Modelocking pulse dynamics ina �ber laser witha saturable Bragg re ectorJ. Nathan KutzProgram in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544,and Bell Laboratories, Lucent Technologies and AT&T Research, Murray Hill, NJ 07974Brandon C. Collings and Keren BergmanDepartment of Electrical Engineering and Advanced Technology Center for Photonics andOpto-Electronic Materials, Engineering Quadrangle, Princeton University, Princeton, NJ 08544Sergio Tsuda, Steven T. Cundi� and Wayne H. KnoxBell Laboratories, Lucent Technologies, 101 Crawfords Corner Road, Holmdel, NJ 07733Philip Holmes and Michael Weinstein*Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544to appear: the Journal of the Optical Society of America BA theoretical model is developed for the pulse dynamics in a �ber laser modelocked by a saturableBragg re ector (SBR) and operating in regimes beyond the scope of the master modelocking equa-tion. An asymptotically valid modelocked evolution equation is derived which includes a heuristicmodel for the SBR dynamics. The model employed allows, for the �rst time to our knowledge,to directly compare (with no free parameters) theoretical predictions of the pulse spectral andtemporal pro�les with experimental results in both the normal and anomalous dispersion regimes.Extensive numerical simulations of the governing evolution equation, an averaged equation, andanalytical solutions are found to be in excellent agreement with experimental results.1. INTRODUCTIONCompact sources of optical pulses near wavelengths of1.55 microns are key enabling technologies for high speed�ber optic communication systems and interconnectionnetworks. Erbium-doped modelocked �ber lasers, cur-rently the subject of much research, provide a poten-tially attractive short pulse source1. Various �ber lasershave been demonstrated using passive modelocking tech-niques operating in both the normal and anomalous dis-persion regimes. Passive modelocking has been success-fully achieved by several groups employing additive-pulsemodelocking (APM) that converts nonlinear self-phasemodulation (SPM) to ultrafast amplitude modulationvia an interferometer in the �gure eight laser2{4 or bypolarization rotation in the ring laser5;6. The modelock-ing force of APM is in essence equivalent to a fast sat-urable absorber. Fiber lasers have also been passivelymodelocked in a linear cavity con�guration that em-ploys a semiconductor structure for the fast saturableabsorber7;8.A recently developed semiconductor saturable ab-sorber device at Lucent Technologies incorporates anepitaxially grown pair of InGaAs quantum wells anda low loss Bragg re ector structure to produce a sat-urable Bragg re ector (SBR)9. This nonlinear mir-ror has been used to passively modelock Cr+4:YAG,Ti:sapphire, Cr:LiSAF, and Erbium-doped �ber lasersnear 1.55 microns8;10. Similar ultrafast semiconductorsaturable absorbers have been developed that incorpo-rate high and low �nesse Fabry-Perot structures and suc-cessfully used to passively modelock lasers at other wave-lengths11. The advantage of semiconductor saturable ab-sorbers over other passive modelocking techniques suchas Kerr lens (KLM) or APM is their extremely low sat-uration energies, enabling e�cient femtosecond mode-locking of low gain coe�cient or short gain length lasers.Erbium-doped �ber lasers modelocked with the SBRhave achieved higher repetition rates (by factors of 3 ormore) than APMmodelocked �ber lasers in a single pulseper round-trip con�guration. These results are promis-ing for high-speed networks that require broadband andstable sources.Although signi�cant strides have been made towarda complete description of femtosecond modelocking insolid state lasers1;12, thus far passive modelocking in a�ber laser has been studied with Haus' master mode-locking equation13{16 or as a soliton system under per-turbation17{19. The master equation is derived underthe condition that nonlinear changes to the intracavitypulse must be small per round-trip. In the soliton per-turbation approach, the pulse dynamics in the cavity are1

