
30
SNR[dB] IFig. 6.3 BER curves (L = 3Ld) for joint influence of the second and the third order dispersions.
16.3 Sech optical pulse IThe sech model of the input optical pulse and the in-band interference I( crosstalk) will now be considered that has following shapes:
set) -
JPo
- cosh (t/To)
eiwot
(6.28)
6 Optical Crosstalk
186
JPi
r
I"i(t) =
cosh(A - b')
(6.29)
ei(wot+¢)
is the interference peak power and b (b = b'To) is the interference shift. The phase shift 1> varies in a random manner due to the temperlature and wavelength variations in the range (0, n). The envelope and phase pf the resulting signal sr(t) at the fiber input are [6] ~here Pi ~ime
ISr(t)1
=
Po
cosh
2
i: +
r
2v'P OP 1 cos 1> t t cosh To cosh( To - b')
(t)
= tan- 1
Pi
+ --2~'-----cosh
(i: - b')
vPi sin> cosh(..L-b') ~ cosh..L TO
TO
vPicos>
+ cosh( ..L-b')
1/2
(6.30)
(6.31)
TO
IA general
expression for the fiber response for an arbitrary input pulse is lalready given in Eq.(6.15), which for the case with the influence of the second land the fourth order dispersions, can be written as in Eqs.(6.22) and (6.23) IWhere now
1
00
l(t)
=
-00
cos
1 h~
1-
2
f cos(wt - b2 w2
-
V(P: Po cos ('b Tow )
b3w3 )
~fi, sin (b'Tow) sin(wt -
1
00
2(t)
=
-00
cos
1 h~
1-
2
f sin(wt - b2 w2
-
b2 w2
-
b3W3)] dw
(6.32)
V(P: Po cos ('b Tow )
b3w3 )
~fi, sin (b'Tow) cos(wt -
b2 w2
-
b3W3)] dw
(6.33)
IBinary data sequences with values 0 and 1 are transmitted through the optical !fiber. The digital bit rate (B) is less than the reciprocal of the broadened pulse Iduration. Depending on the value of interference time shifts, the coherent ~nterference can be left (b < 0) or right (b > 0) shifted in regard to center bf data in binary sequences. The propagation length is expressed via the Idispersion length which for the nth order dispersion is given by Eq.(6.20). IThe pulse shape under the influence of the second and the fourth order Idispersions is shown in Figure 6.4. Strong influence of the interference may [be noticed even for b > To and long trailing ends will unquestionably induce
87
16.3 Sech optical pulse
0.6.------------------------------. SIR=10 dB, b=O L=3Ld , B=100 Obis, P,=10ps2/km, To=lps
0.5
! ~!: .O
.9
([ps] (a)
SIR=10 dB, b=O L=3Ld , B=1.7 Tbls, P,=3.6*10-'ps'/km, To=O.lps
.8
0.7
~
! 0.4 .3
.2
.1
.O~~--~~-.~~-r~~._~~T_~~
1-1.5
-1.0
-0.5
0.0 ([ps]
0.5
1.0
1.5
IFig. 6.4 Eye diagram for individual influence of (a)the second order dispersion and (b) ~he fourth order dispersion in the presence of worst case interferences.
~he inter-symbol interference (lSI). The worst case in the detection process !happens for b = 0 and is seen in Figure 6.5 below. IThe joint influence of the second and the fourth order dispersions is shown ~n Figure 6.6.
6 Optical Crosstalk
188
.6 L=4Ld ,p,=5.6*10-' ps'/km,P2=2 ps2/km To=17 fs, SIR=10 dB
.5
in the absence of interference --b=O _ .. _ .. b=To _._._ .. b= 2To
.4
~
~
-------- b=3 To
0.3
---b=4To
E-..~
0.2
~0.2
-0.1
0.0 t[ps]
0.1
0.2
IFig. 6.5 Pulse shape at the end of the optical fiber (L = 4LD) under the second and the ~ourth order dispersions for SIR = 10 dB and different time shifts b > O.
0.5
SIR= 10 dB, b=0, L=3Ld P4=5.6*104 ps4/km, P 2=2ps2/km, B=4 This, To=17 fs
0.4
(1)
~
0.3
ii>' 0.2 0.1 0.0
IFig. 6.6 Eye diagram for joint influences of the second and the fourth order dispersions (L = 3L d ) in the presence of the worst case interference.
16.4 Super-Sech optical pulse
IA more general form of the sech pulse
(and interference) can be mathematiIcally represented as a super-sech pulse model having following forms:
89
16.4 Super-Sech optical pulse
(6.34)
r
I"i(t) =
JPi
coshm(A - bl )
(6.35)
ei(wot+>l
IThe Fourier transform F(w) of the input pulse for even m (m Ik = 0,1,2, ... ) can be written as (w)
= 2mTo
+
F(m
F(m
+ m-iTow
m-iTow 1 '2'.
m - zTow
m+iTow '2'
1+
m+iTow 2 '
= 2k
where
-1)
2
'
-1)
(6.36)
m+iTow
Iwhere the Gauss' hyper-geometric function is defined as
(
0:,
(3 . . )= r(')') "r(o:+n)r((3+n)zn ,'Y, z r(o:)r((3) ~ r(')' + n) n!
(6.37)
!While if m is odd (m = 2k + 1 where k = 0,1,2, ... ), the Fourier transform is gIVen by
F(w)
=
_1_2-':j'-2e-H(k-1li+2W}
,.fir
1+ e
7rW
B-1
1
4(k -1 -
(~(k -1 + 2iw), ~(3 -
k))]
1
2iw), "2(3 - k) (6.38)
!Where the incomplete beta function is defined as (6.39) !Now, both the Gauss' hyper-geometric function and the incomplete beta [unction can be numerically evaluated with any arbitrary precision. In Figures 16.7(a) and (b), the super-sech optical pulse and crosstalk with m = 2 are Shown.
IA more realistic scenario includes nonlinearities in optical fibers along kYith the dispersion. In such case, the optical fiber transfer function (6.1) is Ino longer valid. The influence of the crosstalk on an optical pulse propagation Ican be determined by solving the NLSE for the in-band crosstalk or by solving la set of coupled NLSE for the out-of-band crosstalk. IThe in-band crosstalk and useful optical signal are approximately at the ~ame frequencies: (6.40)
6 Optical Crosstalk
190
,-, C1\ t ti!!"';'\f'~
L=3Ld , p,2
-·-b-O -'-b=To -o-b=2T
0go
,8,=5.6*10-'ps'/km, ;/\ '\ R
0')'
'/1-
-~~
V
~
.;:::
g
\\
oooo • •
1'\ \ \ \
~ 0,1
p
l' 'P!
..... ~'i.. .~
00 1-0,2
~'o
.,\. " ". -o-b=4To in the absence of .'. crosstalk , \
To=17 fs, SIR=10 dB
• •-
r
'".\\
./
'.,\.0\ '.,~ ~~ ·.. -:.'as., ....
0,0
0,2
(a)
1°,20
SIR=10 dB, b=0,L=3Ld P"=5 .6* 10-' ps'/km, ,8,=2ps'/km, B=41b/s, To= 17fs
8 0,15
.~
;a"
>.
"
° '
10
0,05
O,O~O~~~~~~~,:"",;;-r;;-~::=~~ FO,8 -0,6 -0,4 -0,2 U,U 0,2 0,4 0,6 0,8 t[Ps] (b)
IFig.6.7 (a) Pulse shape at the end of the optical fiber (L = 3L d ) under the second and the ~ourth order dispersions for SIR=lO dB and different time shifts b > 0, (b) corresponding ~ye-diagram.
(6.41) ~here
the pulse propagation is governed by the NLSE that is given by (6.42)
~
= Ar(O,T)
1=
JAi(O, T) + 2A (O, T)A (O, T) 1
2
cos¢> + A~(O, T)
(6.43)
91
16.4 Super-Sech optical pulse
this case, the crosstalk and useful optical signal will propagate superimposed through an optical fiber. IThe propagation of short pulses under the influence of the out-of-band Icrosstalk being at a different wavelength than useful signals is governed by a ~et of two coupled NLSE: ~n
(6.44)
A2z
I
i(322 + -2-A2tt
. = 2')'1
(I A2 12 + 2 IAl 12) A2
(6.45)
Mrhere
DA2
(32. = _ _ _J J
27rC
IWhere j - 1,2 and
lHere ')' is the coefficient of nonlinearity and Aeff = 7rW 2 is the effective core larea. Aeff is typically 10 20 f..lm 2 in the visible region but can be in the range 150-80 f..lm 2 in the 1.55 f..lm region, so')' can vary over the range 2-30 W 1. km 1 Idependmg on n2. lEach of these two coupled NLSE in Eqs.(6.44) and (6.45) is a nonlinear par~ial differential equation which can be solved by using the split-step Fourier iffiethod (SSFM) [1]. The input or useful optical pulse and the out-of-band Icrosstalk SIgnal can be modeled, respectIvely, as
h (0, t) =
Al (0, t) cos (WIt)
(6.46)
~2(0, t) =
A 2(0, t) cos (W2t)
(6.47) (6.48) (6.49)
PI and P 2 are the peak powers of the useful optical pulse and the outpf-band crosstalk, respectively and Ts is the out-of-band crosstalk time shift. !Also, f represents the pulse shape. It could be Gaussian, super-Gaussian pr super-sech as the case may be. It is assumed that ITsl < n/2 since the ~nterest is confined to one bit period n. ~here
6 Optical Crosstalk
192
IThe pulse evolution picture contour plot for the case of the super-Gaussian !pulse is shown in Figure 6.S. The following factors are taken into considera~ion: T FWHM = 12.5ps, Al = 1550nm, bit rate R = 20 Gb/s, PI = 50mW, ISMF is the regime of a normal dispersion (D = 0.2 ps/nm-km) with a par am~ter Aeff = 50 f-lm 2 . The out-of-band crosstalk wavelength is A2 = 1551.5nm. IThe fiber length is 60 km and SIR = 0 dB.
~r 40
30
[l
o
150
200
IFig. 6.8 Pulse evolution picture and corresponding contour plot (super-Gaussian pulse).
IThe influence of the out-of-band crosstalk occurring at the input of the link can be expressed by estimating the eye opening penalty I(EOP). The EOP is a performance measure used very often when considering
~ransmission
IReferences
93
dynamic propagation effects, such as the dispersion, nonlinearities and pther influences that distort the pulse shape. The EOP is defined as a ratio pf the initial eye opening (EObefore) to the eye opening after transmission I(EOafter)' The initial eye opening is measured at the fiber input:
~he
EOP
=
20 log EObefore
(6.50)
EOafter
IThe out-of-band crosstalk occurring at the fiber input greatly affects the eye ppening. The change of the EOP with a crosstalk time-shift and the SIR ~s shown in Figure 6.9. For small values of time-shifts, The EOP has large ~alues which means that the crosstalk induces greater eye closing. When the put-of-band crosstalk optical power is equal to the signal optical power (SIR F 0), the nonlinear effects additionally distort the pulse shape and EOP [becomes greater. A thorough analysis of the crosstalk influence is very useful ~or improving the existence transmission links or designing the new ones. It ~s also very useful in designing the DWDM systems as this type of crosstalks ~s pretty common.
Ii
-SIR=OdB ····SIR=5 dB -----SIR=10 dB Pm=50mW
13 12
~11
o
~
10 9 8 7 6 5 4 3
,..,.,-\., :::~/ ,I
...
'---"=--
\
2
1
O+---.-~--~--~-,.-~--.---~-,--~
I-O.4T.
-0.2T.
O.OT.
0.2T.
O.4T.
'L IFig. 6.9 EOP vs Ts for super-Gaussian optical pulse and different SIR.
IReferences ~.
G. P. Agrawal. Nonlinear Fiber Optics. Academic Press, San Deigo, CA. USA. (1995).
k2. A. Ehrhardt, M. Eiselt, G. GoBkopf, L. Kliller, W. Pieper, R. Schnabel, G. H. We~.
Iher. "Semiconductor laser amplifier as optical switching gate". Journal of Lightwave ITechnology. Vol 11, No 8, 1287-1295. (1993). A. D. Fishman, G. D. Duff & A. J. Nagel. "Measurement and simulation of multipath interference for 1.7 Gb/s lightwave transmission systems using single- and multi-
6 Optical Crosstalkj
Wrcqll(,TlCY laser". JmLT·fw.l of T/ightwa'l!f: Tn:linology. Vol 8, No 0,894-90.5. ("1990).
tJ-. A-H. C;uan &', V-H. 'Vang. "Experiment.al st.udy of interband and intraband crosstalkl WD1\.1 net.works". Optoelectronics Letters. \/01·'1, No 1, ·'12-''1-'1. (2008).1 E. Iannone, IL Sabella, 1-1. AvaUaneo & G. De Paolilol. "Modeling of in-band cWIolIolt.alkl lin \VDT\{ optical networks". Journal of L1.qhJwQ.1w Technoloq1/. Vol 17, No 7. 11:~5-1141. ~n
~.
K1999). A. Panajotovic, D. Milovic & A. Mittie. "Boundary CiJ,se of pulse propagation mmlyticl folution in t.he presence of int.erference and higher order dispersion". TF.:L8TK82005 Conference Proceedings. 5--17-550. Nis-Serbia. (2005)J 17. Y. Point.urier, M. Brandt-Pearce & C. 1. Brm".'n. "Analyticallolt.mly of cwsioltalk pwpa !gat.ion in all-optical networks using perturbation theory". Journal oj T,ighJwa've Tech-I ~wlogy. Vol 2:~. No 12, 4074-408:~. (2005). [8. 1·1. Stefanovic, D. Draca. A. Panajot.ovic & D. l\.1ilovic. "Individual and joint. influence second and third order llilolperlolion on tranlolrnisiolion quality in the prelolence of coherent linterference". Oplik. Vol 120, No 1:~, 636-641.(2009). D. St.ojanovic. D. 1\1. l\.1ilovic & A. Dis\vas. ''Timing shift. of optical pulses due t.o interpmnnel cWIolIolt.alk". PmgTess in Electromagnetics ReseaTch M. Vol 1, 21-30. (2008)1 110 . .J. 1'vT. Senior. Oplic F1.hr:r Comm1J,n1.ca.rl.On.~. Prentice Hall, New York. NY. USA. (1992). ~.
rA
r.
IChapter 7
Rabitov-Turitsyn Equation
rT.l Introduction IThis chapter is devoted to the study of the DM-NLSE in polarization preserv~ng fibers, birefringent fibers as well as DWDM systems by the aid of multiple ~cale analysis. When this technique applied to the DM-NLSE it will convert ~he nonlinear partial differential equation to a nonlinear integro-differential ~quation with a nonlinear non-local kernel. This integro-differential equation ~s known as the Gabitov-Turitsyn equation (GTE) that first appeared in 1996 [16]. Later in 1998, this equation was refined in a simpler format by Ablowitz land Biondini [3]. Later, this equation was extended by Biswas to the cases pf birefringent fibers and DWDM systems in 2001 and 2003, respectively [11-13]. IThe GTE will be the universal asymptotic equation that governs the evo~ution of the amplitude of an optical pulse for a dispersion-managed system ~hat is governed by Eq.(2.41). In the GTE all fast and large variations are ~emoved. It is necessary to note that GTE is equally applicable to the case of lPulse dynamics for a zero or normal value of the average dispersion. The speIcial solution will be considered with Gaussian pulses in a separate subsection. IThe case of super-Gaussian and super-sech pulses are yet to be studied at this point. Finally, the result will be extended to the case of birefringent fibers land DWDM systems. The starting point is same equation as in Eq.(2.41) !which is
(7.1) IThis equation will be studied asymptotically now in the following section.
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
7 Gabitov-Thritsyn Equation
196
rr.2 Polarization-preserving fibers IThis section has been taken from the work of Ablowitz and Biondini that lappeared in 1998 [3]. Equation (7.1) contains both large and rapidly varying ~erms. To obtain the asymptotic behavior, the fast and slow Z scales are ~ntroduced respectively as and !Now, expand the field u in powers of ~((, Z, t)
= u(O)
((, Z,
t)
Za
Z=Z [6,30,31]:
+ ZaU(l)((, Z, t) + Z~U(2)((, Z, t) +...
