Taking up to fourth order dispersion effects into account, we show that fiber resonators become stable for large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for arbitrary small but nonzero fourth order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and opens the way to experimental implementation

Kozyreff, Gregory

Louvergneaux, Eric

Mussot, Arnaud

Taki, Abdelmajid

Tlidi, Mustapha

Vladimirov, Andrei G.

English

Taking up to fourth-order dispersion effects into account, we show that fiber resonators become stable for a large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for an arbitrary small but nonzero fourth-order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and open the way to experimental implementation

Taki, Mustapha

Publications Server of the Weierstrass Institute for Applied Analysis and Stochastics

Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities: Vladimirov, Andrei G.; Taki, Abdelmajid

Taking up to fourth-order dispersion effects into account, we show that fiber resonators become stable for a large intensity regime. The range of pump intensities leading to modulational instability becomes finite and controllable. Moreover, by computing analytically the thresholds and frequencies of these instabilities, we demonstrate the existence of a new unstable frequency at the primary threshold. This frequency exists for an arbitrary small but nonzero fourth-order dispersion coefficient. Numerical simulations for a low and flattened dispersion photonic crystal fiber resonator confirm analytical predictions and open the way to experimental implementation. © 2007 Optical Society of America.Journal Articleinfo:eu-repo/semantics/publishe

Vladimirov, Andrei

Taki, Majid

DI-fusion

Control and removal of modulational instabilities in low-dispersion photonic crystal fiber cavities

Taking up to fourth order dispersion effects into account, we show that
fiber resonators become stable for large intensity regime. The range of pump
intensities leading to modulational instability becomes finite and
controllable. Moreover, by computing analytically the thresholds and
frequencies of these instabilities, we demonstrate the existence of a new
unstable frequency at the primary threshold. This frequency exists for
arbitrary small but nonzero fourth order dispersion coefficient. Numerical
simulations for a low and flattened dispersion photonic crystal fiber
resonator confirm analytical predictions and opens the way to experimental
implementation

Repositorium für Naturwissenschaften und Technik  (TIB Hannover)

Control and removing of modulational instabilities in low dispersion photonic crystal fiber cavities