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Nonlinear, Gain-Saturated Fiber Propagation (Eq. (1))Fig. 1. Schematic of modelocking �ber laser which in-cludes a 1% output coupler, a saturable Bragg re ector(SBR), and a gain-saturated �ber segment.assumed to deviate by small factors from the soliton solu-tion. In many modelocked �ber lasers, these two under-lying conditions do not necessarily hold as the nonlinearphase shift can be as high as several � per round-trip,and the pulse can signi�cantly deviate from the funda-mental soliton solution. This limit has been previouslyconsidered in solid state lasers, where the master mod-elocking equation proved inadequate in describing themodelocking due to the strong nonlinearity and disper-sion1;12;20;21. The e�ects of dispersion on modelocked�ber lasers have also been extensively investigated ex-perimentally and numerically22.In this paper we present theoretical and numericalstudies of a �ber laser modelocked by the SBR. Themodel, unlike the master modelocking equation, is capa-ble of treating signi�cant intracavity pulse changes perround trip. A description of the pulse interaction withthe SBR is also included. To our knowledge, this is the�rst time a direct comparison is made between numericaland analytical results of a theoretical �ber laser modelwith experimental results using no free parameters. Thecomparison is made for �ber lasers operating in boththe normal and anomalous dispersion regimes. Based onthe excellent agreement between theory and experiment,this model provides a sound basis for understanding thecontributions of various physical mechanisms, i.e., dis-persion, nonlinearity, saturable absorption, etc., to themodelocking dynamics.Our model begins with the nonlinear Schr�odingerequation for propagation in �ber and applies to caseswhere the nonlinearities per round-trip are large. Wenumerically solve the full propagation problem, includ-ing the leading-order dynamics of the SBR as a jumpcondition imposed every round-trip. The SBR is mod-elled with three loss factors which include its intrinsicloss, its fast and slow (saturating) responses, and a timeconstant to characterize the relaxation of the latter. Theanalytical model for the temporal response of the SBRmatches closely experimental measurements obtained viapump-probe experiments. Thus, the modelocking modelprovides a means to accurately predict the output tem-poral and spectral pulse pro�les obtained experimentally.In addition, we derive an averaged model that capturesthe SBR dynamics and which may be solved analyticallyin a particular asymptotic regime.

The paper is organized as follows. In Section 2 theformulation of the general modelocking equation is in-troduced. Section 3 describes the SBR model which isutilized in determining the appropriate averaged evolu-tion equation described in Section 4. Section 5 considersanalytic solutions to the averaged equation. In Section6 the experimental results are presented and comparedwith numerical and analytic predictions. Conclusionsare provided in Section 7.2. FORMULATIONA pulse propagating in an optical �ber under the in u-ence of the Kerr nonlinearity and including the e�ects ofdispersion, loss, and a parabolic gain bandwidth pro�lebehaves according to the modi�ed nonlinear Schr�odinger(Ginzburg-Landau) equationi@Q@Z + D2 @2Q@T 2 + �jQj2Q+ i�Q (1)� iG(Z)�1 + � @2@T 2�Q = 0 :Here Q is the normalized electric �eld envelope andD;�; � and � denote the normalized dispersion, non-linearity, gain bandwidth and intrinsic loss respectively.The variable T represents the physical time in the restframe of the modelocked pulse divided by an arbitraryreference time T0 = 1 ps, and the variable Z representsthe physical distance divided by the cavity round-triplength Zcav. These scalings give the following parame-ter de�nitions D = Zcav�20 �D2�cT 20 (2a)� = 2�n2jE0j2Zcav�0Aeff (2b)� = 12T 20 (2c)� = ~ Zcav; (2d)where n2 = 2:6 � 10�16 cm2/W is the nonlinear coe�-cient in the �ber, jE0j2 = 1 kW is an arbitrary powerscaling, Aeff = 60 �m2 is the e�ective cross-sectionalarea of the �ber, ~ = 0:0230 km�1 which correspondsto a power loss rate of 0.20 dB/km, is the full-widthhalf-maximum (FWHM) bandwidth of the �ber ampli-�er, and �0 = 1:55 �m and c are the free-space wave-length and speed of light respectively. The parameter�D (in ps/km-nm) gives the average normal ( �D < 0) oranomalous ( �D > 0) dispersion in the laser cavity. Notethat this model di�ers from the experimental set up forwhich up to three �ber segments are utilized with di�er-ing dispersion values and e�ective cross-sectional areas(see Sec. 6, Fig. 5). Here we have incorporated the e�ectsof all three �ber segments into a single �ber medium (SeeFig. 1). Although corrections to the average dispersioncan be included, numerical and analytical23{25 results