(7.2)
~gain, decompose Eq. (7.1) into a series of equations corresponding to the Idifferent powers of Za. In general, at O(Z;;;-l) yields
f
[u(n)] =
-Pn [u(O),u(l), ... ,u(n-l)]
(7.3) (7.4)
Po
=
(7.5)
0
(7.6)
land so on. The operators Land Fare defined as
L[f] == .af + ~(() a2 f z a(
ISo,
o (l/za)
2
at2
gives (7.8)
kYhile 0(1) gives (7.9) IThe Fourier transform and its inverse are respectively defined as
17.2 Polarization-preserving fibers
97
(7.10) (7.11) ~n this chapter, the convolution theorem of the Fourier transform will be iused. The inner product f * g of the two functions f(x) and g(x) is defined las the integral
~f *g) (x) =
1:
(7.12)
f(x')g(x - x')dx'
land the convolution theorem states that
(7.13) land
(7.14) ITaking the Fourier transform of Eq.(7.8) gives
(7.15) Mihose solutIOn
IS
~Co) ((,Z,w)
= U(Z,w)p(C((),w)
and
C(()
= Co +
l'
(7.16)
6.((')d('
!With arbitrary co. The integration constant U( Z, w) represents the slowly ~arying amplitude of u CO ), while p (C((), w) contains the fast periodic oscil[ations due to large local values of the dispersion. The function U(Z, w) is larbitraryat this stage and its governing equation can be determined at higher prders. One needs to note that if U(Z,w) is real, then C(() represents the Ichirp of u(O) ((, z, w). Now, taking the inverse Fourier transform of Eq.(7.16) land using the convolution theorem of the Fourier transforms give
lu(O)((,Z,t) =
I:
p(x, t)
U(Z,t')p(C((),t-t')dt' 1
it 2
= - - . e2"X V21f2X
(7.17)
(7.18)
!Here, JX is taken on a principal branch with I arg( x) I < 1f. It needs to be !noted that p(O, w) = 1 so that p(O, t) = 8(t), the Dirac's delta function. There-
7 Gabitov-Thritsyn Equation
198
~ore, at (for which C(() = 0, gives u(O)((,Z,t) = U(Z,t) from Eq.(7.17). IAlso, p(x, t) and p(x, w) are even in t and w, respectively. Thus, the parity of ~(O) (., ., t) is determined by the parity of U(·, t); namely, if U(Z, t) is even with ~espect to t, then so is u(O)((, z, t) and vice versa. Now, taking the Fourier ~ransform of the 0(1) equation, namely, Eq.(7.9), gives
r I
iL[u(1)] == A
ait,(l) iw ---a;: + 2~(()u(1) = iF[u(O), v(O)] 2
A
(7.19)
Mrhose solution is
IU(l)((, Z,w)
= p(C((,w) [O(l)(Z,W)
~i ~here
1(
P*(C((,),W)F[U(O),v(O)](("Z,W)d(']
(7.20)
needs to be determined at the next order in the perturbation aVOId secular terms, also known as resonances, III the perturIbation expansion, the integrals must vanish over the dispersion map period IZa that corresponds to ( = 1. By Fredholm's Alternative (FA), [2,3, 11-13] lapplied to Eq.(7.20) implies 0(1)
~xpanslOn. '1'0
(7.21) IThis condition leads to the nonlinear evolution equation for the unknown ~unction U( Z, w). To obtain this equation in the Fourier domain, the equaion representing the FA, namely,Eq. 7.21, is first written down explicitly. Then F[u(O)]((, Z, w) is expressed as a Fourier transform of F[u(O)]((, Z, t). ~lso, substitute u CO ) from Eq.(7.17) to get the following integro-differential lrionlinear evolution equations in the Fourier domain. 2 f.au~ az - 2s: W U + Ua
A
11 00
00
-00
-00
U(Z, w + wdU(Z, w + W2) A
A
(7.22) !where the kernel rex) is given by Ir(x) =
~ (27r)
r g(()eiC(()Xdx Jo 1
(7.23)
IBy taking the inverse Fourier transform of Eq.(7.22), the following integroIdifferential nonlinear evolution equation in the temporal domain is obtained:
17.2 Polarization-preserving fibers
au + i8i2 b a u + g(z) iaz 2
fU*(Z, t + tl ~here
11 00
00
99
U(Z, t
+ tdU(Z, t + t2)
+ t2)R(tl' t2)dtldt2 = 0
(7.24)
the kernel is given by (7.25)
IEquation (7.24) is known as the GTE due to the dispersion-managed solitons ~n optical fibers for a polarization-preserving fiber.
[.2.1 Special solutions ~n this subsection, a traveling wave solution of the GTE that is given by IEq.(7.24), is being sought after. Now, the GTE equation is Galilean invariant,
~amely, if U(Z, t) is a solution to Eq.(7.24), then U(Z, t)e~(vt-~) is also so
~or any real v. Thus, it is sufficient to look for solutions of the form U(Z, t) = IJ(t)d>.2 Z for real and even J(t) and for nonnegative constant A called the ~igenvalue [3]. Then F(w), the Fourier transform of J(t), is also real and ~ven. Thus, by virtue of Eq.(7.24), the following nonlinear integral equation ~or F(w) is obtained:
(7.26) ~t can be argued using scaling analysis that if Fl(W) is a solution to Eq.(7.26) Icorresponding to the eigenvalue )\ - )\1 and the map strength 8 - 81, then IF2(w) = Fl (w/v) is also a solution to Eq.(7.26) corresponding to the eigen!value A2 = VAl and the map strength 82 = 8I/v 2. Alternatively, if FI(W) is a ~olution corresponding to the average dispersion ba = 1 and A = AI, then if Iba > 0, F(w) = V8:'FI(W) is a solution corresponding to the average disper~ion ba and A = JJ;;)l for any fixed value of the map strength 8. Also note ~hat 12(t) = vfIevt) and thus the scaling parameter a is also directly proIportional to the pulse energy and inversely proportional to the pulse width. ISo, 1112112 = V IIfIII2 as in the case of a classical NLSE [3]. IThe following figures shows the direct numerical simulation of (7.26). Figlure 7.1 is the shape of the stationary pulse in the Fourier domain for 8 = 1 and IA = 4, while Figure 7.2 represents the same pulse in the temporal domain.
ocoo
7 Gabitov-Thritsyn Equation
~ -4
S
~ -6
-8 -10
IFig. 7.1 Shape of stationary pulse in the Fourier domain for s = 1, A = 4.
-4 f=3
-2
2
3
IFig. 7.2 Shape of stationary pulse in the temporal domain for s = 1, A = 4.
rr.3 Birefringent fibers IThe material of this section is taken from the work of Biswas in 2001 [11]. IThe dimensionless form of the DM-VNLSE, that appears in the context of !birefringent fibers, is going to be studied in this section by asymptotic analy~is. The version of the DM-VNLSE that will be considered are Eqs.(4.3) and 1(4.4). They are (7.27) (7.28) [Equations (7.27) and (7.28) contains both large and rapidly varying terms. To pbtain the asymptotic behavior, the fast and slow z scales, as in the previous
101
17.3 Birefringent fibers ~ection,
~ollows
are introduced. Now, expand the fields u and v in powers of Za as [11]:
+ Z~U(2)((, Z, t) +...
(7.29)
H(' Z, t) = vCO ) ((, Z, t) + ZaV(l)((, Z, t) + Z;V(2)((, Z, t) +...
(7.30)
~((, Z, t)
~s
= u CO ) ((, Z, t) + ZaU(1)((,
Z, t)
before, decompose Eqs. (7.27) and (7.28) into a series of equations corre-
~ponding to the different powers of Za. In general, at
land
f
[v(n)]
= - Pn
O(z;:-l ),
[v(O) , u(O); v(1), u(1); ... ; v(n-l), u(n-l)]
(7.32)
!where
(7.33) (7.34) Po
f21
u(O), v(O); u(1), v(1)
=0
(7.35)
I
iu~) + !8au~i) + g(z)
1+ a {2Iv(O)12v(1) +
[2I u (O)1 2u(1)
+ (u(O))2(u C1 ))*
(v(O))2(v(1))*}]
(7.38)
land
Ip2 [v(O) , u(O); v(1), u(1)] iv~) + !8av~i) + g(z)
[2IvCO)12vCl)
+ (v CO ))2(v C1 ))*
1+ a {2Iu (O)1 2u(1) + (u(O))2(u(1))*}]
(7.39)
OC02
7 Gabitov-Thritsyn Equation
land so on. Again, the operators Land F are defined as
L[f] : : : .af + ~(() a2 f Z a(
IF[f, h] : : : i~~ +
2
at2
8; ~:; + g(z)(lfI
2
+ alhl 2 )f
ISo, 0 (1/ za), yields (7.40) land (7.41) ~hile at
0(1), (7.42)
land (7.43) ITaking the Fourier transforms of Eqs.(7.40) and (7.41), respectively, gives (7.44)
(7.45) IWhose respective solutions are
~(O) ((,Z,w)
= U(Z,w)fJ(C((),w)
(7.46)
~(O) ((,Z,w)
= V (Z,w)fJ(C((),w)
(7.47)
ith fJ x, wand C ( as before. Here, the integration constants U Z, wand Z, w represent the slowly varying amplitudes of u CO ) and v CO ), respectively. IThe functions U(Z,w) and V(Z,w) are arbitrary at this stage and the equa~ions governing them will be determined at higher orders. Just as in the case pf polarization-preserving fibers, if V(Z, w) is real, then C(() represents the Ichirp of vCO ) ((, Z, w). Now, taking the inverse Fourier transforms of Eqs.(7.44) land (7.45) and using the convolution theorem of Fourier transforms implies
V
luCO)((,Z,t) = land
fCO)(('Z,t)
=
I: I:
U(Z,t')p(C((),t-t')dt'
(7.48)
V(Z,t')p(C((),t-t')dt'
(7.49)
17.3 Birefringent fibers
103
iwhere p(x, t) is defined in Eq.(7.18). Here again, the parity of v(O)(.,., t) is Idetermined by the parity of V(·, t); namely, if V(Z, t) is even with respect to ~, then so is v(O)((, z, t). row, taking the Fourier transform of the O( 1) equations, namely, Eqs. (7.42) land (7.43), respectively, gives
r I
A
iL[u(1)]
8it,(l)
iw 2
8iP)
iw 2
== ----a( + 2~(()u(1) = iF[u(O), v(O)]
land
A
----a( + 2~(()i,(1) = iF[v(O) , u(O)]
-iL[v(l)] == A
I
A
(7.50)
(7.51 )
Iwhose solutions are, respectively,
lu(1)((, Z,w)
=
p(C((,w) [U(l)(Z,w)
i
(1)((, Z,w)
1
z, W)d(']
(7.52)
P*(C((,),W)F[v(O),u(O)](("Z,W)d(']
(7.53)
p*(C(('), w)F[u(O), v(O)](("
= p(C((,w) i
V(l)(Z,W)
1
!Where (;(1) and "(7(1) are to be determined at the next order in the perturba~ion expansion. Again, to avoid secular terms use the FA to Eqs.(7.52) and 1(7.53) that respectively yield (7.54) land
1
p*(C((), w)F[v(O), u(O)]((, Z, w)d(
=0
(7.55)
IThese conditions lead to the nonlinear evolution equation for the unknown [unctions U(Z,w) and V(Z,w). To obtain these equations in the Fourier doain, write explicitly the equations representing the FA, namely, Eqs.(7.54) and 7.55. Then express F u(O), v(O) (, Z, wand F v(O), u(O) (, Z, w as !Fourier transforms of F[u(O), v(O)]((, Z, t) and F[v(O) , u(O)]((, z, t), respecbvely. Also substitute u(O) and v(O) from Eqs.(7.46) and (7.47), respectively, ~o get the following coupled integro-differential nonlinear evolution equations ~n the Fourier domain [3]:
OC04
11
au - 2Oa W U + _ _ iaz 2
00
A
7 Gabitov-Thritsyn Equation
00
r(WIW2)U(Z, WI A
f [U(Z, W + WI)U*(Z, W + WI
+ W2)
+ W2)
1+ aV(Z,w + WI)V*(Z,w + WI + W2)] dwIdw2 =
0
(7.56)
1+ aU(Z, W+ WI)U*(Z, W + WI + W2)] dwIdw2 = 0
(7.57)
land
11
av - 2Oa W V + _ _ iaz 2
I·
00
A
00
[V(Z,w + WI)V*(Z,W
r(WIW2)V(Z, WI A
+ W2)
+ WI + W2)
IBy taking the inverse Fourier transform of Eqs.(7.56) and (7.57), we get ~he foIIowmg coupled mtegro-dIfferentIaI nonlmear evolutIOn equatIOns m the Itime domain:
au + f7fi2 0 a U + g(z) iaz
11 00
00
-00
-00
HU(Z, t + tl)U*(Z, t + tl
R(h, t2)U(Z, h
+ t2)
+ t2)
1+ aV(Z, t + tl)V*(Z, t + tl + t2)]dtIdt2 = 0
(7.58)
(7.59) IThe kernels in Eqs.(7.58) and (7.59) stay the same as in the case of polar~zation preserving fibers. Equations (7.58) and (7.59) are the GTE for DM ~olitons in birefringent fibers.
[7.4 DWDM system IThe material of this section is also taken from the work of Biswas that first lappeared in 2003 [12, 13]. The NLSE for DWDM system is studied in Chap~er 5. It is this equation that will be studied in this section by asymptotic lanalysis. Thus, to start off, the governing equation is
105
17.4 DWDM system
. ZU1,z
D(z) + -2-Ul,ft
+ g(z) {2" + N
L....- aim Iuml
lUll
2} = Ul
0
(7.60)
ml
1 :s; I :s; N. This represents multi-channel WDM transmission of copropagating wave envelopes in a nonlinear optical fiber, including the XPM ~ffect. Just to recall, aim are known as the XPM coefficients. Once again, IEq. (7.60) contains both large and rapidly varying terms. So, to obtain the lasymptotic behavior introduce the fast and slow z scales, as usual, and expand ~he fields Ul in powers of Za as [12,13] ~here
~ I ( (, Z, t ) = Uz(0) ( (, Z, t )
2 (2) ( (, Z, t ) + ... + zauz (1) ( (, Z, t ) + zauZ
(7.61)
ISubsequently, decompose Eq. (7.60) into a series of equations corresponding ~o the different powers of Za. In general, at O(Z;;-l),
II<' [u(n)
r"
u(n)] = -Po [u(O) u(O). u(l) u(1)· . u(n-1) u(n-1)] I'm n I'm' I'm'···' I ' m
f
[u(n)] I
= iu(n) + !~(()u(n) 1,( 2 l,ft Po
=0
(7.62)
(7.63) (7.64)
N
+
g(z)
almlu~)12u~)
L ~
IN
+
g(z)
Lalm
lu~)12(u?))* +ufO)(u~)(ug))* +ug)(u~))*)
Im#l
land so on The notation
land
>l2
(n)
u(n)_~ l,tt at2
(7.65)
7 Gabitov-Thritsyn Equation
OC06
Iwas used. Again, with the operators Land F defined as
IT[f] == .afl Z a(
I Jl [fl' ] h m =- Z.afl az
aat2 fl 2
+ 26a
+ ~(() a 2 fl at 2
2
1 12) fl + g ()(I z fl 12 + ~ L...J aim h m ml
lat 0 (1/ za) gives (7.67) Iwhile at 0(1) gives (7.68) ITaking the Fourier transform of (7.67) gives (7.69) Mihose solutIOn
IS
IU1CO) ((, Z, w)
=
01(Z, w) p (C((), w)
(7.70)
pnce again, the integration constant Ul (Z, w) represents the slowly varying lamplitude of ufO), while p (C((), w) contains the fast periodic oscillations due ~o large local values of the dispersion. The function Ul (Z, w) is arbitrary at ~his stage and the equation governing them needs to be determined at higher prders. Now, taking the inverse Fourier transform of Eq.(7. 70) and using the Iconvolution theorem of the Fourier transforms give
ujO)((,Z,t) =
[00 Ul(Z,t')p(C((),t-t')dt'
(7.71)
k\There p(x, t) is defined in Eq.(7.18). Now, taking the Fourier transform of ~he 0(1) equation, namely, Eq.(7.68), gives
a
A
(1)
"LA [u (1)] =- ~ a l
Z
. 2
+~ 2
(I")Ul (1)
A L.l. ."
