Effectiveness of gain control in EDFAs against traffic with different levels of bursty behaviour

Transcription

1 Effetiveness of gin ontrol in EDFAs ginst trffi with different levels of ursty ehviour M.Krsek, A.Bononi, L.A.Rush nd M.Menif Astrt: Previous pulitions hve ddressed the impt of ursty, self-similr trffi on erium-doped fire mplifiers (EDFA). Severl gin ontrol tehniques hve een suggested to omt the gin flututions of EDFA with this type of trffi. The effetiveness of two suh methods re investigted nmely highly inverted mplifiers nd ll-optil gin lmping, while vrying the prmeters hrterising the urstiness of the soures. While previous pulitions hve foused on the effet of the verge lod or tivity ftor, the pper further investigte the dependene on the reltive vriility of the pket urst lengths (ON times), nd lengths of the interurst idle times (OFF times). Both gin ontrol methods redue the output power nd signlto-noise rtio exursions with respet to stndrd mplifier hin. The uthors find tht, for n eight hnnel WDM system nd sde of six EDFAs, ll-optil gin lmping n redue the vritions y ftor of five, provided dequte power is present in the lmping lser, while highly inverted mplifiers hve vritions redued y ftor of two. The uthors find tht, in lmped hin, the soure tivity ftor does not uniquely determine the required level of lsing power. The uthors lso dedue tht informtion on the reltive vriility of ON nd OFF times is essentil. introdution A serious prolem fing wvelength division multiplexed (WDM) networks with fire mplifier sdes is trnsient ross sturtion or gin dynmis of fire mplifiers. Attention hs een foused primrily on iruit-swithed senrios. When the numer of WDM hnnels trnsmitted through iruit-swithing network vries due to network reonfigurtion or hnnel filure, hnnel ddition/ removl will tend to pertur signls t the surviving hnnels tht shre ll or prt of the route. Although this perturtion will generlly e smll in single mplifier, it will grow rpidly long sde. Power trnsients in the surviving hnnels n use severe servie impirment due to either indequte eye opening or the pperne of optil nonlinerities [ -4, Severl solutions to this prolem hve een proposed nd experimentlly verified. These inlude fst pump ontrol [5, 6, fst link ontrol y insertion of ontrol hnnel [7] nd gin lmping y n ll-optil feedk loop [8, 9. Reent trffi mesurements from working pket networks (inluding Ethernet lol re networks, wide 0 EE, 2000 EE Proeedings online no DO:.49/ip-opt: Pper reeived 6th Jnury 2000 M. Krsek, L.A. Rush nd M. Menif re with the Centre for Optis, Photonis nd Lsers (COPL), Lv University, Deprtment of Eletril nd Computer Engineering, Quee, Cnd GK 7P4 A. Bononi is with the Universit di Prm, Diprtimento di gnegneri dell lnformzione Prm, tly M. Krisek is on leve of sene from the nstitute of Rdio Engineering nd Eletronis, Ademy of Sienes of the Czeh Repuli, Chersk 57, 82 5 Prgue EE Pro.-Optoeletron.. Vol. 47, No. 5. Otoer 2000 re networks, integrted servies digitl network, nd vrile it rte video over synhronous trnsfer mode) hve shown fetures of self-similrity: i.e. relisti pket network trffi looks the sme when mesured over time sles rnging from seonds to minutes nd hours [lo-2. t hs een onluded tht the superposition of mny ON/ OFF soures or pket trins with stritly lternting ONnd OFF- periods with infinite vrine produes ggregte network trffi tht is self-similr. When suh pket trffi is diretly trnsmitted in urst mode on the WDM hnnels, s in the se of nternet protool (P) over WDM, long interurst idle intervls my give enough time to fire mplifiers to reh gins gretly exeeding the verge vlues. This n, in turn, led to signifint vrition in output power nd optil signl-to-noise rtio (OSNR). This effet umultes long sde of fire mplifiers in the sme wy s the fst power trnsients in the iruitswithing senrio. The effet of WDM trffi sttistis on the output power nd optil OSNR swings in sde of five erium-doped fire mplifiers (EDFAs) of stndrd design, hs een theoretilly investigted in [ 3, 4. The results of the simultions indite tht sustntil power nd OSNR swings our t the output of sde when highly-vrile urst-mode trffi is trnsmitted. Power swings in exess of 9dBm nd OSNR swings of more thn 4 db were oserved. The stilistion effet of lmping the gin of the first EDFA y ll-optil feedk loop nd letting the lsing power propgte through the sde of three EDFAs hs een demonstrted experimentlly in ~5. This pper presents theoretil nlysis of output power nd OSNR flututions in sdes of six EDFAs due to the vriility of urst-mode pket-swithed trffi trnsmitted through eight WDM hnnels. The effet of trffi urstiness nd network utilistion ftor on output power nd OSNR swings hs een investigted for three 355