Kutz et al., Modelocking with a Saturable Bragg Re ector 3suggest that they can be neglected with only a minimalamount of error incurred.The gain in the �ber is incorporated through the di-mensionless parameter G = G(Z). We model the satu-rated gain in the �ber15 by lettingG(Z) = 2G01 + kQ(Z)k2=Esat ; (3)where Esat is the saturated energy of the �ber, G0 isthe saturated gain, and kQ(Z)k2= R1�1 jQ(Z)j2dT is thecurrent energy.3. SATURABLE ABSORBER DYNAMICSA comprehensive model of the SBR dynamics requiresa description of the microscopic interactions betweenthe intracavity radiation and the quantum well struc-ture, and consequently, extensive numerical modeling26.Here we simply incorporate the modelocking action ofthe SBR via an empirical model of its nonlinear tempo-ral response. The SBR structure, illustrated in Fig. 2,consists of a broadband AlAs/GaAs Bragg re ector andtwo uncoupled InGaAs/InP quantum wells separated by7 nm and located 15 nm from the top surface of the sam-ple. The complete structure has a �99.5% saturated re- ectivity at 1.55 �m with a bandwidth of �150 nm, asshown in Fig. 2.Since the optical �eld penetrates through only sev-eral microns in the SBR structure, we ignore the e�ectsof the material dispersion in the Bragg re ector, andwe model the steady-state SBR nonlinear re ectivity re-sponse as a jump condition attached to the propaga-tion equation (1). The model includes a \fast" compo-nent representative of the virtual carrier transitions (ACStark) and a \slow" component representative of the realtransitions that incorporates a saturated term and a slowrelaxation time27;28. Thus, the combined response canbe described by the jump conditionQ+ = �1� �l � �f �1� jQ�j2jQ�j2max�� �s 1� R T�1 jQ�j2dTkQ�k2 (4)� exp��H(T � Tmax)T � TmaxTd ���Q�;= f(jQj)Q�where Q� denotes the pulse before ({) and after (+) theSBR interaction, Td is the decay time of the SBR slowresponse which is normalized on T0, �l is the intrinsicloss of the SBR, �f measures the relative strength of theinstantaneous SBR response and �s the relative strengthof the corresponding slow saturation response. Note thatjQ(Tmax)j = jQjmax gives the power maximum and itsrelative position, and H(T �Tmax) is the standard Heav-iside function29 for which H(T � Tmax) = 0 for

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Fig. 2. Saturable Bragg Re ector (SBR) structure andits associated re ectivity spectrum.T�Tmax < 0 and H(T�Tmax) = 1 for T�Tmax > 0. Thejump condition (4) is a phenomenological idealization ofthe steady-state interaction dynamics of the SBR withthe electric �eld. Physically, the SBR is capable only ofresponding to the instantaneous power rather than theintegrated power (kQk2) and maximum power (jQj2max)as per Eq.(4).In addition to the convenient mathematical descrip-tion, Eq. (4) provides a qualitatively correct leading-order model for the SBR dynamics. Thus the SBR ac-tion is localized to the temporal window of the pulse,i.e., far from the pulse the radiation experiences only a�2% absorption. This is in contrast to the master mode-locking equation13 and soliton modelocking models19 forwhich radiation far from the localized pulse can experi-ence nearly �100% absorption.The e�ect of the SBR can be illustrated by consideringits action on a speci�c pulse. In particular, we considerthe case for which Q� = � sechwT exp(i�(Z; T )). Plug-ging this initial ansatz into Eq. (4) gives the appropriatechange in re ectivity of the SBR. In the case of a fastresponse acting alone (�s = 0; �l = 0), the change ofre ectivity of the SBR is given by�f = 1� f(jQj; �s = �l = 0) = �f tanh2wT: (5)Similarly, the case of the slow response acting indepen-dently can be considered (�f = 0; �l = 0) and is foundto give

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Fig. 4. Comparison of the experimental pump and probemeasurements of the SBR at 1.55 �mwith the jump conditionof Eq. (7) which includes a fast response and slow saturationsresponse with relaxation. Here, a � 400 fs hyperbolic secantpulse is assumed for Q� and �f = �s = 0:65� with Td = 14.�s =1� f(jQj; �f = �l = 0) = �s2 (2� (6)(1 + tanhwT ) exp(�H(T )T=Td)) :The fast and slow changes in re ectivity are plotted inFigs. 3a and 3b respectively with w = 1:76, which corre-sponds to a 1 ps FWHM pulse, and Td = 14 so that theSBR relaxation time is 14 ps.In general, the SBR action will be some combinationof the fast and slow responses which depends primar-ily upon the relative position of the quantum well ex-citon peak. In practice the quantum well structures,and its associated exciton peak, are designed around thedesired wavelength of modelocking operation. Time re-solved pump-probe measurements of the re ectivity havebeen made on the SBR sample at 1.55 microns. In thiscase it is observed that both a slow and fast response arepresent so that for a (chirped) hyperbolic secant pulsewe havef(jQj) = 1� �l � �f tanh2wT (7)� �s2 (2� (1 + tanhwT ) exp(�H(T )T=Td)) :