A
=
"FA [(0) u , u m(0)]
Z
l
(7.72)
!whose solution is (1)
u l ((, Z,w) = p(C((,w) Ul A
i
(1)
(Z,w)
r( in p*(C(('),w)F[ujO),u~)]((',Z,w)d(' A
(7.73)
107
17.5 Properties of the kernel
iwhere Oz (1) is to be determined at the next order in the perturbation expan~ion. To kill secular terms, apply FA to Eq.(7.73), thus resulting in (7.74) IThis condition leads to the nonlinear evolution equation for the unknown ~unctions Uz(Z,w). To obtain this equation in the Fourier domain, write explicitly the equations representing the FA, namely, Eq.(7.74). Then express IP[ufO) , u~)l((' z, w) as a Fourier transform of F[ufO) , u~)l((' z, t). Also sub~titute ufO) from Eq.(7.71) to get the following coupled integro-differential !nonlinear evolution equations in the Fourier domain:
loUz 6a w2 I~ oZ - 2 Uz A
I·
[Oz(Z,w
L
Z#m
+
11 00
00
-00
-00
r(wlw2)UZ(Z,Wl +W2) A
+ wdOz*(Z,w + WI + W2)
(XZmUm(Z, W + Wl)Um*(Z, w + WI
+ W2)] dwl dw2 = 0
(7.75)
IBy taking the inverse Fourier transform of Eq.(7.75) the following coupled ~ntegro-differential nonlinear evolution equations in the time domain is ob~amed:
L
Z#m
(XZmUm(Z, t + tl)U:;'(Z, t + tl
+ t2)] dtl dt2 = 0
(7.76)
1 < I < N and the kernels stay the same as in the case of polarization preserving fibers. Equation (7.76) is the GTE for DWDM systems. ~here
rr.5
Properties of the kernel
IThe GTE equations for different types of fibers are the fundamental equations ~hat govern the evolution of optical pulses for a strong dispersion-managed ~oliton system corresponding to the frequency and time domain, respectively. [n the GTE, all the fast variations and large quantities are removed and ~herefore contain only slowly varying quantities of order one. These equations
7 Gabitov-Thritsyn Equation
OC08
lare not limited to the case 5a > 0, however, they are also applicable to the Icase of pulse dynamics with zero or normal values of average dispersion. Ifthe ifiber dispersion is constant, namely, if ~() = 0 and Co = 0, then C() = 0 land so rex) = 1/ (2n)2 and thus, R(iI, t2) = 5(iI)5(t2), a two-dimensional lDirac's delta function. The kernel rex) will now be studied in the following two cases.
rr. ,1;.1
Lossless Case
!For the lossless case, namely, when gee) = 1, note that, the kernel rex) for a ~wo-step map defined in Eq.(2.34) takes a very simple form, namely,
rex) = _1_sin(sx) (2n)2 sx
R(tl,t2)
=
1 ci(lI;2) ~ lsi
(7.77)
(7.78)
IWhere the cosine integral ci(x) is defined as
.()x /,00 cos(xY)dY
Cl
=
1
Y
ISince Ir(x)1 :::; 1/2n, the strength of the coupling between different frequencies ~hat measures the effective nonlinearity of the system is always less than that pf the ordinary vector NLSE.
'11.5.2 Lossy Case !For the lossy case, namely, when gee) -I=- 1, the kernel rex) depends on the !relative position of the amplifier with respect to the dispersion map. Consider ~he two-step map given by ~e() in Eq.e2.34) and define (a to represent the position of the amplifier within the dispersion map. So I(a I < 1/2 and (a = 0 !means that the amplifier is placed at the mid point of the anomalous fiber ~egment. The function gee) given in Eq.(2.42) can then be written as
(7.79) [or (a + n < ( < (a + n + 1, where G = rza /2. The kernel rex) in the lossy Icase is computed in a similar method as in the lossless case. If I(a I < () /2,
IReferences
109
!namely, the amplifier is located in the anomalous fiber segment, the resulting ~xpression for kernel is
L I'
Ge iCox (x) = (27f)2 (sx + 2iG())(sx - 2iG(1 - ())) 1
l [eG(4(a-20+1) sx sin(sx -
2iG(1- ())) sinh(2G)
r
1+ i()ei (4(a -20+1) ¥o' (sx -
2iG(1 - ()))]
(7.80)
Eq.(7.80), unlike the lossless case, the kernel r(x) is complex and is explicitly dependent on the parameters (), r, Za and (a in a nontrivial way. lHowever, one still obtains ~n
. hmr(x)
8---+0
=
1
--2
(7.81 )
1 sin(sx) (27f)2 sx
(7.82)
(27f)
land moreover, . () ~lImrx ---+0
=-----
means that as Za ---+ 0, Eq.(7.77) is recovered. For the particular case 1/2, (a = 0 and Co = 0 which corresponds to fiber segments of equal ~ength with amplifiers placed at the middle of the anomalous fiber segment, !the kernel modifies to ~hich
I()
=
r(x)
=
1 (27f)2
G X2S2
+ G2
sm(sx) sx--sinhG
+ isx
1 _ _co_s-:-(s_x-:,:-) coshG
(7.83)
IReferences ~. ~.
fJ. fi. ~.
~.
F. Abdullaev, S. Darmanyan & P. Khabibullaev. Optical Solitons. Springer, New York, INY. USA. (1993). M. J. Ablowitz & H. Segur. Solitons and the Inverse Scattering Transform. SIAM. IPhiladelphia, PA. USA. (1981). M. J. Ablowitz & G. Biondini. "Multiscale pulse dynamics in communication systems ~ith strong dispersion management". Optics Letters. Vol 23, No 21, 1668-1670. (1998). M. J. Ablowitz & T. Hirooka. "Nonlinear effects in quasi-linear dispersion-managed ~ystems". IEEE Photonics Technology Letters. Vol 13, No 10, 1082-1084. (2001). M. J. Ablowitz, G. Biondini & E. S. Olson. "Incomplete collisions of wavelength~ivision multiplexed dispersion-managed solitons". Journal of Optical Society of IAmerica B. Vol 18, No 3, 577-583. (2001). M. J. Ablowitz & T. Hirooka. "Nonlinear effects in quasilinear dispersion-managed pulse transmission". IEEE Journal of Photonics Technology Letters. Vol 26, 1846~848. (2001).
1110
7 Gabitov-Turitsvl1 Equationl
17. 1'1,'1. J. Ablowitz &: T. Hirooka. "TntrachaTlTlcl pulse interactioTls and timing shifts inl fj,ronglv dispersion-managed transmission svsj,ems". ()P/,],C8 rr:iJr:rs. Vol 2(i, No 2:~.
11M6-1X'lX. (2001). fg. 11. J. Ablowit.z & T. Hiroolm. "Int.rachanncl pubc intcnlz:t.ions in dispersion-managed ~,ransmission systems: energv transrer". Oplics Lellers. Vol 27. No 3, 20:~-205. (2002). r. 1'...1. ,I. Ablowitz & T. llirooka. "l\.lanaging nonlinearity in strongly dispersion-managed r)pticai pulse transmission i Jonrnal oj Optical Society oj America B. Vol 19, No :3, ~25-4:l9. (2002).1 110. ld. ,I. Ablowitz. G. 13iondini. A. 13i,,-was, A. Docherty, T. llirooka & S. (;hakraYarty. "Collii:lion-inlluccd timing shifti:l in dispcrsion-rnmmgcll soliton systems Optics Let~ I',r.,. Vol 27. :318-:320. (2002)1 111. A. Biswas. "Dispersion-managed vector solitons in opt.ical filWfS". Fiber Ilnd Tnlegrnl.rod Optics. Vol 20, No 5, 50:1-515. (200l)J ~2. A. Biswa,s. "Cabitov-Turitsyn equation tor solitons in multiple channels". Journal oj 1F:',drmnognel.ic lV",w.' and A ppZ,cah.on.,. Vol 17. No I I. 10:J9-1560. (20o:l)1 11:1. A. l3iswas. "Gabito\'- T"uritsvn equation for solitons in optical fibers". Journal af Nan-I Vinear Optical Physics and Materials. Vol 12, No 1, 17-37. (2003). 114. D. Breuer, F. Kuppers. A. I'vTat.t.heus, K G. Shapiro, L Gabitov & S. K. Turit.syn. dispersion compensation for standard monomode fiber based communiIolystenm ,,,lith large amplifier Iolpw:.:ing'· Optics Letters. \'01 22, No 13, 9tl2-9tl4.
"~Symmetrical
~ation
KJ(97). 115. D. 13reuer, K. Jurgensen, F. Kiippel'S, A. l\lattheus. 1. Gabito\' & S. K. Thritsvn. "Optimal schemelol tor dilolpcrsion cOHlpcnsatioTi of Ioltandard HlonoHlode fiber based ~inks·'. Oplics Communiraliolls. Vol 40, No l-:~, 15-18. (1997). 116. 1. H. Ciabitov &. S. K. Turitsyn. "Average pulse dynamics in a cascaded transmissiOl~ 8ystem ·wit.h pCl,ssive llilolpersion compensation Optics Letters. Vol 21, No 5, 327-329. K1996). p. I. R. Ciabitov &. S. K. Turitsyn. "Breathing solitons in optical fiber links" JETP !Letters. Vol 63, No 10, 861-866. (1996)1 ~8. T. R. Gabil.ov, E. G. Sllapiro &. S. K. Turil.sYII. "Asymptol.ic brtcathing pulstc in optical [ransm1sslOn svstems wIth (iISperSlOn compensatlOn Physical Revzew Ii. \.01 Eh), NO! 11. 3624-3633. (1997). ~ 9. T. It. Gabitov &. P. TvI. Lushnikov. "Nonlinearity management in a dispersion COlTIpen~ation s.vstem". Optics Lellers. Vol 27, No 2, 1l:~-1l5. (2002). 120. 1. H.. Gabitov, H.. indik, L. 11ollenauer. M. Shkarayev, M. Stepanoy & P. M. Lushnikov. "1 win families of bisolitons in dispersion-managed systems Optic.'! tetter.'!. Vol :~2, 1N0 G, tlOS-tl07. (2007). 121. A-l1. Guan & V-H. Wang. Expenmental stud, ot mtelband and mhaband clOsstalkl ~n vVDI\-1 nctworb". Optoelectmnics Letters. Vol 4. No 1, 42-44. (2008).1 t22. T. Hirooka & A. Hasegawa. "Chirped soliton int.eraction in st.rongl\! dispersion!managed wavelength-division-multiplexing systems". Optics Letters. Vol 2:1, No 10. 176~-770. (1998).1 r2:~. T. Hirooka & S. \Vabni/.;~. "Stabilihat.ion or dispersion-managed soliton transmissionl Iby nonlinear gain". £lectronics Letters. Vol :15, No ~, 655-657. (1999)J 124. T. Hirooka &. S. \iVabnitz. "Nonlinear gain cont.rol of disperlolion-nmnaged Iololit.on aIlllPlit.ude and collisions". Opl.1,ml F1,hr:r Tr:chnology. Vol 6. No 2, 109-121. (2000). 125. V. Eo za.kharov & S. \Vabnitz. Optical .'J'alitons: Theoretical Challenges and Indu.stria4 !perspecfh'es. Springer, Heidelberg. DE. (1999).1
IChapter 8
Quasi-linear Pulses
18.1
Introductjonl
OCn quasi-linear or low-powered Hystems, i:ltrong dispen,ion-management
(DlvJ),
Iwhich uses multiple section of fibers that alternates positive and ncgativd [group-velocity diHpersions, is llsed to compensate the fiber diHpersion, IIlc-lIlag9 ifiber nonlinearity and Huppress t,he int,er-C'hannel crosstalk. The differencel [between the DT\.T Holitol1 tramllnission and the quasi-linear puli:les is that inl Ithe [onner. nonlinearity balances dispersion. while in the latter. nonlincaritvl as managed. In the past few veaTS. there has been guiLe a fe\\' theoretical. ~mIIlcncaI and expenmental stuciIes that were conducted 1Il regards t,o thel hmlHi-linear pulse t.ramllnission. In this chapter, the rnat,hernatical theory of ~,he quasi-linear pub;e tramllnission through optical fibers will be i:>tudied inl klctails. The GTE is considered in all these three cases and the asvmptoLid !analysis is carried out. The significant reduction in the eITective nonlinearitvl ~or large IIlap strengt,h is quantified.1 IThe study of quai:>i-Iinear pulses ii:> split into three i:>ections corresponding) ~,o the polarization preserving fiberi:>~ birefringent fibers as well ai:> the [)\7\;,])1\/1 ~ystems. The mathematical theor\' of quasi-linear pulses was first studied byl p\blowilz cL a1. in 2001 [6, 11] for polarization·preserving fibers. Laler, ill 12004, this study ,vas extended to the caHe of birefringent fibers and D\VDJ'vI ~ystems by Biswas [IU]. The first section of this chapter is therefore takenl ~rom Ablowit7:'s works that first appeared in 2001, while the second and thel Ithird sections o[ this chapter are taken [rom Biswas's vmrk that appeared iI~ 12004 [19].
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
8 Quasi-linear Pulses
18.2 Polarization-preserving fibers IRecall the GTE in the Fourier domain, for pulses in polarization preserving !fibers is given by
fU*(z, W + WI ~here
+ w2)r(wlw2)dwldw2 = 0
(8.1)
the kernel rex) is given by (8.2)
Iwhile the GTE in temporal domain is given by
fU*(z, t + tl ~here the kernel
+ t2)R(tl' t2)dtldt2
=
0
(8.3)
R(tl' t2) is
(tl' t2; s)
=
100 100 ei(Wltl+W2t2)r(WlW2)dwldw2
(8.4)
Eq.(8.3), U(z, t) is the slowly varying amplitude of u(z, t) at the leading prder. ThIs equatIOn IS the umversal asymptotIc equatIOn that governs the ~volution of the amplitude of an optical pulse for a dispersion-managed sys~em that is given by Eq.(2.41). Here in Eq.(8.3), all fast and large variations lare removed. Recall here that Eq.(8.3) is equally applicable to the case of lPulse dynamics for a zero or normal value of average dispersions. [n this section, the details of the nonlinear terms of the GTE, for large s Iwill be analyzed. This analysis will also explain the difference between the Iquasi-linear transmission and soliton propagation. This study will be split into ~wo subsections that deal with the lossless and the lossy cases, respectively. ~n
18.2.1 Lossless system [n a lossless system, namely, g(z) 1(8.4) respectively modify to
r(x;s)
=
1, the kernels given by Eqs.(8.2) and
_l_sin(sx) (27r )2 sx
(8.5)
18.2 Polarization-preserving fibers
land
113
In
~here
1 cie 1t2 ) 21l'
lsi
=
rL(t1 , t2; S)
(8.6)
the cosine integral ci(x) is defined as
.()x Joo cos(xY)dY =
Cl
(8.7)
Y
1
rote that from Eq.(8.5), one can write lim r(x; s)
S=±QQ
=
(8.8)
0
showing that, for large map strength, one obtains the linear evolution ~quations. Assuming that U(z, w) depends weakly on s, the following asymp~otic expansions of the nonlinear terms from the GT equation are obtained: ~hus
.au
ba
a2 u
1
zaz + --a 2 + 2 t 21l's
I
~l(Z' t) =
[(logs - 'Y)J1(z, t) - J2(z, t)]
u (z, t + h) U (z, t + t2) + t2) dtldt2
(8.10)
100 100 log Itlt21 U(z, t + tl)U(Z, t + t2) l'U*(z, t + tl
1+
(8.9)
1: 1:
·U* (z, t + tl h(z, t) =
=0
+ t2)dtldt2
111
(8.11)
1
"2 + "3 + 4 + ... + :;; -logn = 0.57721...
(8.12)
[s the Euler's constant. In the Fourier domain, Eq.{8.9) transforms to 13, 81
.au A
Z- -
2
baW A
-2-U +
r
21 [
IT2(z,w)=;U(z,w) 1 A
A
A]
(logs - 'Y)J1(z,w) - h(z,w)
1
00
-00
=
0
(8.13)
(8.14) 1U(Z,W') 12 A
h(w'-w)dw'
(8.15)
land
(8.16)
P4
8 Quasi-linear Pulses
IThese asymptotic results can be obtained by the stationary phase method lapplied to Eq.(2.41). They can also be derived directly from Eq.(8.1) and lusing the asymptotic expansion of the cosine integral, namely, Ici(x)
-lnx + O(x)
cv - ' ) '
(8.17)
las x ---+ 0. Now, neglecting the 0(1/ S2) term, one can see that the spectral ~ntensity given by IU(z,w)1 2 is preserved during the pulse propagation. Now, ~rom (8.13), (8.18) !Also, the solution of (8.13) is w2 {Oa - i -2-z+i'ljJ [ IU(O,w)1
U(z,w)=U(z,O)exp A
A
A
I
2]}z
(8.19)
!where
'ljJ [IU(z,w)1 2 ]
=
_1_ (logs - ')')IU(z,w)1
2
27fS
11
-;
00 -00
IU(z,w)1 2 h(w' - w)dw A
'J
(8.20)
IThe linear phase shift exp (-iOaW2Z/2) can be corrected by the pre-transmission kJr the post-transmission compensation. After this, the linear phase is reIffioved. However, if the system has a smaIl value of the path averaged dispersion, namely, if « 1, the average dynamics of the quasi-linear pulse ~ransmission through an optical fiber is characterized only by the nonlinear !phase shift
oa
~Ndz,w)
=
'ljJ [IU(O,wW] z.