2 sdes: () sde of six stndrd design, two-stge EDFAs, (6) sde of two-stge, highly inverted EDFAs, nd () ll-optil gin lmped sde in whih only the first EDFA is lmped using ring lser onfigurtion nd the lsing power propgtes long the sde. The nlysis is sed on the pplition of dynmi spetrlly resolved model of n EDFA. n this pper, the uthors first solve the trnsendentl eqution for the stedy-stte vlue of totl numer of exited erium ions in the doped fire of eh mplifier. A trnsient nlysis is then performed sed on the ssumption tht the tomi popultions remin onstnt during time step 6t of severl hundreds of nnoseonds, so tht diret time evolution of signl nd ASE powers n e performed for time vrile input WDM signls. The urst-mode pket-swithed trffi is simulted y generting sttistilly independent ON nd OFF intervls (pkets) with rounded Preto distriution for eh WDM hnnel. The effet of trffi urstiness hs een investigted y vrying the prmeters of Preto distriution for oth the ON nd OFF intervls. 2 Dynmi model of EDFA sdes To study the effet of ross gin sturtion in ontented EDFAs on highly vrile pketised trffi, the uthors will investigte the time-dependent output power nd OSNR flututions in sde of six EDFAs ontented with five trnsmission fires of spn loss L = 20 db. shown shemtilly in Fig. l nd. Time-vrying signls simulting synhronous eight hnnel trffi (hnnels pled etween 547 nd 554nm with nm sping) were rndomly generted in wy similr to tht desried in [4]. To demonstrte the differenes etween the stndrd, strongly inverted nd gin lmped sdes, simultion of the highly vrile input trffi ws performed for three different sdes: Cse (): sde of six equl two-stge EDFAs of stndrd design nd gin equlising filter s shown in the inset of Fig. l (l, = 2m, PP =9mW, 2= 3m, Pp2 = 40 mw); EDFA son loss pumplser diodes OBF VA EDFA 6 Fig. Shemti / EDFA6 J Csde of six two-stge EDFAs, the inset shows single repeter. Gin lmped sde of six EDFAs: OBF=optil ndpss filter; 0 =optil isoltor, VA =vrile ttenutor; C, OC =input nd output ouplers. 356 Cse (): sde of six equl two-stge EDFAs with strongly inverted doped fiers nd gin equlizing filter (ll=lom, Pp=70mW, 2=.3m, Pp2=200mW); Cse (): sde of single-stge EDFAs with gin equlising filter. The first mplifier is gin lmped y ring lser onfigurtion, Fig. l (gin lmped EDFA: l= 30 m, Pp = 65 mw, EDFAs numers 2 to 6: l = 25 m, Pp = 49 mw). The model used for the simultions is sed on the dynmi model of EDFA derived in [6] under the ssumption of homogeneously rodened gin medium nd sene of exited-stte sorption. n ontrst to [6], the mplified spontneous emission is tken into ount in the model used in this pper, in ordne with [7]. The wvelength region from 450nm to 650nm is resolved in A4 ins of onstnt width AA. Let A denote the set of ll ASE ins, nd S denote the set of ins on whih the WDM signls, the gin stilising lsing power (in se of gin lmped sde ) nd the pump fll (the in width is smll enough suh tht, t most, one signl per in is present, loted t its entre). Dynmi ehviour of eh doped fire in the sde is desried y single ordinry differentil eqution for time evolution of the totl metstle level popultion, r(t), referred to s the reservoir, v(t) = Jg"Jr[E2(p, 4, z, t)pdrd+dz, where n,(p, 4, z, t) is the popultion density of the 4Z3,2 metstle level, l is the length of the erium-doped fire (EDF), nd p, 4, z re the ylindril o-ordintes. The eqution desriing the i ime evolution of the reservoir rk(t) t EDF numer k is given y id where z is the spontneous lifetime of the metstle