We can compare this simplemodel with the experimental�ndings obtained via pump-probe measurements. Fig-ure 4 depicts the comparison between experiment andthe analytic results of Eq. (7) where w is chosen to matcha �400 fs FWHM pulse, Td = 14 (a 14 ps decay time),�l = 0:005 for a saturated re ectivity of 99.5%, and�f = �s = 0:65� where � is the maximum change inre ectivity of the SBR. Note the excellent agreement inFig. 4 between the simple analytic model given by Eq. (4)and the experimental results. We estimate from experi-ments that the maximum change in re ectivity is � 2%.This gives � = 2% and �f = �s = 1:3%. These parame-ter values for the SBR will be used in the numerical andanalytical calculations that follow.Before concluding this section, the range of validityof Eq. (4) should be discussed. We can expect the sim-ple jump condition (4) to model the SBR dynamics onlyfor well-behaved, near steady-state modelocking, i.e., asingle pulse per round trip con�guration. When multi-ple pulses or radiation pedestals appear, a more detailedunderstanding of the SBR dynamics will most likely berequired.4. AVERAGED EVOLUTIONIn the previous two sections, the evolution dynamics ofthe pulse in the �ber and the leading order e�ect of theSBR were considered. In this section we incorporate theSBR dynamics into the propagation equation (1) for thepulse evolution in the �ber. The model derived is notsimply a heuristic or qualitative model, but rather anasymptotically valid model for the nonlinear pulse dy-namics. Thus, a direct comparison between the averagedevolution, which resembles the usual master modelock-ing equation (Ginzburg-Landau equation), the full dy-namics given by Eqs. (1), (3), and (4), and experimentcan be made. Finally, it should be noted that averagingdoes not require the self-phase modulation to be smallper round trip. Rather, the physically relevant case of adominant nonlinearity (SPM) can be addressed.The averaged evolution incorporates the e�ect of theSBR and output coupler by implementing them contin-uously over the �ber segment. Since the SBR and theoutput coupler act as quasi-linear and linear loss termsrespectively, we can incorporate them into the propaga-tion equation (1) via the simple linear model@Q@Z = � ZcavQ: (8)Here we neglect the e�ects of dispersion, nonlinearity,bandwidth limited �ber gain, and intrinsic �ber lossesand once again scale the physical distance on Zcav. Thesolution to Eq. (8) is Q(Z; T ) = Q(0; T ) exp(�Zcav Z).In order to make use of Eq. (8) and its solution,we consider the propagation over a single round trip.We then equate the round trip dynamics with an ef-fective jump condition and compare this with Eq. (4).Thus after a single round trip, Z = 1 and we �nd that

Kutz et al., Modelocking with a Saturable Bragg Re ector 5Q(1; T ) = Q+ = Q(0; T ) exp(�Zcav) = Q� exp(�Zcav).Comparing the above relationship between Q+ and Q�with Eq. (4) givesexp(�Zcav) = f(jQj): (9)Solving for � in Eq. (9) yields �Zcav = log(f(jQj)). Wecan further approximate �Zcav by noting that log(1 +�) = � + O(�2) where � � 1. Since �l, �f , �s � 1, itfollows from linearization that�Zcav � ��l � �f �1� jQj2jQj2max�� �s�1 (10)� R T�1 jQj2dTkQk2 exp��H(T � Tmax)T � TmaxTd �! ;with an error which is O(�2f ; �2s; �2l ).Similarly, the output coupler can be incorporated intoEq. (8) by simply noting that the appropriate jump con-dition there isQ+ = RQ� (11)where R is the re ectivity of the output coupler. Thus,the contribution from the output coupler gives�Zcav � (1 � R): (12)which is derived again by comparing the relation betweenQ+ and Q� and noting that (1 �R)� 1.By making use of Eq. (8), Eqs. (10) and (12) can beincorporated into the propagation equation (1) to givean averaged evolution equation for the pulse dynamicsi@Q@Z + �D2 � i�G(Z)� @2Q@T 2 + i( � G(Z))Q+ ��� i �fjQj2max� jQj2Q� i�s R T�1 jQj2dTkQk2 (13)� exp��H(T � Tmax)T � TmaxTd �Q = 0;where = � + 1 � R + �l + �f + �s. Equation (13) issimilar to the master modelocking equation. However,there are signi�cant di�erences due to the addition of theslow response with relaxation, the fast response scaling�f=jQj2max, and the contribution of linear parts in pro-portional to �l, �f and �s.Equation (13) is most accurate under the conditionthat the pulse amplitude and shape does not change sig-ni�cantly per round trip. Since the e�ects of dispersionand nonlinearity can distort the pulse pro�le consider-ably, we would ideally require Zcav=Z0 � 1. This condi-tion is easily met for the picosecond modelocked pulsesof the normal dispersion regime (Zcav=Z0 � O(10�3)).However, ultrashort pulses are generated in the anoma-lous dispersion regime for which Zcav=Z0 � 0:5. Despitethis, the agreement between the averaged evolution andfull equations is remarkable. In addition, the conditions�f ; �s; (1�R)� 1 must hold in order for Eqs. (10) and(12) to apply. We note that these results di�er from theprevious derivation of the master modelocked equation15for which all the parameters are required to be much lessthan unity per round trip.