IThe nonlinear chirp, for small values of the map strength (s), can induce !pulse broadening. However, there is a simple way to compensate for the nonpnear chirp by expanding ¢Ndz,w) in a Taylor series with respect to w: 2
Ndz,w)
=
¢Ndz,O)
2
+ ;¢'IVL(z,O) + ;¢'l!r~(z,O) + ...
~ith IU(0,w)1 2 assumed to be even. !Also, note that large values of s reduce the effective nonlinearity. Therefore, la pulse in a polarization preserved fiber will be able to propagate for much ~onger distances before being distorted by nonlinearity, as opposed to a pulse, ~ith the same energy, in a system with constant dispersion. Thus, strong PM allows the propagation of a stationary pulse in a quasi-linear regime, ~ith energies comparable to that of classical solitons, but at the same time, !much lower than the energy required for the formation of a stable DM soliton
115
18.2 Polarization-preserving fibers
lat 5a
rv
~olitons.
0 for the same value of s owing to the energy enhancement of DM Figure 8.1 shows a quasi-linear Gaussian pulse in the lossless case. 2,0 1,8 1,6 1,4 ~ 1,2
~ ;s 10 ,
~
"" 0,8
I I I
p,6 p,4 p,2
b,o
1-10
,
-8
-6
-4
I
\
\
I I I I I I I I I
"
-2
\ \
\
\ \
\
\ \
\
0
\
\,
2
4
6
8
10
w IFig. 8.1 Shape of the quasi-linear Gaussian pulse after propagation of z = 20 and the linitial profile (dotted curve) for s = 100, (d) = 0, and G = 0 in frequency domain.
18.2.2 Lossy system Wor the lossy case, namely, when ge() -I- 1, the kernel rex; s) depends on the ~eIative position of the amplifier with respect to the dispersion map. Consider ~he two-step map given by ~e() in Eq.e2.34) and define (a to represent the !position of the amplifier within the dispersion map. So I(a I < 1/2 and (a = 0 Iffieans that the amplifier is placed at the mid point of the anomalous fiber ~egments. The function g(() given by Eq.(2.42) can then be written as
~(()
=
2Ge 2G e- 4G ((-n-(a) sinh(2G)
(8.21)
[or (a + n < ( < (a + n + 1, where G = rza /2. The kernel rex; s) in the lossy Icase is computed in a similar method as in the lossless case. If I(a I < () /2, !namely, the amplifier is located in the anomalous fiber segment, the resulting ~xpression for kernel is
8 Quasi-linear Pulses .
[eG(4(a-2&+1) sx sin(sx - 2iG(1- (}))
sinh(2G)
I
1+ i(}e i (4(a- 2&+1)¥e-(sx
- 2iG(1- (}))]
(8.22)
lNote that in Eq.(8.22), unlike the lossless case, the kernel r(x; s) is complex land is explicitly dependent on the parameters (), r, Za and (a in a nontrivial Iway. However, one can still write
. hm r(x; s)
8---+0
land moreover lim r(x;s) I
G---+O
=
=
1 (27f)
--2
(8.23)
_l_sin(sx) (27f)2 sx
(8.24)
!which means that as Za ----> 0, Eq.(7.82) is recovered. The effect of nonlinearity ~m quasi-linear transmission with g(() ¥- 1 is analyzed by the asymptotic ~xpansion of the nonlinear terms in the GT equations for large s. The study !will now be split into the following four cases, for () = 1/2, depending on the lPosition of the amplifiers.
18.2.2.1 Case-I:
Ca -
0
IThis locates the amplifier in the middle of the anomalous GVD segments. In ~his case, the kernels r(x; s) and R(tl' t2; s) are r(x; s)
=
1 G sin sx (27f)2 x 2s 2 + G2 sx sinhG
.
+ 2SX
~+hG sinhG
1 _ cos sx coshG
12
-
12 c coshG
---
(8.25)
(8.26)
(8.27) (8.28) (8.29)
18.2 Polarization-preserving fibers
IT2c(t l,t2;S)
r
=
11 00
00
-00
-00
117
ei (Wlh+ W2t 2) 2SWIW2COS (SWIW2) dw l dw2 G2 + (SWIW2)
(8.30)
IThe asymptotic expansion of the integrals in the kernel R(tl' t2; s) for large IS gives
(8.31) ~here
sgn(x) is the signum function defined as sgn(t) = -:1 Z7r
I
I
=
1
-4 --:---G 7r~
=-
00 Q()
G G -.-+1 ) 27r smG
[e G IG + (3')' + log G -
IG 2
1
P2
=
IG 1
IG 2 =
lNote that
iwt -dw e w
(e-
land
Po
1
)
e- G ] + -~ (2')' + log G)
G
=-
roo ~dx
l
G
x
e X -1
--dx o x
. hm Po(G)
(8.34)
(8.36) (8.37)
=-
I 27r
(8.38)
lim PI(G) = 'Y 27r
(8.39)
=a
(8.40)
G->O G->O
lim P 2 (G)
G->O ~o
(8.33)
(8.35)
4
JG
(8.32)
that these reduce to the lossless case as r approaches zero. Thus, the GT given by Eq.(8.1), for large s, reduces to
~quation,
.au Zaz
I
!Sa a2 u 1 + 27fi2 + :;-[(Po logs - PI)JI(Z, t)
I-POJ 2 (z, t) - iP2 J 3 (Z, t)]
~3(Z' t) =
I: I:
=a
(8.41)
sgn(ht2)U(Z, t + h)U(z, t + t2)
rU*(z, t + tl
+ t2)dtldt2
(8.42)
ps ~n
S Quasi-linear Pulses
the Fourier domain, (8.41) transforms to
au ~.-;:l A
uZ
2
6a W 1 - U + -[(Po logs - P1)J1(Z,w) 2 s A
A
(8.43) !where
(8.44) lNote that in Eq.(8.44), the integral represents the Cauchy's principal value. lNow, from Eq.(8.43) observe that
a
n-IUI uZ IH(z,w)
A
2
=
P2 s
-H(z,w)
= h(z,w)U*(z,w) + J;(z,w)U(z,w)
(8.45)
(8.46)
IFor large s, and moderate z, one can wnte (8.47) land, thus, the total spectral mtensity does not remam constant m thIS case.
18.2.2.2 Case-II: ~n
Ca
= -1/2
this case, the amplifier is positioned in the middle of the normal GVD The kernels of the GT equations, in this case, reduce to
~egment.
sinsx sx-sinhG
.
-2SX
(8.48)
land
(8.49) the parameters are as the same as before. The only difference here is imaginary part is negative. Thus, in the asymptotic state,
~here ~hat
(8.50) IThe intensity satisfies
119
18.2 Polarization-preserving fibers
~IUI2 = _ P2 H(z,w) !Finally, far s
»
8z
s
(8.51 )
1 and moderate z,
IU(z,w) 12 ~ 1U(O,w) 12 A
A
P2 z -s-H(O,w)
(8.52)
ISo, the total spectral intensity does not stay conserved here, too.
18.2.2.3 Case-III:
Ca
= -1/4
lHere, the amplifier is placed at the boundary between the anomalous and parmal GVD segment. In this case, the kernels are given by
r(x; s)
I
=
~ (27f)
X
2
S
G 2G G2 [{ ( . eh G -1)sxsinsx + G COSSX} + sm
~ i { (c:::G -
1) sx cos sx + G sin sx } ]
(8.53)
(8.54)
(8.55)
(8.56) !For large s,
(8.57)
G e- 2G Qo=--27f sinhG
(8.58) (8.59)
Here, also
.
1
hm Qo(G) = G->O 2n
(8.60)
8 Quasi-linear Pulses
~20
(8.61 ) ~o that, once again, as r approaches zero, it collapses to the lossless case. IThe GT equations, in this case, reduce to
.au 6a a u 1 z-a + --a 2 + - [(Qo logs z 2 t s 2
I
QdJ1(z, t) - Qoh(z, t)] = 0
(8.62)
Iwhile, in the Fourier domain, (8.63) ~n
this case, note that the spectral intensity stays constant as from Eq.(8.63): (8.64)
land also from Eq.(8.63), solution (8.19) is recovered where now
I~ [IU(z,w)n
=
~ [(QolOgS -
1- ~o 18.2.2.4 Case-IV:
Ca =
I:
Ql)
IJ1 (z,w)1 2
IJ2(Z,w')1 2h (w - w') dW']
(8.65)
1/4
iHere, the amplifier is placed at the boundary between the normal and anoma~ous GVD segment. The kernels reduce to
ti{ land
F(t
1'
sx sin sx + G cos sx C=:hGG -1 )sxcossx + G sin sx }]
t2; s)
=
~ [{(~ +
i{
C=:hGG -1
1 )hs
(8.66)
+ G1lC }
)12C + Ghs }]
(8.67)
IThe only difference in this case from that of the previous one is the imaginary !part of the kernel with an opposite sign. But, it was shown in the previous ~ubsection that the imaginary part does not make any contribution to the Idynamics of quasi-linear pulses, the sum of the spectral intensities is again
121
18.3 Birefringent fibers
Iconserved in this case during the pulse propagation. Figure 8.2 shows the IPlot of a quasi-linear Gaussian pulse in a lossy case, where G = 0.5 in the ~requency domain. 2,8 2,6 2,4 2,2
-(,=0
1,8 1,6 ~ ~ 1,4 1,2 0,8 0,6 0,4 0,2 0,0 f=3
C'
-2
o
-1
3
2
IFig. 8.2 Shape of the quasi-linear Gaussian pulse after propagation of z = 20 and the linitial profile (dotted curve) for s = 100, (d) = 0, and G = 0.5 in frequency domain.
18.3 Birefringent fibers IThe corresponding GTE, in the Fourier domain, for the DM-VNLSE are
l1~8z au - 26 W2 U + a
A
11 00
00
-00
-00
r(wIw2)U(Z, w + W2) A
1·1 U(z, w + WI)U*(Z, W+ WI
+ W2)
1+ aV(z,w+wdV*(Z,W+WI +w2)] dWIdw2 land
av
i aZ -
I
f
26a W2 v + A
11 00
00
-00
-00
0
(8.68)
+ w2)] dwIdw2 = 0
(8.69)
=
r(WIW2)V(Z, w + W2) A
[V(z,w +wdV*(z,w +WI +W2)
1+ aU(z,w +WI)U*(Z,w + WI
IThe GTE in the corresponding temporal domain are
~22
au + i8i2 b a u + g(z) iaz 2
11 00
00
8 Quasi-linear Pulses
R(tl' t2)U(Z, tl
+ tl)U*(Z, t + tl + t2) f+- aV(z, t + tl)V*(Z, t + tl + t2)] dtldt2
+ t2)
f [U(z, t
land
av + i8i2 b a 2v + g(z) iaz
=
11 00
00
_
(8.70)
0
_ R(tl' t2)V(Z, tl
[V(z, t + tdV*(z, t + tl + t2) 1+ aU(z, t + tl)U*(Z, t + tl + t2)] dtldt2
+ t2)
I·
=
(8.71)
0
~gain, the study will be split into two subsections that deal with the lossless land the lossy cases, just as in the previous section.
18.3.1 Lossless system !Assuming that U(z, w) and V(z, w) depend weakly on s, the following asymp~otic expansions of the nonlinear terms from the GTE equation is obtained:
. av az
2
1 + 2ba aatv2 + 27rS
[
(log S
i: i: i: i:
IJ~l)(Z' t) =
f~l)(z, t) =
i: i:
(2)
+ td U (z, t + t2)
+ t2) dtldt2
log Itlt21 U(z, t
fU*(z, t + tl
(2)]
'Y)J1 (z, t) - J 2 (z, t)
U (z, t
·U* (z, t + tl
~~l)(Z' t) =
-
(8.74)
+ tl)U(Z, t + t2)
+ t2)dhdt2
V(z, t + t2)U(Z, t + h)V*(z, t + tl
(8.75)
+ t2)dh dt2 (8.76)
123
18.3 Birefringent fibers
~l)(Z, t) = [00 [00
i2)(z, t)
100 100
=
J~2)(z, t) =
log t1t21V(z, t + t2)U(Z, t + t1) I
W*(z, t + t1 + t2)dt1dt2
(8.77)
V(z, t + h)V(z, t + t2)V*(Z, t + t1 + t2)dt1dt2 (8.78)
[00 [00 log Ih t21V(z, t + tdV(z, t + t2) (8.79)
i2)(Z, t)
[00 [00 V(z, t + t2)U(Z, t + h)U*(z, t + t1 + t2)dh dt2 (8.80)
=
f~2)(Z' t) =
[ : [ : log
Iht21V(z, t + t2)U(Z, t + td (8.81 )
~n
the Fourier domain, (8.72) and (8.73) respectively transform to
.au az -
6a w2
1[
'(1)
'(1)]
-2- U + ~ (logs -'Y)J1 (z,w) - J 2 (z,w) ,
1+2~S [(lOgS-'Y)Ki1)(Z,w)-K~1)(Z,W)] =0
av 6a w 2 , i az - -2- V
1[ +~
'(2)
(8.82)
'(2)]
(logs - 'Y)J1 (z,w) - J2 (z,w)
1+2~S [(lOgs-'Y)Ki2)(Z,w)-K~2)(Z,W)] =0
(8.83) (8.84)
IJ~l)(Z'W) = ~U(z,w) [ : IU(z,w )1 2 h (w' l
~i1)(z,w) = 1£.-(1) 1 ' r~2 (z, w) = ;U(z, w)
w) dw '
IV(z,w)1 2 U(z,w)
1-0000
1V(z, ' w)I 12 h (w I - w) dw I
IJi2)(z,w) = IV(z,w)1 2 V(z,w) 1' J'(2) 2 (z,w)=;V(z,w)
I
1-0000
1V(z,w) ' I 12 h(w I -w)dw I
(8.85) (8.86)
(8.87) (8.88) (8.89)
8 Quasi-linear Pulses
~24
(8.90)
'~2)(Z,W) = ~V(z,W)
[00 IU(Z,W')1 2 h (w' - w) dw'
(8.91 )
lOne ,can see fro~ Eqs.(8.82) and (8.83) that the total spectral intensity given iby IU(z,w)1 2 + lV(z,w)1 2 is preserved during the pulse propagation, namely,
(8.92) !Also, the solutions of Eqs.(8.82) and (8.83) are , , U(z,w)=U(z,O)exp
I
{6 , 2] z - i -2w2 -z+i'ljJ [ IU(O,w)1 a
(8.93)
, (z, w) = V(z, 0) exp { -
i~Z + i'ljJ [W(O, W)12] z
1+ ia'ljJ [IU(O,wW] ~espectlvely,
z}
(8.94)
where
'ljJ [IU(z,w)1 2] =
zk -
(logs - 'Y)IU(z,w)1 2
~ [ : IU(z,wWh(w'
- W)dW']
(8.95)
fWith a similar expression for 'ljJ [IV (z, w) 12].
18.3.2 lossy system Wor lossy case, the study will be further split into the following four cases, ~or () = 1/2, depending on the positions of the amplifiers.
18.3.2.1 Case-I:
Ca =
0
IThis locates the amplifier in the middle of the anomalous GVD segments. In ~his case, the kernels r(x; s) and R(tl' t2; s) are as in Eqs.(8.25) and (8.26),
125
18.3 Birefringent fibers
The GT equations, given by Eqs.(8.72) and (8.73), for large s, to [8]
~espectively.