level; Q$ is the signl or pump (if i E S) or externl ASE photon flux (if i E d) [ph/s] entering doped fire k t wvelength Lj orresponding to wvelength in i (Qk = Pi,,Jhvi, where hv; is the photon energy nl vi is the frequeny of in i). Signl fluxes re distinguished from ASE fluxes even when oupying the sme wvelength in, in order to e le to evlute the OSNF. t the doped fier output; Gl,,Jzk(t)) = exp(bi,krk(t) - is the gin t Ai of EDFA k, where Ai,k, Bi,k re: nondimensionl oeffiients, dependent on frequeny v, through the erium ions sorption nd emission ross-setiions: Ai,k Bi,k ''(gi + ui>/(padf) p eing the erium onentrtion [mp3], Aef the effetive re of the doped prt of the iire Te [m2],'ek is the, length of the kth doped fire [m]; nd ui = priy nd gj =prior re the sorption nd emission onstnts; Ti is the onfinement ftor t Ai lulted y the overlp integrl etween the rdil distriution of the LP,, mode intensity nd the erium ions doping distriution; 09 nd f[m2] re the sorption nd emission ross-setions t lui, respetively; nspk(rk(t)) = girk(t)/((gi + ui)rk(t) - uipae,,lk) is the spontneous emission ftor: nd the summtion in whih it ppers in eqn. represents the ASE generted inside doped fire k, the ftor 4 representing two polristion omponents for oth forwrd nd kwrd ASE; AV; is the frequeny width [Hz] of wvelength in i. When lulting the signl nd ASE fluxes t the output of EDFA k in sdes () nd (), it is neessry EE Pro-Optodetron., Vol. 47, No. 5, Otoer 2000

3 to differentite etween odd nd even mplifiers. The uthors ssume tht eh repeter is two-stge EDFA with gin equlising filter tht perfetly equlises the gin of the first stge over the signls wvelength rnge. The signl nd ASE fluxes t the output of doped fires k=, 3,..., re given s follows: Q$ (t> = { Qink (t) Gi,k (rk(t)) + 245(t)(Gi,k(rk(t)) - )AVj}F;,k i E A (2).j where Fl,k is the power trnsmission of the gin equlising filter k t wvelength Ai. For the output fluxes t even doped fires: ' Qz:t(t) = Qzlt-_,(t)Gi,k(i"k(t)) k = 2, 4,..., 2 i E S = {Q$L (t)gi,k(rk(t)) + 2ns$(t)(Gi,k(rk(t)) - l)av;} k=2,4,..., 2 ied (3) n the ove equtions note tht only hlf of the ASE generted in the EDFA, orresponding to forwrd ASE, is present t the output of the EDFA nd will e propgted downstrem. Also, the uthors impliitly ssume n isoltor is pled etween the two doped setions in eh twostge EDFA. Finlly, inditing with L the loss of the trnsmission spn etween EDFAs (i.e. etween even nd odd doped fires), the updted fluxes t the input of eh odd doped fire re s follows: Qjnk(t) = Q$lL k = 3,5,..., i E (S, A) (4) For the gin lmped sde () with six EDFAs, Fig. 6, the gin equlising filter is prt of eh mplifier, so tht eqn. 2 holds for k=,...,6. Moreover, eqn. 4 holds for EDFAs k = 3,...,6, while, for EDFA 2, Qif;(t) = Qz';'Cou'L i E {S, A} (5) Here C""' represents the oupling rtio of the output oupler (OC). The equtions desriing the optil feedk nd hrterising the input flux of the ll optil gin lmped (AOGC) EDFA t eh wvelength in i E { S, A} re given s Qjn,(t) = Qk,,i(t)Cin + QYY'(t)(l - Cou')T,p,,(l - Cin) i E (,!?,A) (6) n these equtions, Q2,,i(t) is n externl photon flux representing WDM signl entering the sde of EDFAs t wvelength Ai, whih is zero for i E {A}; Cf" is the oupling rtio of the input oupler (C), Ti is the power trnsmission of the optil ndpss filter (OBF) t wvelength Ai, nd p,, is the ttenution of the vrile ttenutor (VA). The optil ndpss filter effetively suppresses most of the ASE omponents fed k from the output to the input of the AOGC EDFA, so tht lsing tion ours with Q&(Alus) eing the lsing photon flux t the input 'of EDF numer. For the solution of the reservoir equtions t EDFAs 2,...,6 the lser flux is treted s n externl signl flux in eqn.. Numeril solution to the time-dependent eqn. is seprted into two steps: first, the stedy-stte vlue of rk for eh doped fire in the sde is determined for ontinuous-wve (CW) input signls, followed y time evolution of rk(t) with urst-mode WDM trffi t the input to doped fire numer, Qir,(t). To otin stedy-stte vlue of rk, trnsendentl eqution derived from eqn. y setting dr,(t)/dt=o is solved numerilly for k=, 2,..., strting with doped fire numer. n the se of EE Pro.