Numerical Simulations of Femtosecond Pulse. Propagation in Photonic Crystal Fibers. Comparative Study of the S-SSFM and RK4IP

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1 Appled Mhemcl Scences Vol. 6 1 no Numercl Smulons of Femosecond Pulse Propgon n Phoonc Crysl Fbers Comprve Sudy of he S-SSFM nd RK4IP Mourd Mhboub Scences Fculy Unversy of Tlemcen BP.119 Tlemcen Alger momhboub@yhoo.com Tsour Zendgu Scences Fculy Unversy of Tlemcen BP.119 Tlemcen Alger.endgu@homl.com Absrc We nvesge he propgon of femosecond pulse n phoonc crysl fbers PCFs whch s cully of gre neres for sudes. The generled nonlner Schrödnger equon GNLSE descrbes he dfferen physcl phenomen lke dsperson nd some nonlner effecs ncludng he SPM self seepenng nd Rmn scerng encounered when he femosecond pulses propge n he PCF. In our smulon we use he symmerc spl-sep Fourer mehod S- SSFM nd he fourh-order Rnge-Ku nercon pcure RK4IP mehod whose re ofen used o clcule he numercl soluons of he GNLSE. In hs pper we presen our mplemenon lgorhms; nd we wll show h for gven sep se he clcule S-SSFM number of fs Fourer rnsforms FFTs s less hn he one of he RK4IP by fcor 3 n spe of lrge errors. In order o evlue he performnce we hve lso mesured he globl relve error by usng lner nd nonlner chrcerscs of PCF found n he lerure. Our numercl resuls llow showng h for fxed errors he ol number of FFTs of he RK4IP s much smller. The Oh nd Oh 4 orders of he S-SSFM nd RK4IP respecvely confrm hese resuls. Keywords: Generled Schrödnger Equon S-SSFM RK4IP Globl Relve Error PCF

2 584 M. Mhboub nd T. Zendgu 1 Inroducon Snce 199s he mcrosrucure opcl crysl fbers PCFs hve known gre number of pplcons n some domns lke opcl elecommuncons he bo-phoonc sensors nd he lser sources. They llow precse conrol of he chromc dsperson profle over brod wvelengh rnge [1 ] such s shf of he ero-dsperson wvelengh [3] no he vsble rnge nd sngle mode operon over lrge specrl rnge. Ths bly o conrol he mgnude nd wvelengh dependence of he group velocy dsperson GVD nd he sme me o enhnce or reduce he effecve nonlner coeffcens mkes PCF gre vrees of mens for sudes of nonlner effecs nd explos speclly he properes of s lrge opcl nonlneres. PCF wh sold core 1 m n dmeer hs nonlner Kerr coeffcen ~4 W -1 km nm nd vlues s hgh s ~55 W -1 km nm hve been mesured for PCFs mde from mulcomponen glsses [4]. In complee conrs hollow-core PCF hs exremely low levels of nonlnery; fber ws repored wh nonlner coeffcen ~.3 W - 1 km -1 [5 6]. The use of hghly nonlner glsses such s chlcogende glsses for he relon of PCFs llow o ncrese he vlue of lrgely bove housnds of W -1 km -1 [7]. The propgon of femosecond pulses n PCFs s descrbed by he generled nonlner Schrödnger equon GNLSE where lner nd nonlner effecs re consdered [8]. Ths equon conns he hgh order dspersons HOD SPM self seepenng SS nd Rmn scerng effecs whch mke mpossble o resolve nlyclly. The symmerc spl sep Fourer mehod S-SSFM nd he fourh-order Rnge-Ku nercon pcure RK4IP mehod re ofen used o clcule he numercl soluons of he GNLSE [9 1]. These wo mehods re sbles nd effcen. We presen n hs work our mplemenons nd we evlue her performnce by usng he ol number NFFT of fs Fourer rnsform nd he globl relve error for he mesured lner nd nonlner chrcerscs of PCF found n he references [11 1 nd 13]. In secon we descrbe heoreclly he PCF GNLSE model [11] nd we gve he globl relve error formul [14 15] n order o deermne he mplemenon performnce of ech mehod. In secon 3 nd 4 we presen our mplemenons for he S-SSFM nd RK4IP lgorhms nd we clcule he correspondng ol number of FFTs. We evlue he performnces n secon 5. In hs secon we presen numercl soluons n he cse of n nl hyperbolc secn pulse for some lengh of PCF our numercl resuls relng o he number of fs Fourer rnsforms NFFT nd he globl relve errors for some sep ses. Fnlly we gve our concluson for hs work.