~educes
1.8u
6a
82 U
1
(1)
F8z + 2 8t2 +:; [(Po log s - P1)J1 (z, t) ~ PoJ~I) (z, t) - iP2J~I) (z, t)]
+ ~ [(Po log s -
PdKi1) (z, t)
~POK~I)(Z, t) - iP2K~I)(Z, t)] = 0
(8.96)
land
(8.97)
J~I)(Z, t) = J~2)(Z, t) =
~1)(Z, t) = ~2)(Z, t) =
[00 [00 sgn(tlt2)U(Z, t + tl)U(Z, t + t2) fU*(z, t + tl
+ t2)dtldt2
(8.98)
W*(z, t + tl
+ t2)dtldt2
(8.99)
W*(z, t + tl
+ t2)dtldt2
(8.100)
fU*(z, t + tl
+ t2)dtldt2
(8.101)
[00 [00 sgn(tlt2)V(Z, t + h)V(z, t + t2) [00 [00 sgn(tlt2)V(Z, t + tl)U(Z, t + t2) [00 [00 sgn(tlt2)U(Z, t + tl)V(Z, t + t2)
[n the Fourier domain, Eqs.(8.96) and (8.97) respectively reduce to [8]
8U
i --
6a w 2 , 1 ' 1 ' 1 - - U + - [(Po log s - PI) Ji ) (z, w) - Po J~ ) (z, w) '(1)
a
'
(1)
-iP2J3 (z,w)]+-;[(Pologs-PdKl (z,w)
I
. ' (1) -POK' 2(1) (z,w) - zP 2K 3 (z,w)] = 0
I
land
(8.102)
8 Quasi-linear Pulses
~26
l-iP2j~2) (Z, W)] + ~[(PO log s - Pdki 2)(z, w) I-Pok~2)(z,w) - iP2k~2)(z,w)] Mrhere A
~ I ) (z, w)
=
11 00
_
00
=
0
1 --U(z, w + WI)U(Z, W + W2) WIW2 A
_
(8.103)
A
(8.104)
I·V*(z, W + WI
Ik~I)(z,w) = K3(2) (z, W) = A
11 00
00
-00
-00
(8.105)
_1- V (Z,W+WI)U(Z,W+W2)
WIW2
(8.106)
11 -00
+ W2)dWIdW2
1 --U(z, W + WI)V(Z, W + W2) WIW2 A
-00
A
(8.107) ~n Eqs.(8.104)-(8.107), the integrals represent the Cauchy's principal values. row, from Eqs.(8.102) and (8.103) observe that
I:z (IUI2+ W12)
=
~2
{H(I)(z,w)
+ aH(2)(z,w)}
(8.108)
(8.109) (8.110) Wor large s, and moderate z, one can write
IIU(z,w)1 2 + W(z,w)1 2
t
IU(O,wW
+ W(O,wW + ~~z {H(I)(O,W) + aH(I)(O,w)}
(8.111)
land, thus, the total spectral intensity does not remain constant in this case.
127
18.3 Birefringent fibers
18.3.2.2 Case-II: ~n
Ca
= -1/2
this case, the amplifier is positioned in the middle of the normal GVD Here, the kernels are respectively given by Eqs. (8.48) and (8.49). this case also,
~egment. ~n
(8.112) !Finally, for s
»
1 and moderate z leads to
lu(z,wW + !V(z,wW
~
IU(O,w)1 2 + !V(O,w)1 2
-
P;z {H(l)(O,w)
+ aH(l)(O,w)}
(8.113)
ISo, the total spectral intensity does not stay conserved here, too.
18.3.2.3 Case-III:
Ca
= -1/4
iHere, the amplifier is placed at the boundary between the anomalous and lriormal GVD segment and the kernels are the same as in Eqs.(8.53) and 1(8.54). In this case, the GTE given by Eqs.(8.96) and (8.97) reduces to
.au +28t2+-; 6a a u 1[ (1) (1)] (Qologs-QdJ1 (Z,t)-QOJ 2 (Z,t)
ZaZ
I+~ [(QoIOgS - QdKF)(z,t) - QoK~l)(Z,t)] =
°
(8.114)
.av +2 6a a v 1 [ (2) (2)] at2 +-; (QoIOgS-Q1)J1 (z,t)-QOJ2 (z,t)
zaz
I+~ [(Qologs - QdK~2)(z,t) - QoK~2)(z,t)] ~hile,
=
°
(8.115)
in the Fourier domain, ,
2
.au 6a W , 1[ '(1)] ---2-U+(Q o IOgS-Q1)J'(1) 1 (z,w)-Q OJ2 (z,w) ~~ [(QoIOgS - Q1)k~1)(Z,W) - QOk~l)(Z,W)] =
°
land ,
2
.---2-V+av 6a W , 1 [ (Q o IOgS-Q1)J'(2) '(2)] 1 (z,w)-Q OJ 2 (z,w)
(8.116)
8 Quasi-linear Pulses
~28
(8.117) ~n this case, that the total spectral intensity stays constant as from Eqs.(8.116) land (8.117):
(8.118) land also from Eqs.(8.116) and (8.117), solutions (8.93) and (8.94) respectively lare recovered where now 11f!
[IU(z,w)1 2] =
~ [(QoIOgS - Ql)lj~1)(Z,w)12
1- ~o
I:
Ij~l)(z,w'Wh (w -
w') dW']
1+ ~ [(QoIOgS - Ql)lk~1\Z,w)12 (8.119)
1
00 Qo _001J '(2) - --;2 (z,w')1 2 h(w-w')dw'
(8.120)
18.3.2.4 Case-IV:
Ca =
1/4
!Here, the amplifier is placed at the boundary between the normal and anoma~ous GVD segments. In this case, the kernels are given by Eqs.(8.66) and 1(8.67). The only difference in this case from that of the previous one is the amaginary part of the kernels with opposite signs. But, it was shown in the previous subsection that the imaginary part does not make any contribution ~o the dynamics of quasi-linear pulses, the sum of the spectral intensities is lagain conserved in this case during the pulse propagation.
129
18.4 MultIple channels
18.4 Multiple channels IRecall that for DWDM systems, the GTE is given by
.UZ,z
D(z) + -2-Uz,tt + g(z)
{2 2} Uz Iuzi + ~ aZm N
Iuml
L.J
mZ
=
0
(8.121)
Iwhere 1 ::; l ::; N and models for bit-parallel WDM soliton transmission. IAlso, aZm are known as the XPM coefficients. The corresponding GTE for ~he case of multiple channels in the Fourier domain is given by
auZ
!Sa W2 UZ+ ·7);-2 I·
A
11 00
00
r(WIW2)UZ(Z,WI+W2)
[UZ(z, W+ wdUz *(z, w + WI
L
Z#m
A
+ W2)
azmUm(z, W+ WI)Um*(z, W+ WI
+ W2)] dwIdw2 = 0
(8.122)
IWhile in the time domain, the GT equation is
+
L
Z#m
azmUm(z, t + tl)U:;'(Z, t + tl
+ t2)] dt Idt2 = 0
(8.123)
IWhere 1 ::; I ::; N. Once again, as before the study will be split into two ~ubsections namely the lossless and the lossy cases.
18.4.1 Lossless system ~n a lossless system, namely g(z) = 1, the kernels are given by Eqs. (8.5) land (8.6). The following asymptotic expansion of the nonlinear term from !the GTE is obtained:
. aUz
-
!Sa a 2Uz
+f- L
I
1
+ 22 + 2
7rS
m#Z
[(log S
-
(I)
(I)
'Y)JI (z, t) - J 2 (z, t)]
azm[(logs - 'Y)Kim)(z, t) -
K~m)(z, t)] = 0
(8.124)
8 Quasi-linear Pulses
~30
!where
Jil)(z,t) =
11 00
00
UI(Z,t+tl)UI(Z,t+t2)Ut(Z,t+tl +t2)dtldt2 (8.125)
11
J~I)(z, t) =
00
00
log Iht21 UI(Z, t
1: 1:
fUt(z, t + tl Kil)(z, t)
~ aim
=
fU~(z, t
L
IK~I)(z, t) =
1~n
aim
moll
+ h)UI(Z, t + t2)
+ t2)dt1dt2 UI(Z, t
(8.126)
+ t2)Um(Z, t + td
+ tl + t2)dtldt2
11 00
00
log Itlt2IUI(Z, t
(8.127)
+ t2)Um(Z, t + h)
-00-00
the Founer domam, aUI lSa w2 A i - - -UI 2
az
1 A(t) + -[(logs -')')J (z,w) 27rs 1
A(l)
J.
2
(z,w)] (8.129)
JA(I) 2 (Z,W)
=
1 A ;UI(Z,W)
1
-00
(8.130) A '2 IUI(z,w)1 h(w' - w)dw ,
A (I) A " A 2 Kl (Z,W) = UI(Z,W) L.J almIUm(z,w)1
I
(8.131) (8.132)
~
(8.133) [Now, one can see from Eq.(8.129) that the total spectral intensity given by 1L:~1 1UI (z, w) 12 is preserved during the pulse propagation. So,
(8.134) IThus, the solution of Eq.(8.129) is
UI(Z, w) = UI(Z, 0) exp
- i ¥ Z + i'lj; [IUI(O, W)12] Z
131
18.4 MultIple channels
(8.135) Mrhere 11f!
2~S [(lOgS -
[IUl(Z,WW] =
'Y)IUl(Z,WW
~ ~ [ : IUl(Z,w)12h(w' -
(8.136)
W)dW']
18.4.2 Lossy system IThe study in this case, will be similarly split into the following four cases Idepending on the position of the amplifiers.
18.4.2.1 Case-I:
Ca -
0
IThis locates the amplifier in the middle of the anomalous GVD segments. In ~his case, the kernels rex; s) and R(tl' t2; s) as in Eqs.(8.25) and (8.26). The IGT equations for large s reduce to
.aUl ba a Ul 1 Za; + 27fi2 + -; [(Po log s -iP2J~I)(z, t)] +
-L
(I)
O(I)
PdJ1 (z, t) - P J 2 (z, t)
D:lm[(PO logs - PdKim)(z, t)
s~
[00 [00 sgn(tlt2)Ul(Z, t + tl)Ul(Z, t + t2)
~1)(Z, t) =
fUt(z, t + tl + t2)dhdt2
(8.138)
(8.139) [n the Fourier domain, these equations are
aUl A
baW
2 A
i - - --Ul
1 + -[(Po logs -
A(l)
A(l)
P1)J1 (z,w) - P OJ 2 (z,w)
8 Quasi-linear Pulses
~32
l-iP2j~Z\z,w)l + ~ L
azm[(PO logs - pI)kim)(z,w)
rrLE/,
I-pok~m)(z,w) - iP2k~m)(z,w)l
=
0
(8.140)
Mrhere
(8.141)
(8.142) IThe integrals in Eqs.(8.141) and (8.142) represent the Cauchy's principal Ivalue. Now, from Eq.(8.140), (8.143)
(8.144)
!H(m)
(z,w)
=
(m)
K3 A
*
(z,w)Um(z,w) A
+ K3(m)* (z,w)Um(z,w) A
A
(8.145)
IFor large s, one can write for moderate z, rN
LIUz(z,wW rz=I
(8.146) land, thus, the total spectral intensity does not remain constant in this case.
18.4.2.2 Case-II:
Ca
= -1/2
[n this case, the amplifier is positioned in the middle of the normal GVD ~egment. The kernels of the GTE in this case are given by Eqs.(8.48) and 1(8.49). Then, the sum of the intensities satisfy
133
18.4 MultIple channels
(8.147) iFinally, for s
»
1 and moderate z, one obtains
IN
[ : IUl(z,w)1 2 k1 IN
[ : IUl(O,wW - P;z {H(l)(O,w)
+ [: a 1mH(1)(0,w)} m
1=1
(8.148)
I
ISo, the total spectral intensity does not stay conserved here, too.
18.4.2.3 Case-III:
Ca =
-1/4
!Here, the amplifier is placed at the boundary between the anomalous and pormal GVD segments. In this case, the kernels are given by Eqs.(8.53) and 1(8.54). Thus, the GTE, in this case, is reduced to .aUl z
Z-a
Da a Ul 1 +-a 2 + -[(Qo logs 2 t s
I+! [:
(I)
alm[(Qo logs - Q1)Ki m)(z, t) -
s~
~o
(I)
Q1)J1 (z, t) - QOJ2 (z, t)]
QoK~m)(z, t)] =
0
(8.149)
that, m the Founer domam,
I+! [:
alm[(Qologs - QI)kim)(z,w) -
s~
Qok~m)(z,w)] = 0
(8.150)
[n this case, the total spectral intensity from all the channels stays constant (8.151) ~o
that, from Eq.(8.150), it is possible to get
8 Quasi-linear Pulses
~34
I+~ [(Qologs - Ql)lk~1)(Z,w)12
~ ~o 18.4.2.4 Case-IV:
Ca
I:
Ik~1)(Z,w')12h(w - W')dW']
(8.152)
= 1/4
iHere, the amplifier is placed at the boundary between the normal and anoma~ous GVD segments. The kernels are Eqs.(8.66) and (8.67). The only differ~nce in this case from that of the previous one is that here the imaginary lPart of the kernels with opposite signs. But, again, it was shown in the pre~ious subsection that the imaginary part does not make any contribution to ~he dynamics of quasi-linear pulses, and the sum of the spectral intensities is lagain conserved in this case during the pulse propagation.
IReferences F. Abdullaev, S. Darmanyan & P. Khabibullaev. Optical Solitons. Springer-Verlag, INew York, NY. USA. (1993). ~. M. J. Ablowitz & H. Segur. Solitons and the Inverse Scattering Transform. SIAM. IPhiladelphia, PA. USA. (1981). fJ. M. J. Ablowitz & G. Biondini. "Multiscale pulse dynamics in communication systems ~ith strong dispersion management". Optics Letters. Vol 23, No 21, 1668-1670. (1998). fi. M. J. Ablowitz & T. Hirooka. "Resonant nonlinear interactions in strongly dispersion~anaged transmission systems". Optics Letters. Vol 25, No 24, 1750-1752. (2000). ~. M. J. Ablowitz & T. Hirooka. "Nonlinear effects in quasi-linear dispersion-managed ~ystems". IEEE Photonics Technology Letters. Vol 13, No 10, 1082-1084. (2001). ~. M. J. Ablowitz, T. Hirooka & G. Biondini. "Quasi-linear optical pulses in strongly ~ispersion-managed transmission system". Optics Letters. Vol 26, No 7, 459-461. (2001). 17. M. J. Ablowitz, G. Biondini & E. S. Olson. "Incomplete collisions of wavelength~ivision multiplexed dispersion-managed solitons". Journal of Optical Society of IAmerica B. Vol 18, No 3, 577-583. (2001). ~. M. J. Ablowitz & T. Hirooka. "Nonlinear effects in quasilinear dispersion-managed !pulse transmission". IEEE Journal of Photonics Technology Letters. Vol 26, 1846~848. (2001). ~. M. J. Ablowitz & T. Hirooka. "Intrachannel pulse interactions and timing shifts in ~trongly dispersion-managed transmission systems". Optics Letters. Vol 26, No 23, ~846-1848. (2001). ~O. M. J. Ablowitz & T. Hirooka. "Intrachannel pulse interactions in dispersion-managed ~ransmission systems: energy transfer". Optics Letters. Vol 27, No 3, 203-205. (2002). p. M. J. Ablowitz & T. Hirooka. "Managing nonlinearity in strongly dispersion-managed pptical pulse transmission". Journal of Optical Society of America B. Vol 19, No 3, f!25-439. (2002). ~2. M. J. Ablowitz, G. Biondini, A. Biswas, A. Docherty, T. Hirooka & S. Chakravarty. f'Collision-induced timing shifts in dispersion-managed soliton systems". Optics Let~ers. Vol 27, 318-320. (2002). ~.
IHdcrcnccs
~:~.
llit. ~5.
116. 117.
118. 119.
120. 121.
122. 12:1. 124. I2G.
126.
127. t28.
129.