5. PULSE SOLUTIONSEquation (13) admits exact solutions in certain asymp-totic parameter regimes. For the case in which the SBRresponse has no resonant (slow) response, i.e. �s = 0,chirped solutions take the form30Q(Z; T ) = �[sechwT ]1+iA exp(i�(Z)) ; (14)where the parameters � and � are determined from thevalues of w and A byd�dZ = 12Dw2(1� A2) + 2�G0w2A (15a)�2 = w22� �D(2 �A2) + 6�G0A� ; (15b)and the parameters w and A satisfyDw2A� (�w2(1� A2) + 1)G0 + = 0 (16a)3Dw2A� 2�G0w2(2� A2) + 2�f = 0 : (16b)Solving the system given by Eq. (16) gives the quadraticequation for A:A2(�f + B) +A D�G0 ��f + 32B�� (�f + 2B) = 0;(17)where B = G0 � , and whose solution isA� = 1�f +B �� D2�G0 ��f + 32B� (18)� s D24�2G20 ��f + 32B�2 + (�f + B)(�f + 2B) 1A :Once A is determined we �ndw2� = 2�f2�G0(2� A2�)� 3DA� ; (19)and � and � are determined from Eq. (15). Although twosolution branches are given by Eq. (18), these solutionsmust be self consistent in that the right hand sides ofboth Eq. (15b) and (19) must be real and positive. Itshould be noted that the chirped solution (14) of theGinzburg-Landau equation arises in a wide variety ofmodelocking con�gurations15;22;30.We will not make parameter regime studies here. How-ever, we will point out that nothing at this point can besaid concerning the stability of the chirped hyperbolicsecant pulse solutions. A stability analysis of the pulseevolution requires a linearization about the chirped solu-tion. This has been explored numerically for the mastermodelocking equation31, but remains an open questionanalytically. These issues are currently under investiga-tion and will be addressed elsewhere.We also consider the case for which �s 6= 0. In thiscase, there is no longer an exact solution of Eq. (13).However, in the physically relevant limit for which �s �

Kutz et al., Modelocking with a Saturable Bragg Re ector 61, the slow response term can be treated as a perturba-tion to the �s = 0 case. In particular, we examine thecase for which there is no relaxation (e.g. Td = 1) andconsider the travelling wave solutionQ(Z; T ) = �[sechw(T � vZ)]1+iA exp(i�(Z)): (20)As in the previous case with �s = 0, the parameters�; w; � and A satisfy Eqs. (15) and (16), with the addi-tional restriction now thatv(1 + iA) = �s2w (1 + cothw(T � vZ)): (21)Equation (21) cannot be satis�ed unless v = 0 and �s =0, i.e., the case of no slow response considered previously.However, for the case in which v � 1 and we considerthe T � vZ � 1 limit we �ndv � �sw : (22)So although no solutions exist, we obtain a crude esti-mate for the leading order drift of the pulse.In addition to the approximate travelling wave solu-tion found above, care must be taken when comparingthe analytic solution with the numerical results of thefull equation, which includes the slow response and re-laxation. In particular, the value of �s in the de�nitionof must be modi�ed in order to account for the e�ec-tive loss due to the slow response. Simply setting �s = 0gives pulse solutions which are much too large becausethe value ofG0 calculated numerically takes into accountthe loss due to a nonzero �s. The e�ective slow responsecan be found by averaging the slow response in Eq. (6):��s = �s2(t1 + t2) Z t2�t1�2� (1 (23)+ tanhwt) exp��H(t)tTd �� dt � 0:65�s:Here the limits of integration, t1 and t2, are chosen so asto capture the localized structure of the slow responseand relaxation near the pulse (see Fig. 7). Note thatif no relaxation were included, Eq. (23) would simplyintegrate to ��s = 0:5�s.6. NUMERICS AND EXPERIMENTSIn order to verify the analytic predictions made in thepreceding two sections, extensive numerical simulationsare described in this section. The numerical methodutilized employs a fourth-order Runge-Kutta method inspace (Z) and a �ltered pseudospectral method in time32(T ). This procedure combines the advantages of split-step33 and explicit Runge-Kutta34 methods, giving a rel-atively simple fourth-order scheme with improved nu-merical stability.In the simulations that follow, we chose an unchirpedhyperbolic secant pulse as the initial condition:

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SMFFig. 5. Experimental setup of modelocking laser cavitywith a 1% output coupler, SBR and average dispersion in theanomalous regime. The 980 nm pump is coupled through awavelength division multiplexer (WDM) into the laser cavitywhich consists of three �ber segments: a single mode �ber(SMF), an Erbium-Ytterbium �ber (Er/Yb) and �nally adispersion shifted �ber (DSF).Q(0; T ) = �0 sech(1:76w0T ): (24)Here �0 and w0 are the initial pulse amplitude and theinverse of the initial FWHM width respectively. We setw0 = 0:1 which corresponds to a 10 ps pulse. The initialamplitude �0 is then determined by the initial energy inthe cavity. It should be noted that these initial condi-tions are arbitrary as we have observed numerically thata very wide range of pulse-like initial conditions mode-lock to the same steady-state pulse solution.We begin by considering the modelocking pulse dy-namics in the normal dispersion regime. For this case,the experimental laser cavity consists of three �ber seg-ments of length 13 cm, 19 cm, and 95 cm with dis-persion values of 17.0 ps/km-nm, -9.1 ps/km-nm and-51.6 ps/km-nm respectively (see Fig. 5). Thus the cav-ity length is 127 cm (Zcav=254 cm) with an averagecavity dispersion of �D � �38 ps/km-nm. It should benoted that the - 9.1 ps/km-nm (19 cm) segment is theErbium-Ytterbium gain �ber. The saturated intracav-ity power measured experimentally is Esat=21.3 mW sothat a value of G0 = 0:03256 is required. Finally, we as-sume the Erbium-Ytterbium �ber has a 25 nm FWHMgain bandwidth and take �0 = 0:1144 so that the initialenergy is half the saturated energy of the cavity.We begin by considering the full governing evolutionas given by Eqs. (1), (4) and (11). In Fig. 6 we show theevolution over 1000 round trips of the �ber cavity. InFig. 6a we observe that the pulse settles to a �13.5 ps(FWHM) pulse which is consistent with experimentalestimates. It is very interesting to note the evolution ofthe spectrum in Fig. 6b as it evolves into a square shapein the frequency domain. This supports the notion of theformation of chirped-pulse structures alluded to in theprevious section. The evolution of the pulse as governedby the averaged equation (13) gives similar quantitativeresults.In order to test the validity of the proposed evolutionmodel and its various approximations, we compare themodel predictions directly with experimental results. InFig. 7 we compare the experimental output pro�les withthe steady-state (numerical) solution of the full govern-ing equations (Eqs. (1), (4) and (11)), the averaged equa-tion (Eq. (13)) and its exact solution (Eqs. (14)-(19) with