3 Numercl smulons of femosecond pulse propgon 5843 PCF GNLSE model Ulrshor pulse propgon n PCF s descrbed heoreclly by he generled nonlner Schrödnger equon GNLSE. Ths equon gven by he equon 1 conns enuon hgh order dsperson HOD nd nonlner effecs ncludng he self phse modulon SPM self seepenng nd Rmn scerng.! mx T R = = ω β α 1 Where s he mplude of he vrble feld β s he h-order dsperson coeffcen he pump frequency ω α s he enuon coeffcen nd s he nonlner coeffcen of he SPM due o opcl Kerr effec. R T s he Rmn me consn esmed from he slope of he Rmn gn specrum smuled Rmn scerng SRS. The quny g v = ' s he rerded me where s he poson long he fber ' s he physcl me nd g v s he group velocy he cener wvelengh λ. Ths equon cn be wren s N D = Wh lner operor D nd nonlner operor N defned s T N D R = = = 1! mx ω β α 3 As he nlycl soluon of he forml equon s dffcul o rele severl pproches hve been developed o deermne he numercl soluon of h equon. We presen n our mplemenon wo lgorhms he symmerc spl sep Fourer mehod S-SSFM nd he fourh order Rnge Ku nercon pcure RK4IP mehod. In boh mehods he me prl dervves re clculed n he specrl domn by usng he Fourer rnsform. The dfferenl operor s replced by ω ω s he frequency nd ech prl dervve of order n should be represened s [ ] F n F n n ω where F denoes he

4 5844 M. Mhboub nd T. Zendgu Fourer rnsform operon. In he followng we use he FFT nd IFFT o represen he fs Fourer rnsform nd nverse respecvely. In order o evlue he mplemenon performnces we use he globl relve error δ gven by he followng equon: An A δ = 4 A Where A s he fne numercl soluon he end of he fber compued for very smll nd consn vlue of he sep se h; he norm A s defned by: 1/ A A = d 5 3 S-SSFM soluon for GNLSE The symmerc spl sep Fourer mehod gven by he Srng formule [16 17] subdvdes he globl propgon dsnce no seps of lengh h nd supposes h he effecs of dsperson nd nonlnery c ndependenly long ech sep. Effecs of nonlnery re nsered he mddle of ech sep h/ of he fber. hd hd hn h e e e 6 The nonlner erm s solved n me domn wheres he dsperson erm s solved n he frequency domn nd requres FFT rounes. To clcule he self seepenng nd he Rmn erms we use some FFTs nd IFFTs. Ths pproch bsed on he S-SSFM s gven by he followng lgorhm: For fber wh lengh L he number p of seps s p = L/h. n = nh n = p - 1 hd 1 = IFFT e FFT n = e hn 1 1 = IFFT e hd n 1 FFT Where D nd N re clculed s followng by neglecng he enuon nd where only dspersons up o order sx hve been consdered β β3 3 β4 4 β5 5 β6 6 D = ω ω ω ω ω N 1 = 1 IFFT[ ω FFT 1 1 ] TRIFFT[ ω FFT 1 ] ω 1 Th mplemenon needs four FFT nd IFFT per sep. Thus ol number NFFT of FFTs-IFFTs s NFFT = 8 p. 7