135
C. D. Ahrens. TvI. J. AhloV\rit:>;. A. Docherty. V. Oleg, V. Sinkin, V. Gregorian & C. IH . .l\lenyuk. "Asymptotic analysis of collision-induced timing shifts in return-to-zerol buasi-linear systems with pre- and post-dispersion compensation". Optics Letters. Voll ~O, 20.56-205~. (200.5). N. N. Akhmediev &: A. Ankiewic~. 80lztons. Nonlinear Pu.l.~es and Beams. Chapmanl land llall, London. UK. (1997).1 A. Bis. .vas. "Dispersion-managed solitons in optical couplers i Journal oj Nonlinear' Opliral Physics and Maleria1s. Vol 12, No L 4G-74. (200:3). A. 13is\vas. "Ciabitov-Turitsyn equation for solitons in multiple channels" . .ioll..Tnal ot I,Electmmagnetic Waves and l1pplications. Vol 17, No 11, 15:39-1560. (200:3)1 A. Riswas. "Gabi1,ov- Turil.syn {'quaUon for solitons in optical fibers". JOILrnal oj Non-I ~inmr Opliral Physics and Maleria1s. Vol 12, No 1, 17-:37. (200:3). A. 13iswas. "Dispersion-l\.lanaged solitons in multiple channels" .ioll..Tnal ot Nonlinear Optical Physics and Materials. Vol 1:3, No 1, 81-102. (2004). A. Riswas. "Them}' of quasi-linear pulses in opl.ical fibers". Opliral Fiber Technology. IVollO, No :1, 2:12-259. (200/1). I. R. Gabit.ov & S. K. Turit.syn. "1\verage pulse dynalllicl:l in a cascaded tranl:lrnisl:lionl f},stem with passive dispersion compensation". Opl1,r.~ rpiJprs. Vol 21, No G. :327-:329. K1996). I. R. Gabit.ov & S. K. Turitl:lyn. "Breathing I:lolitom in optical fiber linkl:l". JETP IrpiJprs. Vol G:3. No 10,861-866. (199G).1 1..1:(. Gabitov, E. Ci. Shapiro & S. K. Thritsvn. "Asymptotic breathing pulse in optical ~ranl:lrnisl:lion I:lYI:lt<.'llll:l with disperl:lion cOlllpemation". Physical Rcuicw E. Vol 5.5, N(~ ~, 3624-:363:3. (1997). 1..1:(. C;abitov & P. 1\,1. Lushnikov. "Nonlinearity management in a dispersion compen~ation I:lYl:lklll". Optics Letters. Vol 27, No 2, 113-115. (2002). T. R. Gabil.ov, R. Indik, L. ]'vlollenauer. ]'v1. Shkarayev, ]'v1. Stepanov &. P. ]'v1. Lushnikov. "Twin families of bisolit.ons in dispersion-managed svst.ems" Optics Letters. Vol :32. ~I\jo 6, 605-607. (2007. A-H. Gual! &. V-H. \Vang. "Experiment.al st. uri)' uf il!terbaml ami inj,rabaml crosstalkl ~n 'VDr. . ! networks". OpLoelecLmmcs Lellers. Vol ,t, No 1, ,t2-'H. (2008).1 T. llirooka & A. llasegawa. "Chirped I:lolit.on interaction in strongly disperl:lion~llanaged "lavciength division multiplexing systems Optics fJetters. Vol 23, No 10, 17G8-770. (1998).1 T. llirooka & S. Wabnitz. ''St.a.biliza.tion of dispersion-managed soliton transmissionl Iby nonlinear gain". F:leci'rnnics JA,tters. Vol :~5, No 1), 6.5·5-6.57. ("1999)1 T. Hirooka K, S. \Vabnit~. "Nonlinear gain control of dispersion-managed soliton amplit.ude and collisions". Optical Fiber Technology. Vol 6, No 2, 109-121. (2000). V. E. Zakharov &. S. \iVabnitz. Optical Solitons: Theoretical Challenges and Iruln.~tT'ia~ Ippr8pedi've.~. Springer, Heidelberg. DE. (1999).1
IChapter 9
Higher Order Gabitov-Turitsynl !Equations
!U Introductjonl rrhis chapter deab: with the Gabitov-Turib:,:yn at the next higher order. This ~ncans thaL the multiple-scale expansion that was seen in Chapter 7, \",ill nowl ~H-) carried onto t,o O(z~). This gives a better accura(,y and to the approxirn<-1~.ion that is deHcribed by the GTE Heen in Chapter 7. In this te(,hnique, thel lPuli:le in the Fourier domain will be decomposed into a slowly evolving ampliItude and a rapid phase that describe the chirp of the pulse. The fast phasel as calculated explicit! v that is driven by the large variations of the dispcrsiod ~lbollt the average. rhe amplItude GvolutIOIl ",vIII be descrIbed by t,hc nonlo~:al integro-different.ial evolution eqllatioml that is kIHHvn as the higher orderl pabitov-Cj\lritsyn equation (HO-GTElJ IThe HO-GTE was 111'8(. published by the remarkable work 01 AblowiLL; cL al. ~n 2002 [9]. This work was [or the case of polarization preserving fibers. Subschnent.ly, in 2007, Dis"\vas extended this work to t,he C<-1HeS of birefringent fibers ~nd DWDIvl systems [14-161. In this chapter, the section on polarizationpreserving fiberi, is taken from Albowitz et a1.'s work while the sections onl birefringent fibers and D\VD1\I svstems arc Laken [rom Bisvms's vmrks. H beeds to be noted that although the HO-GTE is useful for studying thel Ntnu:tllre and properties, it. is, h(l\vever, inconvenient. for numerical cornpnt.a~,iol1s becaui:le of the presence of the tour-told integrali:l that are given in thel !GIza) terms of the HO-GTE, as will be seen.
~,2
Polarization preserving fibersl
ITo obtain the al:)vrnptotic behavior for ~ast and slow
11
polarization prmcrving fiber. the!
z sea les are jntroduced asl
A. Biswas et al., Mathematical Theory of Dispersion-Managed Optical Solitons © Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg 2010
9 Higher Order Gabitov-Thritsyn Equations
z=Z
and IThe field q is expanded in powers of
as
Za
jEquating coefficients of like powers of
(9.1)
Za
gives (9.3)
(9.4)
(9.5)
I:
lNow, the Fourier transform and its inverse are respectively defined as
j(w)
If(t) = ~
=
~
1
00
27f_ 00
f(t)eiwtdt
(9.6)
j(w)e-iwtdw
(9.7)
IAt 0(1/ za), Eq. (9.3), in the Fourier domain, is given by
(9.8) whose solution is
~(D) ((,Z,w)
Q(D)
= Q(D)
(Z, w)
land
C(()
=
(Z,w)e-i~2C(')
= q(D)
(0, Z, w)
l' ~ ((')
d('
(9.9) (9.10)
(9.11)
~t 0(1), Eq. (9.4) is solved in the Fourier domain by substituting the solution given by Eq.(9.9) into Eq.(9.4). This gives
139
19.2 Polarization preserving fibers >l '(1)
._uq_
B(
2
_ ~~(()q(1)
2
=
.
2
-e-'~ C(()
(9.12) !Equation (9.12) is an inhomogeneous equations for q(l) and with the homogepeous parts having the same structure as in Eq. (9.3). For the non-secularity Icondition of q(l), it is necessary that the forcing terms are orthogonal to the ladjoint solution of Eq.(9.3), a condition that is comII}only known as FredIholm's Alternative (FA). This gives the condition for Q(O)(Z,w) as (9.13) Iwhich can be simplified to
I·Q(O) (Z, w + WI) Q(O) (Z, w + WI + W2) dW1dw2 = 0 ~here,
(9.14)
the kernel ro (x) is given by (9.15)
!Equation (9.14) is the GTE for the propagation of solitons through polariza~ion preserving fibers as seen in Chapter 7. !Equation (9.4) will now be solved to obtain q(l)((, Z, t). Substituting Q(O) ~nto the right-hand side of equation (9.12) and using Eqs.(9.9) and (9.13) give
k
[iq(1)e¥C(()] =
1[00
_ g(()e
g(()e¥C(() Iq(O) 12q(0)e iwt dtd(
i~2 C(() [ : Iq(O) 12q(0)e iwt dt
(9.16)
Iwhich integrates to
(9.17)
9 Higher Order Gabitov-Thritsyn Equations
!where
10(1) (Z, w)
=
iq(1) (0, Z, w)e i,:;2 C(O)
(9.18)
!Also, O(I)(Z,w) is so chosen that (9.19) ~hich is going to be an useful relation for subsequent orders. Applying IEq.(9.19) to Eq.(9.17) gives
10(1)(Z, w)
=
11(,[: I-~ 11[: 1
g((')e i,:;2 C((") Iq(O) 12q(0)eiwt dtd(' d( g(()e¥c((,) Iq(O) 12q(0)e iwt dtd(
(9.20)
1N0w, Eq.(9.17) by virtue of Eq.(9.20) can be written as A(I)((, Z,w)
= ie¥C((')
[1 [:
_iorior1
00
l
g((,)e¥c(n Iq(O) 12q(0)e iwt dtd('
g((,)ei,:;2c(nlq(0)12q(0)eiwtdtd(,d(
-(( -~) 1[: -00
g(()e i,:;2 C((,) Iq(O) 12q(0)e iwt dtd(] (9.21 )
!Which can also be rewritten as A(I) ((, Z, w) = ie-¥C((') [00[00 0(0)* (w
+ S?I + S?2)
I·Q(O) (w + S?d Q(O) (w + S?2)
r(, g((')eifM12 C((,,)d(' - inr inr(, g((')eifM12C((, ,)d('d(
. in
1- (( -~)
l
11
g(()eW1 f]2 C((,)d( } dS?IS?2]
(9.22)
IThus, at O(Za), (9.23) IMoving on to the next order at O(z;), one can note that the GTE given by IEq. (9.14) is allowed to have an additional term of O(za) as
19.2 Polarization preserving fibers
-----az - w oQ(O)
2
26'aQ(0)
100 100
+__
141 TO
(WIW2)
Q(O)
(Z,
WI
+ W2)
fQ(O) (Z, W + wd Q(O) (Z, W + WI + W2) dWldw2 1= Zan(Z,W) + O(Z;)
(9.24)
IThe higher order correction n can be obtained from the suitable non-secular Iconditions at O(z;) in Eq.(9.14). Now, Eq. (9.5), in the Fourier domain, is 1
0 [iq(2)e i~2 C()]
~
+ n + e i~2 C()
+ e-'-'?C(()g(() [00
[2Iq(0)12q(1)
(i Oq(1)
M
_
w 2 6' q(l))
2
a
+ (q(0))2 q (1)*] eiwtdt = 0
(9.25)
IBut, again, Eq.(9.19) gives (9.26) IApplying the non-secularity condition (9.26) to Eq.(9.25) gives
iUsing Eqs.(9.9) and (9.21),Eq.(9.27) can be rewritten as
In = [ : [ : [ : [ : TI (WIW2, ill il2 ) f [2Q(0) (w
+ wd Q(O)* (w + WI + W2) Q(O) (w + W2 + ill)
fQO) (w + W2 + il2) Q(O)* (w + W2 + ill + il2)
F Q(O) (w + WI) Q(O) (w + W2) Q(O)* (w + WI + W2 + ill) fQ(O)* (w + WI + W2 - il2) tQ(O)* (w + WI + W2 + ill -
il2) ] dWldw2dilldil2
(9.28)
!where, the kernel Tl(X,y) is given by
(9.29)
9 Higher Order Gabitov-Thritsyn Equations
IEquation (9.24) represents the HO-GTE for the propagation of solitons ~hrough polarization preserving optical fibers.
~.3
Birefringent fibers
IThe equations that describe the pulse propagation in birefringent fibers are pf the following dimensionless form:
~Uz + D~Z) Utt + g(z)(luI 2 + 0:1v1 2 )u = 0
(9.30)
+ D~Z) Vtt + g(z)(lvI 2 + 0:1u1 2 )v =
(9.31)
livz
0
~here 0: represents the XPM coefficients. The fields u and v are expanded in !powers of Za as
+ Z~U(2)((, Z, t) +...
(9.32)
H(' z, t) = v(a) ((, Z, t) + ZaV(l)((, Z, t) + Z~V(2)((, Z, t) +...
(9.33)
~((, Z, t)
= u(a) ((, Z, t) + zau(l)((,
jEquating coefficients of like powers of
Z, t)
Za
gives
P(:a) :
(9.34)
p(L) :
(9.35)
(9.36)
(9.37)
19.3 Birefringent fibers
I. [2I u (0) 12 u(1)
In
~t
143
+ (U(O) )2u(1)* + a{2Iv(0)
.aV(2)
~(() a 2V(2)
i(Za):
z-----a( + -2-7fi2 +
I. [2I v (0) 12 v(1)
+ (V(O) )2v(1)* + a{2Iu(0)
2 v(1)
1
+ (V(O) )2 v (1)*}] } 8a a 2v(1)
{ . aV(l) Z az 12 u(1)
(9.38)
+ 27fi2 + g(z)
+ (U(O) )2 u (1)*}] }
(9.39)
0(1/ za), Eqs.(9.34) and (9.35), respectively, in the Fourier domain are
given by !:j'(0)
land
(9.40)
a'(O) 2 i_v_ _ ~~(()v(O) = 0
(9.41)
a(
I
~hose
_ ~~(()u(O)
2
a(
2
respective solutions are
Z, w)
=
00 (Z, w) e- i';( C(()
(9.42)
~(O) ((,Z,w)
=
Vo (Z,w)e-i~2c(e)
(9.43)
~(O)
~t
2
=0
i_uu_
1
((,
~o (Z, w)
= uCO) (0, Z, w)
(9.44)
Vo (Z, w)
=
v CO ) (0, Z, w)
(9.45)
0(1), Eqs. (9.36) and (9.37) are solved in the Fourier domain by substi-
the respective solutions given by Eqs.(9.42) and (9.43) into Eqs.(9.36) land (9.37). This gives ~uting
'(1)
.au a(
2 2 ,(1) - w -~(()u = -e _ iw2 C(e)
2
'
aUo az
-
-
w2 ' -8 Uo -g(() 2 a
land . av'(1) a(
Z-- -
I
w2 , 1 _ -~(()v( ) = -e 2
iw 2 2
C
(e)
aVO ( ' --
az
w ' 2 a
2 ) -8 Vo - g(()
9 Higher Order Gabitov-Thritsyn Equations
!Equations (9.46) and (9.47) are inhomogeneous equations for u(1) and v(1), ~espectively, with the homogeneous parts having the same structures as in !Eqs.(9.34) and (9.35), respectively. FoAr the non-sec~larity conditions of u(1) land vel), FA gives the conditions for Uo(Z,w) and Vo(Z,w) respectively as
r1 h-oo
2 auo - W -OaUO +
1
A
A
-
I
az
2
00
iw2C W g(()
e"""2
(9.48) land
2
avo _ ~OaVO + A
I
r1 e <;;2C(()g(() Jo-oo 1
az 2 f (Iv(0)1 2 + alu(0)1 2 )
00
i
v(O)eiwtdtd( = 0
(9.49)
jEquations (9.48) and (9.49) can be respectively simplified to
auO w2 0aUO + az - 2: A
11 00