Weierstraÿ-Institutfür Angewandte Analysis und Stohastikim Forshungsverbund Berlin e.V.Preprint ISSN 0946  8633Control and removing of modulational instabilitiesin low dispersion photoni rystal ber avitiesMustapha Tlidi1, Arnaud Mussot2, Eri Louvergneaux2, GregoryKozyre1, Andrei Vladimirov3, Abdelmajid Taki2submitted: 17th January 20071Optique Nonlinéaire Théorique,Université Libre de Bruxelles,C.P. 231, Campus Plaine,B-1050 Bruxelles,BelgiumE-mail: mtlidiulb.a.beGregory.Kozyreifo.es2Laboratoire de Physique des Lasers,Atomes et Moléules, UMR-CNRS 8523 IRCICA,Centre d'Etudes et de ReherhesLasers et Appliations (CERLA) FR CNRS 2416,Université des Sienes et Tehnologies de Lille,F - 59655 Villeneuve d'Asq, FraneE-mail:Arnaud.Mussotphlam.univ-lille1.freri.louvergneauxuniv-lille1.frAbdelmajid.Takiphlam.univ-lille1.fr3Weierstrass Institute for Applied Analysis and Stohastis,Mohrenstrasse 39, D - 10117 Berlin, GermanyE-Mail: vladimirwias-berlin.deNo. 1201Berlin 20072000 Mathematis Subjet Classiation. 78A60,37K45.Key words and phrases. photoni rystal bers, modulational instability, forth order dispersion.1999 Physis and Astronomy Classiation Sheme. 42.65.-k, 42.65.Wi, 42.65.Sf.Edited byWeierstraÿ-Institut für Angewandte Analysis und Stohastik (WIAS)Mohrenstraÿe 3910117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprintwias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstratTaking up to fourth order dispersion eets into aount, we show thatber resonators beome stable for large intensity regime. The range of pumpintensities leading to modulational instability beomes nite and ontrollable.Moreover, by omputing analytially the thresholds and frequenies of theseinstabilities, we demonstrate the existene of a new unstable frequeny at theprimary threshold. This frequeny exists for arbitrary small but nonzero fourthorder dispersion oeient. Numerial simulations for a low and atteneddispersion photoni rystal ber resonator onrm analytial preditions andopens the way to experimental implementation.Instabilities in non-equilibrium systems are drawing onsiderable attention bothfrom fundamental as well as from applied point of views [1, 2℄. One suh instabilitygives rise to periodi self-modulations and is referred to as Modulational Instabil-ity (MI) in temporally dispersive media [3℄ and Turing instability [4℄ in spatiallyextended systems. In optial bers, MI results from the interplay between hro-mati dispersion and the intensity-dependent refrative index. In the usual salarfree propagation, the phase mathing of the underlying four-wave mixing proessrequires anomalous dispersion [3℄. However, phase mathing an also be ahieved innormal dispersion region by onsidering extra degrees of freedom suh as polariza-tion in birefringent [5℄ and isotropi [6℄ bers, bi-modal bers [7℄, working aroundthe Zero Dispersion Wavelength (ZDW) [8, 9, 10℄ or inserting the ber within aavity [11℄. Salar MI in free propagation through a single mode optial ber isusually desribed by the Non Linear Shrödinger Equation (NLSE) in whih thepropagation onstant is expanded in a Taylor series in the frequeny domain. It hasbeen shown that only even order terms ontribute to the MI gain and that devel-opment up to the fourth order must be onsidered when the pump wavelength islosed to the ZDW. In this ase, salar MI is possible in normal dispersion regionif the fourth order dispersion term is negative and a seond frequeny of instabilityan be generated if the fourth order dispersion term is positive [8, 9, 10℄. To ourknowledge, intra-avity MI leading to a single frequeny has only been studied inrelatively strong dispersion regions where models inluding up to the seond orderdispersion term are relevant to desribe its dynamis.In this letter, we show that it is neessary to take into aount up to the fourthorder dispersion term to apture the full MI dynamis of a passive ber resonator,espeially when proeeding lose to the ZDW. To this end, we extend the modeldeveloped by Lugiato-Lefever [12℄ (LL model) up to the fourth order dispersion term.We then demonstrate that, however small the fourth order dispersion oeient1FiberLASEREEiBSR, TFigure 1: Experimental setup. BS : Beam Splitter.is, a seond frequeny of instability an be observed at the primary threshold ofstationary state destabilization whih adds to the single one predited and observedup to now [11℄. Moreover, we demonstrate that the MI proess has a nite domain ofexistene delimited by two pump power values, allowing for the stationary state torestabilize at large powers. We investigate the evolution of the MI frequenies withinthe existene domain from their rise up to their disappearane. Finally, in view