5 fs Numercl smulons of femosecond pulse propgon RK4IP soluon for GNLSE In hs subsecon we presen n pproxme soluon of he equon 1 by usng he RK4IP mehod. In hs mehod he me prl dervves re lso clculed n he specrl domn. However he spl dervves re clculed n he me domn s he S-SSFM lgorhm. The RK4IP mehod s deled n he references [9 1]. hd I k1 k k3 k4 n 1 = e n Where hd I n = e n hd k 1 = he N n n I h I k 1 k = h N n k1 n 1 I h I k 3 k = h N n k n hd hd I [ ] I k4 = h N e n k 3 e [ n k3]. Operors D nd N re clculed from he equon 8. We use FFT-IFFT s descrbed by he equon 7 o mplemen he RK4IP. The ol number NFFT whch we hve compued s NFFT = 4 p. Ths number s greer by fcor 3 hn he one of he S-SSFM mplemenon. 5 Numercl smulons In our smulon we consder propgon of n nlly T = 1 fs hyperbolc secn sech pulse 155-nm wvelengh long PCF wh dsperson coeffcens β = -11. ps β 3 = -.44 ps 3 β 4 = 1.6 ps 4 β 5 = -6.3 ps 5 β 6 = -1.9 ps 6 nd wh nonlner coeffcen = m -1 W -1 nd T R = 5.4fs. The nl pulse wh pek power P = 5kW nd some smuled oupu pulses fer propgon dsnce of L =.1 m.3m nd.4m for h = 1-5 m re depced n Fg. 1. We show h he S-SSFM nd RK4IP re effcen nd gve prcclly he sme oupu pulses. To compre her performnce we repor n ble 1 he ol numbers NFFT nd globl relve errors correspondng o ech mehod for some vlues of he sep se h nd for L=.1m.

6 5846 M. Mhboub nd T. Zendgu 5 x Inpu sech pulse 6 x 16 5 S-SSFM RK4IP Inensy mw 3.5 Inensy mw T/T T/T Inl hyperbolc secn sech pulse b L =.1 m.5 x 17 S-SSFM RK4IP 16 x S-SSFM RK4IP 1 Inensy mw Inensy mw T/T T/T c L =.3 m d L =.4 m Fgure 1: Numercl soluons for some lenghs L of he PCF for h = 1-5 m. Tble 1. Globl relve error δ S-SSFM nd δ RK4IP for some vlues of he sep se h for L=.1m. h x1-3 m δ S-SSFM x NFFTS-SSFM δ RK4IP x NFFTRK4IP

7 Numercl smulons of femosecond pulse propgon S-SSFM error RK4IP error Globl relve error Slope = h km Slope = Fgure : Globl relve errorδ for some sep ses h nd L =.1 m. The lner vrons of he globl relve errors δ S-SSFM nd δ RK4IP versus he sep se h re represened n log-log grph of Fg.. The esmed slopes re respecvely 1 nd 4 for he S-SSFM nd RK4IP. The globl relve error order of he S-SSFM depend essenlly on he erm used o pproxme n he nonlner operor N [18]. Therefore n our mplemenons δ S- SSFM nd δ RK4IP orders re respecvely Oh nd Oh 4. These orders re n good greemen wh he resuls presened n he lerure [18]. We repor n fgure 3 he ol numbers NFFTS-SSFM nd NFFTRK4IP necessry o ccomplsh he S-SSFM nd RK4IP lgorhms for dfferen vlues of he globl relve error δ gven n ble. We show h hs numbers evolves lnerly for boh mehods nd her slopes re respecvely -1 nd -1/4. We predc some vlues of he NFFTS-SSFM by lner regresson for he sme δ RK4IP. Ths lrge predced numbers nd he gre moun of he NFFTS- SSFM/NFFTRK4IP ro confrm he mporn effcency of he RK4IP for he wek errors.

8 5848 M. Mhboub nd T. Zendgu Tble. The S-SSFM nd RK4IP ol number NFFT for dfferen vlues of he globl relve error. δ x e-6 NFFTS-SSFM NFFTRK4IP NFFTS-SSFM/ NFFTRK4IP e e e6 * e e7 * e e8 * e e8 * e e9 * * Predced numbers S-SSFM RK4IP NFFT Slope = Slope = -1/ Globl relve error Fgure 3: Tol numbers NFFTS-SSFM nd NFFTRK4IP for some vlues of he globl relve errorδ.