00
-00
-00
To (WlW2) Uo (Z, WI A
+ W2)
[00 (Z,W + WI) 0 0 (Z,W + WI +W2)
I·
avo w2 0a az - 2: Vo + A
11 00
00
-00
-00
TO (WlW2) Vo (Z, WI A
+ W2)
[Va (Z,w + WI) Va (Z,w + WI + W2)
I·
jEquations (9.50) and (9.51) are the GTE for the propagation of solitons hrough a birefringent fiber as seen in Chapter 7. Equations (9.46) and (9.47) ill now be solved to obtain U(l) (, Z, t and vel) (, Z, t . Substituting 00 and 1V0 into the right-hand side of Eqs. (9.46) and (9.47), respectively, and using !pairs (9.42)-(9.43) and (9.48)-(9.49) give :
[iu(1)e~C(()] =
11 g(()e~CW (lu(0)1 +alv(0)1 1
00
1- g(()e <;;2C(() i
i:
2
2)
(lu(0)1 2 +alv(0)1 2 )
u(O)eiwtdtd( u(O)eiwtdt
(9.52)
145
19.3 Birefringent fibers
land
1:( [iv(1)ei~2C(()] 1=
11[: g(()ei~2C(() (Iv(O)1 2+alu(O)1 2) [00 (Iv(O)12 + alu(O)12)
_ g(()eif-C(() ~ntegration
v(O)eiwtdtd(
v(O)eiwtdt
(9.53)
of Eqs. (9.52) and (9.53) yields
1= (h(Z, w) + (
1[:
g(()eif-C(()
g((')e i~2 C(n
V1 (Z,w) + (
-1 [:
11[: 1[:
(lu(O)1 2+ alv(O) 12) u(O)eiwtdtd('
g(()eif-C(()
g((')e i~2 C(n
(lu(O)1 2+ alv(O) 12) u(O)eiwtdtd( (9.54)
(Iv(O)1 2+ alu(O)1 2) v(O)eiwtdtd(
(Iv(O) 12 + alu(O) 12) v(O)eiwtdtd('
(9.55)
(9.56) (9.57) !Also, U1 (Z,w) and V1 (Z,w) are so chosen that
r ifP)e i~2 C(()d( = 0 1
io
(9.58)
land
(9.59) !which are going to be useful relations at subsequent orders. Applying Eqs.(9.58) land (9.59) to Eqs.(9.54) and (9.55), respectively, yields
9 Higher Order Gabitov-Thritsyn Equations
land
I-~
11[:
g(()e i<;;2C(()
(Iv(O)1 2 +alu(O)1 2) v(O)eiwtdtd(
(9.61)
row, Eqs. (9.54) and (9.55), by virtue of Eqs.(9.60) and (9.61), can be re~pectively written as ~(l)((,Z,W) =
ie"?C(()
[1 [:
r r1 Jo Jo 1
00
g((')e"?C((')
g((')e i<;;2C((')
(( -~) 1[:
(lu(O) 12 + alv(O) 12) u(O)eiwtdtd('
(lu(O)1 2 +alv(O)1 2) u(O)eiwtdtd('d(
-00
g(()e i<;;2 C(()
(lu(O) 12 + alv(O) 12) u(O)eiwtdtd(] (9.62)
~(1)((,Z,w)
_(_ ~ 1[00 1
g(()e"?C(()
(Iv(O)1 2 + alu(O)1 2) v(O)eiwtdtd( (9.63)
[Equations (9.62) and (9.63) can now be respectively rewritten as '11,(1)((, Z,w)
= ie-"?C(()
[ : [ : U~ (w
+ .01 + .02 ) {Uo (w + .od
I·Uo (w + .02 ) + aVo (w + .01 ) Vo (w + .o 2 )}
19.3 Birefringent fibers
I· {
1(
g( (')eifM12C((') de,' -
1- (( - ~) land
Iv(1) ((, Z, w) =
ie- i<;;2 C(()
11
11( 1
147
g( (')eifM12C((') de,' d(
g(()eifM12C(()d( }df.hSl2]
[l:l:
Vo* (w + Sll
(9.64)
+ Sl2) {Vo (w + Sll)
f"Vo (w + Sl2) + a/fo (w + Sll) Uo (w + Sl2)}
f{1( g((')eifM12C((')d(' -1 1( g((')eifllfl2C((')d(' d( 1
~
(( -~) 11
g(()eifllfl2C(()d( }dSl1Sl2]
(9.65)
IThus, at O(za), (9.66) (9.67) IMoving on to the next order at O(z~), one can note that the GTE given by IEqs. (9.50) and (9.51) are allowed to have an additional term of O(za) such las
auo - 2/ja w Uo + 1 1 az 2
00
00
-00
-00
A
TO
HUo (Z, w + wI) Uo (Z, w + WI
-
I~~ ~/ja Vo +
I: I: TO
(W1 W2) Uo (Z, WI A
+ W2)
+ W2) + aVo (Z, w + WI)
(W1 W2) Vo (Z, WI
+ W2)
1·[Vo (Z, w + WI) Vo (Z, W + WI + W2) + aUO (Z, W + WI) 2 I,U (Z, W + WI + W2)]dw1dw2 = Zan2(Z, W) + O(Za) O A
A
(9.69)
IThe higher order corrections iLl and iL2 can be obtained from suitable non~ecular conditions at O(z~) in Eq. (9.38) and Eq.(9.39), respectively. Now, [Eqs. (9.38) and (9.39), in the Fourier domain, respectively, are
+ e~C(()g(()
100
9 Higher Order Gabitov-Thritsyn Equations
[2I u (0)1 2 u(l)
+ (U(0))2 U (I)*
1+ a { 2Iv(0) 12v(1) + (v(O) )2v(1)* }] eiwtdt = land
0
(9.70)
[ifP)ei<;;2C(()] +n2+e i<;;2C(() (i 8fP ) _ I~ 8( 8Z
+ e~C(()g(()
[00 [2Iv(0)12v(1) +
w2 8
2
a
i,(l))
(v(O) )2v(1)*
1+ a {2Iu(0) 12u(1) + (U(0))2 u (1)* }] eiwtdt = 0
(9.71)
IBut, again, Eqs.(9.58) and (9.59) give
(9.72)
(9.73) ~pplying the non-secularity conditions (9.72) and (9.73) to Eq.(9.70) and !Eq.(9.71), respectively, gives
nl =
-1 [: e~C(()g(()
F
a { 2Iv(0) 12v(1)
n2 =
+ (U(0))2 U (I)*
+ (v(O) )2v(1)* }] eiwtdtd(
-1 [00 e~C(()g(() ~ a {2Iu(0)12u(1)
[2Iu(0)12u(1)
[2I v (0)1 2v(1)
+ (u(0))2 u (1)*}]
(9.74)
+ (v(0))2 v (1)*
eiwtdtd(
(9.75)
[Using pairs (9.42)-(9.43) and (9.62)-(9.63), Eqs.(9.74) and (9.75) can respec~ively be written as
Inl =
[ :[ :
[:
rl
[ (WIW2, :
DI D2 )
+ WI) (;~ (W + WI + W2) (;0 (W + W2 + D I ) I· Uo (W + W2 + D 2 ) U~ (W + W2 + DI + D 2 ) f-- Uo (W + WI) Uo (W + W2) U~ (W + WI + W2 + Dd
I· [{2(;0
(w
149
19.4 DWDM systems
U~ (W + WI + W2
I·
1+
a { 2Vo (W
-
il2 )
U~ (W + WI + W2 + ill -
+ WI) Vo* (W + WI + W2) VO (W + W2 + ill)
I· Vo (W + W2 + il2 ) Vo* (W + W2 + ill + il2 ) f--- VO (W + WI) VO (W + W2) Vo* (W + WI + W2 I·Vo* (W + WI + W2 - il2 ) Vo*
I·
land
(W
+ WI + W2 + ill -
1:1:1:1:
In2 =
f [{2Vo
(w
il2 ) }
+ ill) (9.76)
il2 )}] dWldw2dilldil2
rl (WIW2,il l il2 )
+ WI) Vo* (W + WI + W2) VO (W + W2 + ill)
Va (W + W2 + il2 ) Vo* (W + W2 + ill + il2 )
f
F Vo (W + wI) VO (W + W2) Vo* (W + WI + W2 + ilI) fVo* (W +WI +W2 - il2 ) Vo* (W +WI +W2 + ill - il2 )} 1+
a
{2Uo (W + WI) U~ (W + WI + W2) UO (W + W2 + ill)
+ W2 + il2 ) U~ (W + W2 + ill + il2 ) F Uo (W + wI) UO (W + W2) U~ (W + WI + W2 + ilI) f[r~ (W + WI + W2 - il2 ) f UO
(W
!Equations (9.68) and (9.69) represent the HO-GTE for the propagation of ~olitons through birefringent optical fibers.
~.4
DWDM systems
IThe solitons propagating through a DWDM system can be modeled by the [ollowing N -coupled NLSE in the dimensionless form:
lqiZ) I"
+
D~Z) q~!) + g(z){lq(l) 12 +
t
azmlq(m) 12 }q(l)
=
0
(9.78)
m-/-Z
kvhere 1 < I < N. Equation (9.78), once again, models bit-parallel WDM ~oliton transmission. Here, aZ m are known as the XPM coefficients. The fields 1m are expanded in powers of Za as
OC50
9 Higher Order Gabitov-Thritsyn Equations
~( 1 (, Z, t ) =
2(2)( (, Z, t ) + ... ql(0)( (, Z, t ) + zaql (1)( (, Z, t ) + zaql
IEquating coefficients of like powers of
0 (
Za
gives
.aqjO) ~(() a2qjO) z7i( + -2-7fi2
z1a):
1
(9.79)
=
0
(9.80)
. aqjl) ~(() a2qjl) { .aqjO) oa a2qjO) z7i( + -2-7fi2 + Z az + 27fi2
0(1) : 1
N
+ g(z) (lqjO) 12 + ~I Ctlmlq~)12 )qjO)} = 0
L
Ctlm{2Iqg) 12qg)
+ (qg))2 qg)*}] }
=
(9.81)
0
(9.82)
m-/-I
~t
0(1/ Za), Eq. (9.80), in the Fourier domain, is given by (9.83)
Mihose solutIOn
IS
(9.84)
R?fOl (Z, w)
=
qfOl (0, Z, w)
(9.85)
!At 0(1),Eq.(9.81) is solved in the Fourier domain by substituting the solution given by Eq.(9.84) into Eq.(9.81). This gives
aQ (0) A
2
_ 1 _ _ ~o Q(O) 2 a 1
az
151
19.4 DWDM systems
~quation (9.86) is an inhomogeneous equations for qY) and with the homogeIneous parts having the same structure as in Eq.(9.80). For the non-secularity Icondition of qjl), FA gives the condition on afO)(Z,w) as
. (lq}O) 12
N
+ L almlq~)12 )q~)eiwtdtd( = 0
(9.87)
ml
jEquation (9.87) can be simplified to
(9.88) !Equation (9.88) is the GTE for the propagation of solitons through multiple Ichannels or DWDM systems as seen in Chapter 7. Equation (9.81) will now ~e solved to obtain q?)((, Z, t). Substituting into the right-hand side of !Eq. (9.86) and using Eqs.(9.84) and (9.87) give
aiD)
!which integrates to .,(1)
2ql e
1
iw2C(() 2
1= af 1 )(Z,w)
(1° 1 1(1 ° 1
00
g(()e i <;;2C(() (lqjO) 12
+ :talmlq~)12)qjO)eiwtdtd( moll
-00
00
-
1
-00
g( (')e i<;;2 C((') (lqjO) 12
+
:t
moll
aim
Iq~) 12) q}O) eiwtdtd(' (9.90)
OC52
9 Higher Order Gabitov-Thritsyn Equations
!where (9.91 ) !Also,
Qf
1 )(Z,w)
is so chosen that
1 1
°
. ,(1)
zql e
2
""---C()dr 2
"
=0
(9.92)
is going to be an useful relation for subsequent orders. Applying IEq.(9.92) to Eq.(9.90) gives ~hich
R?)(Z,w) =
IqjO) 12 + L>.l:lmlq~)12 qjO)eiwtdtd('d( 111(100 g((')e'~. ° ° -00 L~ (11°O g(()ei~2C(()(lqjO)12+ talmlq~)12)qjO)eiwtdtd( Jo -00 m#l 2
,
C(n
(
N
m
)
I
I
(9.93) row, Eq.(9.90), by virtue of Eq.(9.93), can be written as
1° 1(° 1-0000 1
g( (')e
i~2 C(n (Iq~) 12 + L
fq}Ol eiwtdtd(' d(
~
(( _~) 11: 1
g(()e
HO) eiwtdtd(1 !which can be fmther rewritten
aIm
Iq~) 12)
m#l
i~2 C() (lqjO) 12 + ~ almlq~) 12) (9.94)
as
153
19.4 DWDM systems
I· { QjO) (w
+ S?1) QjO) (w + S?2)
N ,,'0)
L.J almQI (w ml I· {
1(
+ S?d Q'(0) l
11
+ S?2) }
11( 1
g( (')eifM12C(/) d(' -
1- (( - ~)
(w
g( (')eifM12C(/) d(' d(
g(()eifhfl2C(()d( } dS?l S?2 ]
(9.95)
IThus, at O(Za), (9.96) IMoving on to the next order at O(z~), one can note that the GTE given by IEq.(9.88) is allowed to have an additional term of O(za) as
+ L almQ~) (Z, w + wd Q~) (Z, w + WI + W2) ] dW1dw2 m-l-l
IThe higher order correction nl can be obtained from the suitable non-secular Iconditions at O(z~) in Eq.(9.82). Now, Eq. (9.82), in the Fourier domain, is
+ e~C()g(() [ : + 1
t
m-l-l
aim
2IqjO)12q?)
+ (q}0))2qj1)*
{2Iq~) 12q~) + (q~))2q~)* } ] eiwtdt = 0
IBut, again, Eq.(9.92) gives
1 1
o
(9.98)
2
- 0 ql,(1) e ""---C()dr 2 ." -
!Applying the non-secularity condition (9.99) to Eq.(9.97) gives
(9.99)
OC54
r l
9 Higher Order Gabitov-Thritsyn Equations
=
-10
1 [:
e i <:;2C«)g(() [2 Iq }0)1 2qF)
N
+L
aIm
+ (q}0))2qF)*
{2Iq~) 12qg) + (q~))2qg)* }] eiwtdtd(
(9.100)
ml
iUsing Eqs.(9.84) and (9.94), Eq.(9.100) can be written as
r = 1:1:1:1: l
I [{
(0)
2Q l
r
A
(W
+ wd Q l(0)* (W + WI + W2) Q l(0) (W + W2 + ill) A
A
+ W2 + il2) Qz(0)* (W + W2 + ill + il2) ~a}O) (W + WI) a}O) (W + W2) a}O)* (W + WI + W2 + ild ra}O)* (W + WI + W2 - il2) a}O)* (W + WI + W2 + ill - il2) } I
0)
rl (WlW2, ill il2 )
t Qz (W A
rrv
1+ L
aIm {
A
2a~) (W + wd a~)* (W + WI + W2)
~
+ W2 + ill + il2) - Q~) (W + WI) Q~) (W + W2) f Q~)* (W + WI + W2 + ild Q~)* (W + WI + W2 - il2) f Q~)* (W
(9.101) !Equation (9.97) represent the HO-GTE for the propagation of solitons ~hrough multIple channels or DWDM systems.
IReferences M. J. Ablowitz & H. Segur. Solitons and the Inverse Scattering Transform. SIAM. IPhiladelphia, PA. USA. (1981). f;!. M. J. Ablowitz, G. Biondini, S. Chakravarti, R. B. Jenkins & J. R. Sauer. "Four-wave ~ixing in wavelength-division-multiplexed soliton systems: damping and amplifica~ion". Optics Letters. Vol 21, No 20, 1646-1648. (1996). ~. M. J. Ablowitz & G. Biondini. "Multiscale pulse dynamics in communication systems ~ith strong dispersion management". Optics Letters. Vol 23, No 21, 1668-1670. (1998). fi. M. J. Ablowitz & T. Hirooka. "Resonant nonlinear interactions in strongly dispersion~anaged transmission systems". Optics Letters. Vol 25, No 24, 1750-1752. (2000). ~. M. J. Ablowitz & T. Hirooka. "Nonlinear effects in quasi-linear dispersion-managed ~ystems". IEEE Photonics Technology Letters. Vol 13, No 10, 1082-1084. (2001). ~. M. J. Ablowitz, G. Biondini & E. S. Olson. "Incomplete collisions of wavelength~ivision multiplexed dispersion-managed solitons". Journal of Optical Society of IAmerica B. Vol 18, No 3, 577-583. (2001). ~.
IHdcrcnccs
, ;;5
17. 1"v1. J. Ablowitz &: T. Hirooka. "Nonlinear effects in qllasilinear dispersion-managed Iplllse t.ransmission". TF.:F.:F.: Jonrna.l of Phol.orJic.~ Technoloq1/ T,eUer8. Vol 2fl. IR4(j-
11MR. (2001)1
fg. 1-1. J. Ablowit.z & T. Hirooka. "I\-1anaging nonlinearit.y in Iolt.rongly dilolpersion-managed bpt.ical pulse transmission" .i01t.rnal of Optical Society of America 13. Vol 19, No 3. ~.
~O.
~l.
112. II:~.
~4.
~5.
116.