-Optoeletron., Vol. 47, No. 5, Otoer 2000 the gin lmped sde (), the trnsendentl eqution for EDFA is more omplited due to the optil feedk loop implemented in this mplifier. n ddition to the stte vrile r,, the lser flux Qj:s hs to e found. An itertive solution ws used to solve the trnsendentl eqution for the stedy-stte vlue of r,, while the trnsendentl equtions for stedy-stte rk, k = 2,...,6 re solved diretly. For the dynmi nlysis, the power of eh WDM hnnel is modulted ording to n ON/OFF slotted renewl proess. Using rndom numers q., (i =, 2,...8), with uniform distriution on [0,, sttistilly independent ON nd OFF ursts (omposed of rndom integer numer of slots) were generted for eh WDM hnnel with rounded Preto distriution where yoff nd yon re prmeters inditing the urstiness of the trffi (ssumed to e the sme for ll WDM hnnels) nd x is the floor funtion. 3 Numeril results n this nlysis, the uthors onsider typil Luent Tehnologies EDF pumped t 480nm. The ASE power spetrum is tken into ount over the region from 450 to 650nm, sudivided into M=200 ins of equl width 6A = nm. The gin equlistion filter with n insertion loss of 0.5dB hs een onsidered [8]. The oupling rtios of the input nd the output ouplers of sde () were ssumed to e C'" =0.9 nd Co"'=0.5, respetively. The time slot durtion ws set to 2.82,~~ (orresponding to 53 ytes t it rte of 50Myte/s). The pkets filling the ON slots were smpled t points per slot, so tht the time step for the solution of eqn. 2 ws &=0.282p.s. Simultion of the highly vrile input trffi ws performed over more thn ten million time steps, orresponding to 3 s of trnsmission time. t ws ssumed tht '' nd '0' its within eh pket re eqully likely. The verge utilistion of the ggregte WDM trffi is given y the utilistion ftor U: where f?[ti,on], E[Ti,OFF] re the men vlues of the ON nd OFF ursts (in slot units) of individul WDM hnnels. All three sdes hve een optimised for n ggregte utilistion of 50%. The lengths nd pump powers of individul doped fire setions hve een seleted to otin EDFA gin of 20dB, for CW power t the input of the first mplifier, Pi:, = - 20 dbdhnne, in se of sdes () nd (), nd -23 dbm/hnnel, in se of sde (), to ompenste for the spn loss L=20dB nd keep the signl level onstnt long the sde. The first EDFA of the gin lmped sde () hs een designed to hve gin of 23 db to ompenste for the dditionl 3 db loss of the output oupler. The lsing wvelength ws set t ;ls = 558 nm, the EDF length e= 30 m nd pump power power Pp = 65 mw of the gin lmped EDFA give the intrvity lsing power Pi'$' = 20mW. As Put = 0.5, one hlf of this power propgtes long the sde () together with the highly vrile WDM trffi nd stilises the gin of susequent mplifiers. 35

4 3. Chrteristis of the input trffi The rounded Preto distriution used y the uthors to generte the highly vrile input trffi is hevy - tiled hyperoli distriution, very similr to the Zipf (or Zet) distriution [ 9. Mny properties of hevy-tiled distriutions re qulittively different from more ommonly enountered distriutions. f < y < 2, then the Preto distriution hs infinite vrine, nd if y then even the men is infinite. As per eqn. 7, the uthors hve used the rounded Preto distriution to generte ON nd OFF ursts of the WDM trffi. Fig. 2 ompres the theoretil vlues of the men nd vrine of the ON urst length s funtion of y ginst the vlues ttined in the uthors relistion of the rounded Preto distriution. The men nd vrine re reported in units of slots. The disrete points orrespond to simultion results, while the ontinuous urves orrespond to theoretil vlues for the men nd vrine of rounded Preto distriution, i.e. men = l/(k)') nd vrine = :=, (2k- )/(ky), where the summtions hve een rrested to N terms, where NE O7 is the totl numer of simulted time slots. Note tht the summtion for the vrine would diverge to infinity for N+ CO. t is noted tht the men of the rndomly generted ursts mthes the theory well, while the vrine of the simulted ursts is lower thn in theory, mening tht the simultion hs slightly more enign trffi thn plnned. The self-similrity