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(b)Fig. 6. Numerical simulation of the pulse evolution asgoverned by Eqs. (1), (4), and (11) in both the time (a) andfrequency (b) domain for a �ber cavity with an average nor-mal dispersion value of �D=-38.1 ps/km-nm. In (a) the pulseis seen to modelock to a � 13:6 ps chirped pulse whose spec-tral evolution is given by (b).�s replaced by 0:65�s as per (23)). Note the excellentagreement among all four. In the time domain, for whichwe do not have an experimental autocorrelation trace,the pulse width is estimated to be just over 10 ps. Thiscompares favorably with the theoretical predictions of�13.5 ps.In this case, the parameters for the analytic solutiongiven by Eq. (14) are calculated using Eq. (23) so that��s = 0:65�s = 0:00845 and = �+1�R+�l +�f + ��s.Plugging in our parameters values yields �+ = 0:1393,w+ = 0:1128 and A+ = 11:5204 (the second solutionbranch of Eq. (18) is not self consistent in that it yieldsimaginary values for w� and ��). This con�rms thevalidity of the asymptotic approximations made in de-riving the averaged equation (13) and further helps toilluminate the usefulness of the analytic solution givenby Eq. (14) in determining the range of operation of the�ber laser even when the slow response with relaxationis present.

Similarly, we can consider the case for which the av-erage dispersion in the �ber laser is in the anomalousregime. Again the laser cavity is comprised of three�ber segments of length 24 cm, 17 cm and 49 cm withdispersion values of 17.0 ps/km-nm, -9.1 ps/km-nm and17.0 ps/km-nm respectively. The cavity length is then90 cm (Zcav=180 cm) with an average dispersion valueof �D � 12 ps/km-nm. In this case the output couplingis 2% and the intracavity power is estimated from mea-surements to be � 10 mW. Here we chose �0 and w0 sothat initially we have a 1 ps pulse with the exact initialenergy of the saturated cavity. In addition, we take an

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Fig. 7. Comparison of the experimental results with�D �-38 ps/km-nm and the full evolution equations (Eq. (1),(4) and (11)), the averaged equation (Eq. (13)), and the ap-proximate solution with no slow response (Eqs. (14)-(19)).Note the remarkable agreement between all three models andthe experimental results. The 13.5 ps pulse width is consis-tent with experimental observation.

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(b)Fig. 8. Numerical simulation of the pulse evolution asgoverned by Eqs. (1), (4), and (11) in both the time (a)and frequency (b) domain for a �ber cavity with an aver-age anomalous dispersion value of �D �12 ps/km-nm. In (a)the laser is experimentally seen to modelock to a � 420 fschirped pulse with a time-bandwidth product of � 0:4 andwhose spectral evolution is given by (b).e�ective �ber bandwidth (FWHM) of � 45 nm. Al-though this bandwidth is broader than that of the 20-25 nm Erbium-Ytterbium �ber, it is a reasonable as-sumption given that the pulse propagates a majority ofthe time in essentially bandwidth unlimited �ber. Thusthe larger bandwidth allows the SPM to have a strongerbroadening e�ect in the �ber cavity. All other parame-ters, including �f = �s = 1:3% and �l = 0:5%, remainas in the normal dispersion case.In Fig. 8 we show the evolution as given by Eqs. (1)(4) and (11) over 1000 round trips of the �ber cavity forthe parameter values given in the previous paragraph. InFig. 8a we observe that after several hundred round trips,the pulse settles to a � 500 fs (FWHM) pulse. In con-trast with the spectral evolution of the normal dispersioncase, the spectrum in Fig. 8b evolves into a broadbandpulse in the frequency domain. The experimental time-bandwidth calculation yields a value of � 0:4 so that

the pulse is nearly bandwidth limited (i.e., A � 1 inEq. (14)). The evolution of the averaged equation (13)gives similar quantitative results as those in Fig. 8.The validity of the proposed evolution model and itsvarious approximations are once again tested by compar-ing predictions directly to experimental results. We cancalculate the analytic solution of the averaged equation(Eqs. (14)-(19) with (23)) given the appropriate param-eters. Calculating the chirped soliton parameters gives�+ = 0:2858, w+ = 2:9789 and A+ = 0:1566 (� 1).Thus a slightly chirped pulse is indeed the steady-statesolution. As before, the second branch of these solutionsyields an imaginary value for ��. In Fig. 9 we comparethe experimental output pro�les with the steady-state

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Fig. 9. Comparison of the experimental autocorrelationmeasurements (scaled with the assumption that the pulseshape is a hyperbolic secant) and output spectrum with thefull evolution equations (Eq. (1), (4), and (11)), the averagedequation (Eq. (13)), and the approximate analytic solutionwith no slow response (Eqs. (14)-(19)).