In. 118. ~9.
fI25--1:N. (2002).1 M. J. Ablowitz. T. Hirooka & T. Inoue. "Higher-order asymptotic analysis of fbsperslon managed transmIssIon syst.ems: solutIons and t.helr charad.enstlcs· . JonrrJa.~ 10J Optical Society oj America 13. Vol 19, No 12, 2876-28R5. (2002)J C. D. Ahrens, M. J. Ablm,,-,itz, A. Dochert.y, V. Olcg, V. Sinkin, V. Gregorian & C. fR. T\Tenyuk. "Asyrnpt.otic analysis of collision-induced t.iming shirt.s in return-tn-zerol huasi-linear syst.ems with pre- and post-dispersion compensation". Oplic8 rr:iJr:r8. Voll pO, 2056-20GR. (2005). A. Biswa,s. "Cabitov-'Iuritsyn equation tor solitons in multiple channels". Journal oj I"lcrlmm-ngndic IVmw., and A pphml,.on.,. Vol 17. No I I, 15:)9-1560. (200:))1 A. l3iswas. "Gabito\'- Turit.svn equation for solit.ons in opt.ical fibers". Journal of Non-I Vinear Optical Physics and Materials. Vol 12, No 1, 17-37. (2003). A. Biswa.s. "Dispcrsion-Tvlanaged solit.ons in multiple channels". Journal oj NrmlineaT Optical Physics and Materials. \/0113, No 1, 81-102. (200"1). A. Bilol. .vas. "Higher-order Cabitov-Turitsyn equation for disperlolion-managed vector folit.ons in birefringent fibers". Tnl.r:rnoJiona1 Jonrnal oj Theorr:l.1,ml Phy.~ic.~. Vol 46, ~ 0 12, ;J:l39-:135-1. ( 2007). A. Biswas. "'Higher-order Cabitov-Turitsyn equation tor dilolpersion-managed solitons'. Oplik. Vol 118, No :~, 120-1:~3. (2007). A. 13is\vas &. Eo Zerrad. "Higher-order {jabit.ov-Turitsyn equation tor dispersion~nanaged solitons in multiple channeb i international Journal oj Mathematical Anal ~8i8. Vol I, No 1:2, Guo G82. (:2007)1 I. R. Ciabitov &. S. K. Turitsyn. "Average pulse dynamics in a cascaded transmissiOl~ 8ystem wit.h piJ,ssive llilolpersion compensatiorf'. Optics Letters. Vol 21, No 5, 327-329. P 990). I. R. Ciabitov &. S. K. Tllritsyn. "Breathing solitons in optical hber links". JETP !Letters. Vol 63, No 10, 861-866. (1996)1 T. Hirooka & A. Hasegm..r1t. "Chirped soliton int.eraction in st.rongly dispersion-
wavelength-division-multiplexing systems". Optics LeLLeTs. Vol 2:~, No 10. 1768-770. (1998)·1 120. v. E. Zakharov & S. \iVabnit:>;. Optical Solitons: Th.eon~tical Challenges (J.rul TrulnHtri(J.~ !perspectives. Springer, Heidel berg. DE. (HJ99)J ~nanaged
Iindex
II\T coupled NLSK U9
Bend-induced birefringence, 28
(NA, 8 IV-Tlumber,8 j,h order dispersion, 80, 81
In
KHZ), 10
lA.bsorption coefficient, 7
13eta function, 34 KirctringcTI(:c, 27, 45 Birefringence paramet.er, /16 Birefringent. fiber, '16, ~t9, 9r;, 100, HH, B7, 1,12,1,H 13irefringent nonlinear fibers, 46
fA"""""'"n,n"'I",",t""",d,-,d";~~""C,,,,';"o"""c.'I,I_ _~--=c-____ 13=icce-=cci'c,,,gceCn"tco~p7t
diabatic evolution e uations, 21. G5 Bit error rate BErt 8 Adiabatic parameter, 21 Bit rate R 8] li\diahatic variation, 19 Bit-error rat.e (BER), 28, 78 [Amplification. 9 Bright solItons, ~ ~A.:rnplification period, 12, D c IArnplifh'Ci spontaneous emission (ASE), 3D ~A.mplified spontaneous emission (ASE~ inoise, 9 Carrier 'wave, 8 IArnplifi{,l'. II.'), 127, 12~, 131 Cauchy's principal value, 118. 126, l:,t2J IAmplifier spans, 10 Channel (TossLalk, 3 ~A.:mplifiers, 116 Chirp, 2, 1'1, 17, 40 IArnpliLud{' jitter, 11 Chirped fiber gratings. 10 IAnalvtical rncLhod, 7S Chromatic dispersion. 6 bA..nisotropic stress, 27 Cladding. 81 ~\nisotropy, 45 Classical solitons, 1141 IAnomalous and normal GVD segmenLs, Cohercnt crosstalk 77, ~o 1:J:l Cohecent intecfecence, 81, 8iJ ~A.:nomalous dispersion, :~, 109 CohCl'{'nj. l{'ngLh, 7~ IAnomalous liI)Cr segmcnts, 115 Collective variables approach, 2 IAllomalolls GVD s{'grncllts. 124, BI Collision length. 2 ~A.:nomalous path averaged dispersion, 1 Collision-induccd frcquency and timing] ~~\C'~'l~'n~,Cp~tCoCti~c~.aC' ,~,ca"l,c"C'isc,,c2~---'-----'-------':":":=I~';71,t~,c~,,=,~I':"I IAttenllat.ioll, 28 ~A.:ttenuation coefficient, 7R IAn;ragc dispersion, 95, 99
[Rase st.ation (BS). 77
COllo-,ervatiull law. 11) Conserved quantity. 17 Constant of motion, 15 OTitiTlUUHI rut iation, 65 Convcntional solitons, 9, 1 'I Core, 8, 27
Index
IJ 58
ICoupled N LSF.,
6:~,
91
hnergy or VI/HoVe
~IC~,·~o~,,~·_~pTh~a,~·,~,'cn~,o'cu~u';I'ca;ti~o~n~(RXT-p~~.TI)','9·,'6'·:3~--~E"O~P~,"i9~3
pross-phase modulation coemdent, 46 ICrosstalk, 28 ICrosstalk level, tl1
ID
POWCL
16
Erbium-doped llber amplillers (EDFA), 9 Euler's constant, 1131 huler's leHlTna, 79 Euler-Lagrange's (EL) equation, 2q Extinction, 28 Eye diagram,78, 8:1 Eye opening penalty (EOP), 92
lDamping, 65 [Damping and periodic amplilkation, II, 131 iIJark solitons, 1 iIJemultiplexer, 78 lOense yvavclength-division multiplexed FA, 107, 151 Fiber core radins, 8 (D\VDT\T) systems, iIJensity, 15 Fiber dispersion (chromatic and material llJielectric constant, 7 dispersion), 71 [DispersioTl, 9, 11-1 li'iber losses, 2 FIber nonlmeanty, 10, II IDlsperslon compensaj,lOn, 10 [Dispersion lengt.h, 81. 8:3, 86 Fiber optic resonator, :191 ilJispersion managed vector nonlinear Fiber response, tl§ Schrodinger's equation (D1-IFiber-optic sensing, 28 Field radius, 8 IVNLSE), 46, 51. !)5, 100 ilJispersion management. 12 Flux, 15, 16 !Dispersion map, 2, 108, 115 Four-yvave mixing, 1,31 lIJispersion parameter, 9 Fourth order dispersions, 87 IDispersion slope, 9 Fredholm's Alternative (FA), 1:39 lDispersion-compensated svstems, 10 Fundamental fiber mode, 8 lDispersion-compensating fibers (DCF), 2 Fundamental mode, ~ llJispersion-Hattened optical fibers, 63 _Fundamental soliton, 2, 40 ;ID~i~,~p~e~rs~i~o~H'c,=,'c,"'c,=,"=g~e~d;-;(f)~,~.r~)=,~o~l~ij,=o~H=,~l~l----~F~\~\r~~~I=,~'';;O lDispersion-managed NLSE (Dl\l-NLSE), G IQIspersIOn-managect nonIlllear Schrodil1ger's equation (D11-NLSE),1 1,2,1:3.1.5,17,28,9.5 iDIspersIOn managed optIcal solIt.ons, I lDispersion-managed soliton, 1:1, 99, 107 lDispersion-management(DM), 1, 5, 111,
114 lDispersion-shin.ed, 6:~ lDispersion-shifted fiber, 1,10 !Dispersive, :3 !Dispersive pulse broadeniTlg, ]0 iDIstortlOn, Y IO-f-.J soliton, 1, 2, 11, 1:1, 18, 29, 111, 'Jl4,'Jl5 IDvVIHd, 70, 9:3, 95, 104, 107, 111, 129, 137, H\)
lD\vDl\I systems, 10
(Effective core area, 8 IEL equation, 20, 21, 36, 51, 6tl, 71 IEla..<:;tic stresses, ,'15
Cl GabitO\-Turitsyn equation (GTE), 95, 99, 107,112,11:1, 116, 117, 120, 125, 127, 129,132, -131, ]:33, 137, ]:39, 140, 1441 Galilean invariflTlt, 99 (;auss' gcncrali;-;cd hypcr-gcomd.ric function, ;14 Causs' hyper-geomet.ric functioIl, 8~ Gaussian and SG pulses, 52 Gaussian distriblltion, M Gaussian function, 81 Gaussian pulse, 49, 67, 7:3, 74, 80, 83,95 Gaussian soliton, 30, :36, 671 Gaussian Lype, 6q Gaussian yvhite noise, 84 Cordon-Haus (CH) timing jitter, 9,10 Gordon Raus effect, 11 Gordon Hans t.iming jitter, 3 Group delay dispersion, :] Group velocity, 9, /16 Group velocity dispersioIl (GVD), 2, III GVD, \)-11
Iindex
[H ~lamilton 's principle, 20 iHigher order dispersion, 78, 79 iHigher order Gabit.ov-Turitsyn equation (HO-GTE), 137, 149 ~ligher order modes, 7 !Higher order dispersion, 31
Maxwell s equation, ,~ T\·'fodal birerringence, 4fi Mode-locked liber laser, :3 l\:lode-Iocked laser, :3 T\·Toment method, 2 MultI mode, 8 MultI phot.on absorptlOn, {) l\'lultiple-scale perturbation method, I>dultiplexer, 78
q
[
~~=~~_____________ ~n- band,
78 ~n-band crosstalk, 78, 83, 84, 89 OCn-band int.erference. 80 OCn-band optical crosstalks, 77 OCncoherent crosstalk, 77 Iincomplete beta tUTl(:tion, 89 ~nitial pulse energy, 2 ~ntegral of motion, 15 ~nt.egml operator, 19
~Nr'~,S~F~""TI5, ]6, 70, 78, 89, 99, ]04, 10El
N Olse, 10 Non-ret.urn 1.0 zero (NRZ), 10 Nonlinear chirp, 114 NonliTieaT effects, 3, 781 Nonhnear llltegral eqllatlOn, 9U1 Nonhnear lllteract.lOn, 1(1 Nonlinear modes, 6,,1 Nonlinear polarization, § Nonlinear Schrodinger's equation (NLSE),
"11~nTt,e",~,C"",C,y;-im~o~il","d~aT,C,HT'n;;-Cd",","eOcTl-d;cc;;eTt,e".cc.lc,,~oTn;-------,"'-,"9',-rer"1
(11\1-DD), R'1 OCnter-channel crosstalk, 77, III ~nt.er-sYIllbol interference (lSI), 83, 87 ~nterrerometry, :28 OCntemal birefringence, 28 ~nt.ernal stresses (stress-induced bircfringtclH:tc),
27
OCntrachannel crosstalk. 77 OCnverse scattering transform(IST), 1, H, 50 OCsotropic, ,) Iisotropic fiber, 45 OCSOj.roPIC matenaL ,It)
Nonlinear wave equation, ~ Nonlinearity, 10, 11, 111 Nonzero dispersion fibers, 10 Normal and anomalous CVD segment, 120. 128 Normal chromatic dispersion, :3 Normal GVD StcgllltcllL, "127, n~ Normal path average dIspersIOn, ~ Normal- and anomalous-dispersion fibcrsj Iq Nllmencai apertllre, 8
a
IK
lLagrangian. 20. :35. 51. 68 lLangevin equations, 41 1La..<:;er amplihers. n lLinear momentum, 17, 65 lLoss, 17 ILoss coefficieTlt, ]] ILmvest order II mdamental mode, 8
Optical add-drop multiplexers (OADMs)j 771 Optical amplifiers, 2 Opt.ical baTl(lwidt.h, 31 Optical cross-connects (OXeS), 78 Optical losses, 7 Optical mode, 7 Opt.ical pube propagat.ioTl, 78 Optical s\vit.ches, 77 Orthogonal polarization states, 61 Out. of band cro~st.alk, 89, 931 Out. of band optical crmst.al ks, 77
Ilvl
p
IKlap period, 1, 2 IMap strength, 2, 1:2, 99, III, 11;3, 114
Painlcvc test of integrability, Pa..<:;Slve mode locklllg, 4Q
lKeT'Tlel, W7
160 Path-averaged constant, 12 Path-averaged dispersion, 12 Periodic dispersion map, 2 Permeability, 6 Permittivity, 6 Perturbation parameter, 19, 21, 55 Perturbation terms, 74 Perturbation theory, 18 Perturbed DM-NLSE, 21 Perturbed NLSE, 19 Phase constant, 78 Phase shift, 80 Photo-elastic effect, 45 Plasma, 5 Plasma index, 5 PM fibers, 28 Poehammer symbol, 34 Polarization, 45 Polarization maintaining (PM) fiber, 27, 45 Polarization mode dispersion (PMD), 27 Polarization multiplexing, 27 Polarization preserving optical fiber, 13 Polarization preserving (PP)fiber, 27, 28, 95, 96 , 99, 104, 112, 137 Polarization preserving optical fiber, 142 Polarization reserving fiber, 28 Post-transmission compensation, 114 Pre- and post-compensation, 10 Pre-transmission, 114 Probability density function, 84 Propagation constant, 8 Propagation delay, 80 Propagation length, 81, 86 Pulse, 17 Pulse broadening, 11, 27, 114 Pulse distortion, 2 Pulse interaction, 2, 36
Q Quasi-linear Gaussian pulse, 115, 121 Quasi-linear pulse transmission, 111 Quasi-linear pulses, 11, 120, 128 Quasi-linear regime, 114 Quasi-linear transmission, 116 Quasi-monochromaticity, 19
R Random perturbation, 39 Refractive index, 7 Return-to, 10 Root-mean-square (RMS), 18
Index
s Sech,85 Second order dispersion, 81, 83 Self-phase modulation, 1, 9 SG,17 SG pulse, 32, 41, 42, 49, 58, 67, 70, 74 SG solitons, 36, 57 SG type pulse, 65 Shepherd pulse, 64 Signal-to-noise ratio (SIR) 2 10 81 Single-mode fiber,7, 9, 27, '28' , Soliton dragging logic gates, 46 Soliton mode locking, 3 Soliton perturbation theory, 2 Soliton trapping, 46 Soliton-like RZ pulses, 11 Soliton-soliton interaction, 2 Spatio-differential operator, 19 Split-step Fourier method (SSFM) 91 Spot size, 8 ' SS pulse, 34 Step-index, 8 Step-index fiber, 7 Stochastic phase factor, 41 Stochasticity, 38, 39 Super-Gaussian, 29,91 Super-Gaussian (SG) pulse, 30, 35, 69, 92 Super-sech (SS), 32,91 Super-sech pulse, 17, 34, 88 Super-sech type pulses, 29 Susceptibility, 6 T
TE and TM modes, 28 Third order dispersion, 81, 83, 85 Timing jitter, 2 Total dispersion, 10 Transfer function, 78 Transmission distance, 84 Transmission rate, 84 Traveling wave solution, 99 Two-step dispersion map, 13 Two-step map, 12, 108
v Variational approach, 2 Variational principle, 5, 21, 36, 50, 68 Variational problem, 20
w Wave equation, 6 Wave number, 8
161
Iindex
IVVavckTlgt.h rlivisiOTI TTlultipkxiTig (\iVD1\:f), If3;3, 77, lOG, 129
IWidth,2
zI
IX IX[)I\L 100, 129
Zero disperSIOn wavelength, I q
IN onlinear
Physical Sciencel
I(Senes F;dztors: Albert C.l.
LUG.
Nail H. Tbmgimov)
INail. H. Ibragimov/ Vladimir. F. Kovalev: Approximate land Renonngroup Symmetries IAbdul-Majid Wazwaz: Partial Differential Eqnations and SoliItary Waves Iheory IAlbert C.J. Luo: Discontinuous Dynamical Systems on Time~rarying DOlnain~
IAnjan Biswas/ Daniela Milovic/ Matthew Edwards: Math~matical Theory of Dispersion-Managed Optical Solitons IHanke,Wolfgang/ Kohn, Florian P.M./ Wiedemann, Meike: Self-organization and Pattern-formation in Neuronal Systems llnkler Conditions of Variable GravitYI !Vasily E. Tarasov: Fractional Dynamics !Vladimir V. Uchaikin: Fractional Derivatives in Physics IAlbert C.J. Luo: Nonlinear Deformable-Body Dynamics andl [Waves [yo Petras: Fractional Order Nonlinear Systems IAlbert C.J. Luo /Valentine Afraimovich (Editors): Hamil~onian Chaos beyond the KAM Theory IAlbert C.J. Luo/ Valentine Afraimovich (Editors): Long~ange Interaction, Stochasticity and Fractional Dynamicsl