ofan experimental implementation, we perform numerial simulations for a realistiexperimental onguration with a attened dispersion photoni rystal ber andnd exellent agreement with the analytial preditions.The ber resonator is shematially depited in Fig. 1. A ontinuous wave of powerE2i is launhed into the avity by means of a beam splitter, propagates inside the berand experienes dispersion and Kerr eet. At eah round trip the light inside theber is oherently superimposed with the input beam. This an be desribed by thefollowing boundary onditions E(z = 0, τ+tR) = T×Ein(τ)+R×E(L, τ)exp(−iΦ0)and by the following extended NLSE ∂zE(z, τ) = (−iβ22 ∂τ2 + β36 ∂τ3 + iβ424∂τ4 +iγ|E|2)E, with tR the round-trip time, Φ0 the linear phase shift, T 2 (R2) the in-tensity mirror transmissivity (reetivity) and L the avity length. The eletrield inside the avity is denoted E. β2,3,4 are the seond, third and fourth orderdispersion terms respetively. γ is the nonlinear oeient, z the longitudinal oor-dinate and τ the time in a referene frame moving at the group veloity of the light.This innite-dimensional map an be simplied to the following single normalizedequation by applying the mean eld approximation:∂ψ∂t′= S − (1 + i∆)ψ + i |ψ|2 ψ − iβ2 ∂2ψ∂τ ′2+B3∂3ψ∂τ ′3+ iB4∂4ψ∂τ ′4(1)where t′ = tT 2/2tR, τ′ = τ (T 2/L)1/2, ψ = E√2γL/T 2, S = 2/T (2γL/T 2)1/2Eithe normalized input eld, B3 = β3T/√9L, B4 = β4T2/12L, and ∆ = 2Φ0/T2isthe avity detuning. We arry out the analytial study in a low dispersive ber20 0.5 1 1.5 2x 1014Ω(b)2.5 0 10 20 30 40 50 60 70012345I(a)β 4=0ΩUΩCΩLI1mI2mPP2mP1mFigure 2: (a) Marginal stability urve for the steady state solution against MI.Blak urve for β4 6= 0 and grey one for β4 = 0.(b) Evolution of the avity intensitystationary state I = |ψS |2 versus the input intensity P = |S|2 (the dashed lineorresponds to the unstable ase). γ = 10W−1.km−1, Φ0 = 1.98pi, T = 0.35, L =10m, β2 = −3× 10−28s2/m, β3 = 0 and β4 = 6.4× 10−54s4/m.with a small dispersion slope. Thus, B3 an be negleted. The steady state (SS)response ψS of Eq. (1) satises SS = [1+ i(∆− |ψS|2)]ψS. This solution is identialto the one of the LL model leading to a monostable (bistable) regime for ∆ <√3(>√3). Its stability with respet to nite frequeny perturbations, i.e. of the formexp(iΩτ ′+λt′) shows that the MI frequenies that an be destabilized at the primarythreshold I1m = |ψ1m|2 = 1 areΩ2L,U =−β2 ±√β22 + 4(∆− 2)B42B4, (2)and it is immediate to see that two frequenies an be destabilized at the primarythreshold for suitable hoie of β2 and ∆. Thus, taking into aount β expansionup to the fourth order in Eq. (1) evidenes the existene of a seond frequeny ofinstability, whih had not yet been reported experimentally nor theoretially whenworking in quite strong dispersion regions [11℄.This is illustrated by the losed marginal stability urve in Fig. 2(a), where twodestabilization frequenies (ΩL and ΩU ) exist at the primary threshold I1m in themonostable regime [Fig. 2(b)℄. The nite extent of the MI domain is also evidenedby the lower and upper values of avity power |ψ1m|2 = I1m = 1 and |ψ2m|2 = I2m =(2∆eff +√∆2eff − 3)/3. The lower value xes the minimum input power requiredfor the MI proess to our, while the upper one an be tuned as a funtion of thephysial parameter ∆eff = β22/(4B4)+∆. The ritial value of the frequeny at theupper bifuration point I2m is given by Ω2c = −β2/2B4 and we note that it satisesthe averaging relation Ω2c = Ω2L+ Ω2U . This result strongly ontrasts with the usualavity modulational instability where the instability domain is not bounded as shown3on Fig. 2(a) by the gray lines. So the two main results of this stability analysis are(i) two instabilities at frequenies ΩU and ΩL our simultaneously at the primarythreshold (I1m) and (ii) it is possible to restabilize or reover the stationary stateby driving the system to the large intensity regime (I > I2m).0 1 2Maximum gain0 0.5 1 1.5 2x 1014012345ΩI(a) (b)2.5IC1IC2ΩUΩCΩLI1mI2mFigure 3: Evolution of the maximum temporal gains (blak and grey ontinuouslines) versus (a) the frequeny Ω and (b) the output intensity I.In view of the above analysis, an important question arises: how do the rst twofrequenies ΩL and ΩU evolve and onnet to Ωc upon inreasing the input intensityP = |S|2? The linear stability analysis an give us some insight on this pointthrough the evolution of the most unstable frequenies of the SS, as shown in Fig. 3.at I ≥ 1 the SS undergoes a bifuration leading to small-amplitude modulationsat frequenies ΩL and ΩU . The two orresponding bands of unstable frequenieswidens with growing I, until it reahes the value Ic1 at Ω = Ωc [Fig. 3(a)℄. Thissignals the merging of the two