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Fig. 10. Typical evolution of a pulse governed by thefull evolution equations (1), (4), and (11) in the anomalousdispersion regime �D � 12 ps/km-nm. Note that for the es-timated experimental parameter values, the pulse begins tobreak up into a group of multi-pulses. In this simulation wehave taken �s = 0 and �f = 2:6%.(numerical) solution of the full governing equations(Eqs. (1), (4)) and (11)), the averaged equation(Eq. (13), and the analytical chirped solution. Theseresults once again con�rm the validity of the asymptoticapproximations made in deriving the averaged equa-tion (13) and further demonstrate the usefulness of thesolutions (14)-(19)Finally, we simulate the the pulse evolution as gov-erned by Eqs. (1), (4) and (11) for the case in which wedouble the cavity length. Thus we have an intracavitypower of � 10 mW, a bandwidth of 25 nm and a cavityof length � 180 cm (Zcav=360 cm) with average disper-sion �D � 12 ps/km-nm. In this simulation, we assumethe strongest modelocking possible by allowing the SBRresponse to be purely instantaneous. This is also chosenso that the multiple-pulsing behavior observed numeri-cally no longer contains a slow response since the slowresponse as given by Eq. (4) is completely unphysical inthis multi-pulsing regime. Thus we take �s = 0:0 and�f = 2:6%. Figure 10 depicts the time evolution of thepulse through just 30 round trips. Note that the pulsebegins to evolve in a complicated fashion and appearsto be breaking up into a chain of solitary waves. Thisis most likely due to the length of the cavity which al-lows the build up of self-phase modulation which can nolonger be overcome by the intensity dependent loss ofthe SBR. This multi-pulse dynamics has been observedexperimentally but has not yet been fully quanti�ed. Itwill be explored further elsewhere. It should be notedthat similar dynamics have been previously observed inGinzburg-Landau evolution equations22.7. CONCLUSIONSWe have developed a theory for the modelocking pulsedynamics in a �ber laser with a saturable Bragg re ector.

The theory applies in both the normal and anomalousdispersion regimes and provides a model which has beencompared directly with experimental results. Quantita-tive agreement is achieved in both dispersion regimes.The theoretical model begins with the propagationequations in the �ber cavity with appropriate jump con-ditions imposed at the cavity ends where the SBR andoutput coupler are placed. The full evolution is approxi-mated by folding the SBR and output coupler dynamicsinto the governing equations and an averaged equation isderived. In the limit of no slow response and relaxationfor the SBR, the averaged equation yields exact chirpedpulse solutions. Each level of approximation is asymp-totically justi�ed and shown to be relevant for the exper-imental parameters used here. In addition, comparisonis made between the three levels of approximations andexperiment.In this �ber laser con�guration, the modelocking dy-namics are shown to be strongly driven by the \fast" ACStark shift27 of the quantum wells on the Bragg re ectorstructure. This allows modelocking of pulses as short as400 fs in the anomalous dispersion regime. Pump-probemeasurements con�rm the existence of the instantaneousAC Stark. It should be noted that stable modelockingoccurs when the pulse spectrum lies just below the ex-citonic bandgap, where the AC Stark shift is strongest.In addition to the instantaneous fast response, which isthe primary mechanism for modelocking, the slow re-sponse and relaxation of the SBR are shown to cause apulse drift and slight asymmetrization of the pulse. Weemphasize that this SBR model accurately treats the ra-diation far from the localized pulse so that only �2% isat most absorbed. In contrast, the master equation15,and thereby the soliton modelocking model, fails to cor-rectly capture this physical response of the semiconduc-tor saturable absorber. Our model suggests that theideal modelocking mechanism is a purely instantaneousresponse which both stabilizes the modelocked pulse andattenuates the dispersive radiation.The modelocking theory presented extends beyond thequalitative description of the master modelocking equa-tion. We provide a quantitative model which can beutilized as a powerful design tool in evaluating the per-formance of �ber lasers modelocked in the normal andanomalous dispersion regimes.ACKNOWLEDGEMENTSWe would especially like to thank James P. Gordonfor several valuable discussions, M. Koch for the ex-perimental pump-probe measurements of Figure 4, andW. Y. Jan, R. Pathak, and J. E. Cunningham for theSBR sample. JNK and PH acknowledge support from aNational Science Foundation (NSF) University-Industrypostdoctoral fellowship (DMS-9508634). KB acknowl-edges support from the NSF (ECS-9502491). PH ac-knowledges additional support from the Department ofEnergy (DE-FG02-95ER.25238.A000) andMW acknowl-edges support from the NSF (DMS-9500997).* Permanent address: Department of Mathematics, Uni-

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