9 Numercl smulons of femosecond pulse propgon 5849 Conclusons In hs work we hve nlyed he S-SSFM nd he RK4IP lgorhms o sudy he propgon of femosecond pulses n he PCFs. The wo mehods re sbles nd effcen o deermne he numercl soluon of he GNLSE where he hgh order dspersons nd he nonlner effecs lke SPM self seepenng nd Rmn scerng re consdered. In order o evlue he performnce we hve mesured he globl relve error for ech mehod nd for dfferen sep se. We wll show h for gven globl relve error he ol number of FFTs of he RK4IP s much smller nd for gven sep se he S-SSFM number of FFTs s less hn he one of he RK4IP by fcor 3 n spe of lrge errors. The Oh nd Oh 4 orders of he S-SSFM nd RK4IP respecvely confrm he numercl resuls. Therefore we rele from hs nlye h one of hese wo mehods could be chosen o deermne he numercl soluons ccordng o s complexy nd he errors needed n ech pplcon. References [1] J. C. Kngh T. A. Brks P. S. J. Russell D. M. Akn All -slc snglemode opcl fber wh phoonc crysl clddng Opcs Leers [] T. A. Brks J. C. Kngh nd P. S. J. Russel Endlessly sngle-mode phoonc crysl fber Opcs Leers [3] J. K. Rnk R. S. Wndeler nd A. J. Sen Opcl properes of hgh-del r slc mcrosrucure opcl fbers Opcs Leers [4] P. Peropoulos H. Ebendorff-Hedeprem V. Fn R. Moore K. Frmpon D. J. Rchrdson nd M. Monro Hghly nonlner nd nomlously dspersve led slce glss holey fbers Opcs Express [5] F. Lun J. C. Kngh P. S.J. Russell S. Cmpbell D. Xo D. T. Red B. J. Mngn D. P. Wllms nd P. J. Robers Femosecond solon pulse delvery 8nm wvelengh n hollow-core phoonc bndgp fbers Opcs Express [6] C. J. Hensley D. G. Ouounov A. L. Ge N. Venkrmn M. T. Gllgher nd K. W. Koch Slc-glss conrbuon o he effecve nonlnery of hollow-core phoonc bnd-gp fbers Opcs Express [7] M. Lo C. Chudhr G. Qn X. Yn C. Ko T. Suuk Y. Ohsh M. Msumoo T.Msum Fbrcon nd chrceron of chlcogendeellure compose mcrosrucure fber wh hgh nonlnery Opcs Express

10 585 M. Mhboub nd T. Zendgu [8] G. P. Agrwl Nonlner Fber Opcs 4h ed. Acdemc Press Sn Dego 7. [9] J. Hul A Fourh-Order RungeKu n he Inercon Pcure Mehod for Smulng Super-connuum Generon n Opcl Fbers Journl of Lghwve Technology [1] Z. Zhng L. Chen X. Bo A fourh-order Runge-Ku n he nercon pcure mehod for coupled nonlner Schrödnger equon Opcl Socey of Amerc 1. [11] W. H. Reeves D. V. Skrybn F. Bncln J. C. Kngh P. ST. J. Russell F. G. Omeneo A. Efmov & A. J. Tylor Trnsformon nd conrol of ulr-shor pulses n dsperson-engneered phoonc crysl fbres Nure [1] B. Ung nd M. Skorobogy Chlcogende mcroporous fbers for lner nd nonlner pplcons n he md-nfrred Opcs Express [13] H. P. L X. J. Zhng J. K. Lo X. G. Tng Y. Lu nd Y. Z. Lu Specrl compresson of femosecond pulses n phoonc crysl fber wh nomlous dsperson Proc. of SPIE-OSA-IEEE As Communcons nd Phooncs SPIE I-1-6. [14] C.R. Menyuk J. Zweck O.V. Snkn R. Hollöhner Opmon of he spl-sep Fourer mehod n modelng opcl-fber communcon sysems Journl of Lghwve Technology [15] A. Renk T. Tolsno F. A. Cllegr D. F. Gros nd H. L. frgno Uncerny relon for he opmon of opcl-fber rnsmsson sysems smulons Opcs Express [16] G. Srng On he consrucon nd comprson of dfference schemes SIAM Journl on Numercl Anlyss [17] S. Yu S. Zho G. W.We Locl specrl me splng mehod for frs- nd second-order prl dfferenl equons Journl of compuonl Physcs [18] J. Jvnnen J. Ruosekosk Journl of Physcs A: Mhemcl nd Generl 39 6 L Receved: June 1

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