of the ggregte trffi t the input to the first EDFA with independently generted ON nd OFF intervls ws tested using the ggregte vrine method (AVM) [20]. The time series X= {X,, i=, 2,...,N} representing 3 s of trffi generted y summing the instntneous input signl powers over ll eight WDM hnnels, t eh smpling point, ws divided into loks of size m. The verge power within eh lok ws determined nd new time series X(m)= {q = l/m 'r- k=, 2,..., N/m} generted. The vrine ws plotted ginst m in log-log sle. A stright of km\ line with negtive slope p greter thn - is inditive of self-similrity, nd the Hurst prmeter, H, is given y H = + j/2. For self-similr trffi with long-rnge dependene, 2 < H<. As H+, the degree of oth selfsimilrity nd long-rnge dependene inreses. Results of the AVM nlysis of 3 s of ggregte input trffi re shown in Fig. 3, for three different degrees of urstiness. The slopes of the log(vrine) ginst log(m) were estimted using lest-squres regression s = , nd , yielding estimtes for H= 0.994, - 5 m E 2 6- C B 4- E o Preto y prmeter Fig. 2 Dependene of men nd vrine of pket length on Preto distriution prmeter y Disrete points lulted from lo7 rndomly generted pkets, lines otined vrine= x;=,(zk - )/(@), N= O'. from men= xf=,l/(p), C._ 2 g g 0 L $ 80, ' 6' ' time, ms Fig. 3 Results of AVM n,vsis oj"3s of ggregte input trji AVM plot for 3 seonds of ggregte eight hnnel WDM trffi; Time evolution of ms of trffi 0 yon =., H=0.994; 0 YON =?OFF =.30, H=0.896; W YON = yoff =.80, H= nd 0.779, for yon=yoff=.,.30, nd.80, respetively. Prt of the ggregte input trffi (time series X) for 5 ms < t < 6ms is shown in Fig. 36. The proility tht extly i hnnels re simultneously ON n e esily found to e: so tht, for instne, for U =30% the proility of ll eight hnnels eing OFF the sme time is out 5%. Owing to the rndom nture of the input trffi in individul WDM hnnels, output power nd OSNR flututions used y gin ross sturtion were evluted sttistilly. Histogrms of output power nd OSNR were lulted t eh EDFA long the sdl: for ll eight WDM hnnels nd proility mss hntioii (PMF) hs een plotted s funtion of output power or OSNR. The OSNR is evluted over n optil ndwidth of nm. The effet of trffi vriility nd network utilistion on output power nd OSNR flututions hs een investigted y vrying yon, yoff prmeters of the input trffi. The spred of output power nd OSNR proility mss funtion t level of O-' hs een seleted s riterion to quntify the spred resulting from pketised ursty WDM trffi. 3.2 Csde of two-stge EDFAs of stndrd design: se () The uthors egin in Fig. 4 with the evolution of stedystte output power long sde (), when ll eight WDM EE Pro.-Optotdetron.. lhl. 47. No. 5, Otoer 2000

5 ~ EDFA lo ' EDFA numer -6, output power t 574nm, dbm t EDF numer Fig. 4 Evolution of stedy stte output power long sde () CW input power P;, = - 20 dbdhnne; nm nm -A- 547nm nm nm -U- 550nm evolution of stedy-stte reservoir t individul EDFAs -A- 55nm nm hnnels re trnsmitted with CW power t the input of the first EDFA P:l; = - 20 dbdhnne. t is seen tht the gin of eh EDFA ompenstes the loss of the trnsmission fire. Owing to gin equlising filters the interhnnel power spred etween the wekest hnnel t 55 nm nd the strongest one t 545 nm is only 0.39 db. Fig. 4 shows the orresponding vlues of stedy-stte normlised reservoir (i.e. the verge inversion) of individul EDFAs. The verge inversion is lmost onstnt for oth doped fires of eh EDFA nd equl to The plots of output power nd OSNR evluted t the output of EDFAs, 3, 5 nd 6 for the hnnel t wvelength of 547 nm re given in Figs. 5 nd 5 respetively, for input trffi with utilistion of 50 YO nd yon = =.20. The plots re otined y tiling the poweriosnr rnge of interest with ins of equl size, nd y umulting the numer of ourrenes in eh in. Finlly, the ount in eh in is divided y the totl in ount, so tht n estimted proility mss funtion (PMF) is otined. t is seen tht the men vlue of output power does not hnge long sde (), the men OSNR dereses from 22.5 db t EDFA to 8.5 db t EDFA 6. For oth the output power nd the OSNR, the flututions grow long the sde, with spreds hving proilities lrger