bands into a single, larger one. This new band ofunstable frequenies is now haraterized by the existene of three frequenies withpositive gain, as an be seen from Fig. 4(b). When further inreasing I, the twomost unstable lateral frequenies merge into the ritial one Ωc at I = Ic2. Thispoint indiates an outstanding feature leading to an exhange of the maximum gainbetween ΩL (ΩU) and Ωc [Fig. 3(b)℄. Finally, one then an expet from Fig. 3 thatabove this power value (I > Ic2), the dynamis is dominated by the frequeny Ωcuntil the upper limit of the instability domain is reahed (I = I2m).These results should be experimentally observable using a ber whose dispersionurve is low and as at as possible at the working wavelength (β3 ≈ 0). We nu-merially heked our preditions by integrating the extended NLSE with boundedonditions by using the split step Fourier method with an input ontinuous wave.We inluded realisti third and fourth dispersion order term values in our simula-tions (see aption of Fig. 2). Indeed, we did not take exatly β3 = 0 but a verylow value (DS = 0.001 ps/nm2/km i.e., β3 = 2.10−42s3/m) [13℄ to math with re-alisti onguration. We have heked in all our simulations that the nal statewas reahed (≃∼400 round trips). We show on Fig. 4(a) that two frequenies (0.984Figure 4: (a) Evolution of the frequeny of instability versus the pump power withsame parameters listed in Fig. 2 exepted for β3 = 2×10−42s3/m. Cirles numerialsimulations and full lines analytial results. (b), () and (d) power spetra for 30mW, 400 mW and 900 mW of pump power respetively.THz and 3.63 THz) are destabilized (irles) just above the rst pump threshold(20 mW) [Fig. 4(b)℄ in exellent agreement with analytial results (1.1 THz and3.6 THz). By inreasing the pump power they merge together leading to a singlefrequeny of instability arround 300 mW [Fig. 4()℄. This unique frequeny thendisappears just above the seond pump threshold orresponding to a reovering ofthe stationary state of the avity. Thus, the two main preditions of our analytialstudy are numerially veried. This linear stability analysis provides an exellentinsight of the frequeny evolution senario within the instability domain exept for50mW < I < 300mW [Fig. 4(a)℄. In this last region only a nonlinear analysis asin [14, 15℄ will gure out the dynamial evolution of the system. This work is inprogress.To summarize, we presented an analytial and numerial study of a oherently drivenphotoni rystal ber resonator. We showed that it is neessary to take into aountdispersion up to the fourth order to apture the full temporal dynamis of thesystem. Namely, there exist two frequenies at the primary MI threshold, and theirdomain of existene is nite or bounded suh that the stationary state is reoveredfor high enough intensity pumping. In addition, numerial simulations, arried outfor realisti experimental parameters, provide the evolution of these instabilities withthe input eld. They onrmed our analytial results and onstitute a step towardsa future experimental demonstration.M.T and G.K. reeived support from the Fonds National de la Reherhe Sientique(Belgium). This work was also partially supported by the Interuniversity AttrationPole program of the Belgian government. The IRCICA and CERLA are supported5in part by the Conseil Régional Nord Pas de Calais and the Fonds Européen deDéveloppement Eonomique des Régions.Referenes[1℄ M.C. Cross, and P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993).[2℄ G. P. Agrawal, Nonlinear Fiber Optis (San Diego, CA: Aademi, 1995).[3℄ K. Taï, A. Hasegawa and A. Tomita, Phys. Rev. Lett. 56,135 (1986).[4℄ A.M. Turing, Philos. Trans. R. So. London, Ser. B 237, 37 (1952).[5℄ J. E. Rothenberg, Phys. Rev. A, 42, 682, (1990).[6℄ P. Kokaert, M. Haelterman, S. Pitois, and G. Millot, Appl. Phys. Lett. 75,2873 (1999).[7℄ G. Millot, S. Pitois, P. Tehfo Dinda, and M Haelterman, Opt. Lett., 22, 1686,(1997).[8℄ S. B. Cavalanti, J. C. Cressoni, H. R. da Cruz, and A. S. Gouveira-Neto, Phys.Rev. A 43, 6162 (1991).[9℄ S. Pitois and G. Millot, Opt. Comm. 226, 415 (2003).[10℄ J. D. Harvey, R. Leonhardt, S. Coen, G. Wong, J. C. Knight, W. J. Wadsworth,and P. St. Russell, Opt. Lett. 28, 2225 (2003).[11℄ S. Coen and M. Haelterman, Phys. Rev. Lett. 79, 4139 (1997).[12℄ L.A. Lugiato and R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).[13℄ K. Hansen, Opt. Express 11, 1503-1509 (2003).[14℄ G. Kozyre, S.J. Chapman, and M. Tlidi, Phys. Rev. E 68, 015201(R) (2003).[15℄ G. 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Control and removing of modulational instabilities in low dispersion photonic crystal fiber cavities

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