thn of.52, 3.03, 3.68 nd 3.93dB, for output power fter repeter, 3, 5 nd 6, respetively. Spreds in EE Pro -0ptoeletron. Vol 47, No 5, Otoer OSNR t 547nm, db Fig. 5 Plots of output power nd OSNR Proility mss funtion of signl power t. = 547 nm t the output of repeter, 3, 5 nd 6: sde (), yon =yoff =.20, U = 50%; Proility mss funtion of OSNR t A = 547 nm t the output of repeter, 3, 5 nd 6: sde (), yon = yoff =.20,~ = 50% EDFA 6... EDFA 5 EDFA ~ 3 OSNR fter repeter, 3, 5 nd 6 were 0.88,.0,.43 nd.54 db, respetively. To illustrte the dynmi ehviour of individul WDM hnnels, Figs. 6 nd 6 show the PDF of output power nd OSNR t the output of EDFA 6 for hnnels t 545, 547, 549 nd 55 nm. The dynmi interhnnel power spred orresponds to the CW spred plotted in Fig. 4. Spreds in OSNR proility mss funtion of individul WDM hnnels t the level of lop2 re lmost identil. The effet of the utilistion ftor nd the urstiness of the ggregte WDM trffi re demonstrted in Fig. 7. n Fig. 7 the output power exursions verged over ll 8 WDM hnnels re plotted ginst the Preto y vlues of oth the ON nd the OFF ursts, for fixed tivity ftor U. n the Figure, three vlues for the tivity ftor hve een onsidered: U = 30% (irles), U = 50% (squres) nd U = 70% (tringles). Consider, for instne, the urves joining tringles. The urve with upwrd tringles refers to the y of ON ursts, yon, vrying from.04 to.3. For eh upwrd tringle there is n ssoited downwrd tringle tht indites the vlue of the y of OFF ursts, tht gives u=70%, s per eqn. 8. The ssoited?off vlues vry from Severl onlusions n e drwn from these grphs. Clerly, the exursions vry with the tivity ftor. Lrger exursions our with lower tivity ftor. This is due to longer times with no power eing trnsmitted on hnnel, 359

6 ~ 547nm r -2 LL -3 z output power t EDFA 6, dbm t -,,,.o gmm ftor m U -3 - z s! d lo OSNR t EDFA 6, db Fig. 6 llustrtion of the dynmi ehviour of individul WDM hnnels Proility mss funtion of signl power t the output of repeter 6 for hnnels t 2 = 547, 549, 55 nd 553 nm: sde (), yon = yoff =.20, U = 50%; Proility mss funtion of OSNR t the output of repeter 6 for hnnels t.=547, 549, 55 nd 553nm: sde (), yon=yoff=.20. u=50% nm nm 553 nm nd, therefore, more time for the reservoir to hrge nd provide higher gins. Lrger tivity ftors led to hnnels eing more densely 'ON' nd therefore leve the reservoir drined to the nominl stedy-stte vlue. However, there is lerly seond dependene on the reltive vlues of the gmm prmeters. The smller the gmm prmeter, the more ursty the trffi nd the greter vriility of ON nd OFF times reltive to their verge vlues. When gmm exeeds two, the trffi is no longer hrterised y infinite vrine. n Fig. 7 with tivity ftor 30 % nd YOFF vrying (symol 0), it n e seen tht, for fixed tivity ftor, the exursion vries steeply with the gmm ftor, wheres it vries less drmtilly with tivity ftor 70 %. n order to sort out the dependene on yon ginst yoff, nd, in order to tke into ount eqns. 7 nd 8, Fig. 7 shows plots of the output power swings t the output of sde () ginst the rtio yon/yofe.. To the right, there re lrger vlues of yon (lrger vriility in the OFF times) nd to the left lrger vlues of YOFF (lrger vriility in the ON times). The urve for tivity ftor 50 YO is vertil line t rtio one. As the gmm ftor grows from.2 (high vriility) to.8 (moderte vriility), the exursions fll s the uthors would expet. The importne of onsidering not only the tivity ftor o rtio of gmrn ftors yon/yoff Fig. 7 Plots to illustrte effet of utilistion ftor nd the urstiness of ggregte WDM trfi Output power flututions with proility greter thn lo-' s funtion of y prmeter, se (); -0- yon, U = 30% -0- yofp, U = 30% U YOFF, YON, =50% -A- yon, u=70% -0- yoff, U = 70% Output power flututions with pro.ility greter thn O-' s funtion of rtio yonyoff, se () -A- u=70% -0- u=50% -V- u=30% ut lso the level of vriility of the trffi, in order to ssess the mgnitude of the vriility, n now e ppreited. 3.3 Csde of two-stge EDFAs of strongly inverted design: se () The prmeters of the sde of strongly inverted EDFAs, se (), hve een seleted to otin stedy-stte popultion inversion in individul doped fires equl to rk =0.755, k=, 2,..., 2. This inversion is lose to the limiting vlue rlim = ~J~(A~),'(O~(A,,) + o,(ap)), whih, for Ap = 480nm, is rli,(480) = Owing to higher inversion in the doped fires, the sde is less sensitive to ursty trffi thn sde (), s is pprent from Figs. 8 nd 8, where the output power exursions with proility higher thn lo-* verged for ll eight hnnels t the output of sde () r shown in the sme wy s in Figs. 7 nd 7 for sde (). n omprison with sde (), flututions of output power nd OSNR t the end of the sde (6) re redued y ~50%. The dependene of these flututions on trffi prmeters follows the sme rules s in the se of sde (): (i) the higher the utilistion ftor, the nrrower is the spred EE Pro.-Opteletron., Vol. 47, No. 5, Otoer 2000

7 r e t,.o gmm ftor m U 5- K- 4-._ e - 0 ; ~-B;===p~-.e-v,.o gmm ftor m U - 5 Q $ y= y=.5,a-a - A-A y=.8 - v g 4-.- e 0 2 & y=l.8 y=l.5 y=.3 C,,,, o rtion of gmm ftors yon/yoff Fig. 8 Output powerjututions with proility greter thn As funtion of y prmeter, se (); -0- yoff, U = 30% -0- yon, u=30% U- YOFF, YON, = 50% -A- yon, U = 70% -V- yoff, U = 70% As funtion of rtio yon/yoff, se (6) -A- u=70% -0- u=50% -0- u=30% in PDF, (ii) smller spred in output power nd OSNR n e expeted for trffi with the sme U ut greter YOFF o rtio of gmm ftors yon/yoff Fig. 9 Output powerjututions with proility greter thn lo-' As funtion of y prmeter, se (); -0- yoff, U = 30% -0- yon, U = 30% -G YOFF,YON, U = 50% -0- yoff, U = 70% -A- YON, U = 70% As funtion of rtio yon/yoff, se () -A- u=70% -0- u=50% -V- u=30% lmost independent of the utilistion ftor nd of the y prmeters. 4 Conlusion 3.4 Gin lmped sde of single-stge EDFAs: se () The first mplifier of this sde is gin lmped using ring lser onfigurtion. Under stedy-stte onditions (eight WDM hnnels with CW input power of Pyi = - 23 dbm/hnnel), lsing power of p":! = dbm propgtes long the sde. Popultion inversion in individul mplifiers under stedy-stte onditions is equl to r, = 0.69, rk = 0.703, k= 2, 3,...,6, lmost the sme s in se (). Signl power flututions of individul hnnels used y gin ross sturtion nd ursty WDM trffi in the other hnnels re sustntilly redued due to the stilistion effet of the lsing power, whih effetively 'sors' the vriility on the input signls. Vrile urstiness of the ggregte trffi hs een nlysed for the sme set of yon, yoff nd U prmeters s for sdes () nd (). Corresponding output power flututions with proility greter thn OW2, verged over ll eight WDM hnnels, re summrised in Figs. 9 nd 9. Both the output power nd OSNR flututions re five times smller in omprison with sde () nd re EE Pro.-Optoeletron., Vol. 47, No. 5, Otoer 2000 A numeril model of sde of erium-doped fire mplifiers, sed on the solution of one ordinry differentil eqution for time-dependent gin of eh doped fire setion, hs een used to simulte the effet of gin ross sturtion on pket-swithed urst-mode trffi in WDM link. The uthors hve investigted the effet of trffi vriility on output power nd optil signl-tonoise rtio swings in three different sdes of EDFAs: () sde of six equl two-stge EDFAs, with stndrd design; () sde of six equl two-stge EDFAs with strongly inverted doped fires; nd () gin lmped sde of six single-stge EDFAs with the first mplifier gin-lmped in ring onfigurtion. An eight hnnel WDM system hs een onsidered, the input trffi on eh hnnel eing modelled s suession of ursts of pket trnsmission (ON periods) lternted with ursts of intivity (OFF periods). The pkets hd fixed durtion of 2.82ps, orresponding to 53 ytes t it rte of 50Myte/s. The ON ursts nd OFF ursts (expressed in pket units) were modelled s independent rndom vriles, eh hving hevy-tiled rounded Preto 36

8 distriution, nd the ggregte trffi exhiited selfsimilr ehviour. To pture the hevy tils of the output power nd OSNR swings, the uthors simulted out million time slots, representing 3 seonds of trffi. Output power nd OSNR flututions used y gin ross sturtion were evluted sttistilly. Normlised distriutions (PMF) of output power nd OSNR were lulted t the output of eh EDFA long the sde for ll eight WDM hnnels, nd plotted s funtion of output power or OSNR. The effet of trffi urstiness hs een investigted y vrying the prmeters of the rounded Preto distriution for oth the ON nd OFF ursts. The simultion results demonstrte tht swings in oth the output power nd noise figure n e redued y 50 YO when high popultion inversion is mintined in ll mplifiers of the sde. For sdes () nd (), the dependene of output power nd OSNR flututions on trffi prmeters follows the sme rules: (i) the higher the utilistion ftor, the nrrower the spred in PMF; (ii) smller spred in output power nd OSNR n e expeted for trffi with the sme U ut greter yoff. n omprison with n unlmped sde of six onventionl EDFAs, the vrition in output power nd OSNR t the proility level of lo-*, t the end of ginlmped sde (), is redued y ftor of five if the lsing power is only db ove the ggregte signl power. n ontrst to sdes () nd (), output power nd OSNR flututions re lmost independent of network utilistion ftor nd of y prmeters determining the urstiness of the ggregte trffi. 5 Referenes SUN, Y., SRVASTAVA, A.K., ZYSKND, J.L., SULHOFF, J.W., WOLF, C., nd TKACH, R.W.: Fst power trnsients in WDM optil networks with sded EDFAs, Eletron. Lett., 997,33, pp ZYSKND, J.L., SUN, Y., SRVASTAVA, A.K., SULHOFF, J.W., LUCERO, A.J., WOLF, C., nd TKACH, R.W.: Fst power trnsients in optilly mplified multiwvelength optil networks. Pro. OFC 96 Tehnil Digest, Sn Jose, CA, Pper TD KARASEK, M.: Fst power trnsients in ontented PG-doped fluoride fier mplifiers, Lightwve Tehnol., 998, 6, pp HAYEE, M.., nd WLLNER, A.E.: Trnsmission penlties due to EDFA gin trnsients in ddidrop multiplexed WDM networks, EEE Photonis Tehnol. Lett., 999,, pp SRVASTAVA, A.K., SUN, Y., ZYSKND, J.L., nd SULHOFF, J.W.: EDFA trnsient response to hnnel loss in WDM trnsmission system, EEE Photonis Tehnol. Lett., 997, 9, pp KARASEK, M., nd VAN DER PLAATS, J.C.: Anlysis of dynmi pump-loss ontrolled gin-lo king system for erium-doped fier mplifiers, EEE Photonis Tehnol. Lett., 998,, pp SRVASTAVA, A.K., ZYSKND, J.L., SUN, Y., ELLSON, J., NEWS- OME, G., TKACH, R.W., CHRAPLYVY, A.R., SULHOFF, J.W., STRASSER, T.A., WOLF, C., nd PEDRAZZAN, J.R.: Fst-link ontrol protetion of surviving hnnels in multiwvelength otil networks, EEE Photonis Teiinol. Lett., 997, 9, pp DA, H., PAN, J.Y., nd LN, (2.: All-optil gin ontrol of in-line erium-doped fier mplifiers for hyrid nlog/digitl WDM systems, EEE Photonis TehJiol. Lett., 997, 9, pp RCHARDS, D.H., JACKEL, J.L., nd AL, M.A.: Multihnnel EDFA hin ontrol: A omprison of two ll-optil pprohes, EEE Photonis Tehnol. Lett., 998,, pp BERAN, J., SHERMAN, R., TAQQU, M.S., nd WLNGER, W.: Long-rnge dependene in vrile-it-rte video trffi, EEE Puns. Conzmun., 995, 43, pp WLNGER, W., TAQQU, M.S., SHERMAN, R., nd WLSON, D.V: Self-similrity through high-vriility: Sttistil nlysis of Ethernet LAN trffi t the soure level. EEE/ACM Trns. Netw., 997, 5, pp CROVELLA, ME., nd BESTAVROS, A.: Self-similrity in World Wide We trffi: Evidene nd possile uses, EEE/ACM Puns. Netw., 997,5, pp BONON, A., TANCEVSK, -., nd RUSCH, L.A.: Lrge power swings in doped-fier mplifiers with highly vrile dt, EEE Photqnis Tehnol. Lett., 998,, pp TANCEVSK, L., BONON, A., nd RUSCH, L.A.: Output power nd SNR swings in sdes of EDFAs for iruit- nd pket-swithed optil networks, J Ligthwve Tehnol., 999, 7, pp BONON, A., POT, L., nd.4zzn, A.: Mesurement of power spred histogrms in hins of EDFAs fed y multimedi urst-mode pket trffi. Pro. Europen Conf. Opt. Commun., 999, Nie, pp SUN, Y., LUO, G., ZYSKND, J.L., SALEH, A.A.M., SRVASTAVA, A.K., nd SULHOFF, J.W.: Model for gin dynmis in erium-doped fire mplifiers, Eletron. Lett., 996, 32, pp GEORGES, T., nd DELEVAQUE, E.: Anlytil modeling of highgin erium-doped fier mpliliers, Opt. Lett., 992, 5, pp. 3-5 ROCHETTE, M., GUY, M., LAROCHELLE, S., LAUZON, J., nd TREPANER, F.: Gin equliztion of EDFAs with Brgg grtings, EEE Photonis Tehnol. Lett., 999,, pp ROSS, S.: A first ourse in proility (Prentie Hll, 998, 5th edn.) TAQQU, M.S., TEVEROVSKY. V, nd WLLWGER, W.: Estimtors for long-rnge dependene: n empiril study, Frtls, 995,3, pp EE Pro.-Optodetron.. Vol. 47, No. 5, Otoer 2000