We propose a model for passive mode-locking in quantum dot laser and
report on specific dynamical properties of the regime which is characterized
by a fast gain recovery. No Q-switching instability has been found
accompanying the mode-locking. Bistability can occur between the mode-locking
regime and zero intensity steady state

Bandelow, Uwe

Mandel, Paul

Viktorov, Evgeny

Vladimirov, Andrei

English

Repositorium für Naturwissenschaften und Technik  (TIB Hannover)

Weierstra-Institutfur Angewandte Analysis und Stohastikim Forshungsverbund Berlin e.V.Preprint ISSN 0946 { 8633A model for mode-loking in quantum dot lasersEvgeny A. Viktorov1, Paul Mandel 1, Andrei G. Vladimirov2 andUwe Bandelow 2submitted: 9th February 20061 Optique Nonlineaire Theorique,Universite Libre de Bruxelles,Campus Plaine CP 231,B-1050 Bruxelles, BelgiumE-Mail: evviktorulb.a.bepmandelulb.a.be2 Weierstrass Institute for AppliedAnalysis and Stohastis,Mohrenstrasse 39D - 10117 Berlin GermanyE-Mail: vladimirwias-berlin.debandelowwias-berlin.deNo. 1098Berlin 2006W I A S2000 Mathematis Subjet Classiation. 34K60,78A60.Key words and phrases. mode-loking, quantum dot lasers, delay dierential equations.1999 Physis and Astronomy Classiation Sheme. 42.55.Px 42.60.F 42.65.Sf.Edited byWeierstra-Institut fur Angewandte Analysis und Stohastik (WIAS)Mohrenstrae 3910117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprintwias-berlin.deWorld Wide Web: http://www.wias-berlin.de/AbstratWe propose a model for passive mode-loking in quantum dot laser andreport on spei dynamial properties of the regime whih is haraterized bya fast gain reovery. No Q-swithing instability has been found aompanyingthe mode-loking. Bistability an our between the mode-loking regime andzero intensity steady state.Lasers and ampliers based on self-assembled quantum dots attrat signiant at-tention due to redued threshold urrent, low hirp, weak temperature dependene[1℄ and a redued sensitivity to optial feedbak [2℄ at teleom wavelengths. Amongthe regimes displayed by multimode lasers, passive mode-loking (ML) is a powerfulmethod to generate short pulses for time domain multiplexing and for optial ombgenerator. The rst experiments have been reported on ML in quantum dot (QD)lasers at 1:3 m up to 50 GHz [3, 4℄ and demonstrated their superiority to quantumwell lasers for network appliations [5℄. The physis of the passive ML in a QD laserremains the same as in other lasers: the absorbing medium saturates faster thanthe amplifying one, and, therefore, a short window of net gain emerges for the pulseampliation [6℄. However, the nonlinear dynamis of ML depends on the speiharateristis of QD material and deserves detailed theoretial analysis in order tooptimize QD lasers for pratial appliations.In this Letter we onstrut a delay dierential model to desribe ML in quantumdot lasers using the approah proposed in [7, 8℄. In our model the dynamis of eahof the two quantum dot laser setions, gain and absorber, is governed by two rateequations for the time evolution of the quantum dot oupation probabilities andthe arrier densities in the wetting layer. We nd that ML in quantum dot lasersexhibits spei dynamial properties whih we assoiate with the esape and theapture proesses in the quantum dot material. In partiular, we explain the abseneof the low frequeny Q-swithing instability and the bistability with the nonlasingstate.Let us onsider a ring laser onsisting of three setions. The rst setion with thelength Lq; and the seond setion with the length Lg, ontain the saturable absorberand the gain medium, respetively. The third setion ats as a spetral lter thatlimits the bandwidth of the laser radiation. The equations desribing the evolutionof the eletri eld envelope A (t) at the entrane of the absorber setion an bewritten in the form [7℄: 1tA (t) + A (t) = pe(1 ig)G(t T )=2 (1 iq)Q(t T )=2A (t  T ) (1)1where A (t) is the normalized omplex amplitude of the eletri eld,  is the di-mensionless bandwidth of the spetral ltering setion, g (q) the linewidth en-hanement fators in the gain (absorber) setion. The delay parameter T is equalto the old avity round trip time. The attenuation fator  < 1 desribes totalnon-resonant linear intensity losses per avity round trip.The variables G (t) and Q (t) are the dimensionless saturable gain and absorption:G (t) = 2ggLg [2g(t)  1℄ ; (2)Q (t) =  2gqLq [2q(t)  1℄ ; (3)where the variables g (t) and q (t) desribe the oupation probabilities in a dotloated either in the amplier or absorber setion, respetively. A detailed derivationof Eq. (1) and the relations among the input and output elds in the amplier andabsorber setions are given in [8℄. Using these relations we write the following fourequations for the amplifying and absorbing setions:tg =  gg + Fg (g; Ng)  e Q eG   1 jAj2 ; (4)tq =  qq + Fq (q; Nq)  s 1  e Q jAj2 ; (5)tNg = Ng0    gNg   2Fg (g; Nq) ; (6)tNq = Nq0    qNq   2Fq (q; Nq) : (7)Here the variables Ng;q (t) desribe the arrier densities in the wetting layers, saledto the QD arrier density. The parameters gg;q,  g;q, and g;q are, respetively, thedierential gains, the arrier relaxation rate in the wetting layers and the arrierrelaxation rates in the dots. The dimensionless parameters Ng0 and Ng0 desribepumping proesses in the amplier and the absorber setions. The parameter s =gq=gg is the ratio of the saturation intensities in the gain and absorber setions. Thefator 2 in Eqs. (2), (3) and (6),(7) aounts for the twofold spin degeneray inthe quantum dots energy levels. The funtions Fg;q (g;q; Ng;q) desribe the arrierexhange rate between the wetting layers and the dots. In the most general form,the arrier exhange an be written as [9℄:Fg;q (g;q; Ng;q) = Rapg;q (1  g;q)  Resg;qg;q; (8)where 1   g;q is the Pauli bloking fator, Rapg;q = Bg;qNg;q desribes the arrierapture from the wetting layer to the dots with the rate Bg;q. Resg;q is a temperature-dependent oeÆient dening arrier esape from the dots to the wetting layer.Together with Eq. (1), the equations (4)-(7) onstitute a losed set of equations.The purpose of this Letter is not to model a spei laser onguration, but todesribe some generi properties of the ML regime in quantum dot lasers. Therefore,some of the parameters are taken idential for the amplier and the absorber forsimpliity. We onsider low linewidth enhanement fators g;q = 2 and relativelyhigh dierential gains. Carrier apture times in the dots were reported to be from21ps to 100ps. We onsider apture rates Bg;q = 2 whih orrespond to 5ps and takeesape rates as Resg;q = 0:1 Bg;q. The devies for ML have usually a long ampliersetion and a short absorber setion with a typial ratio 6 to 1 [10℄. The otherparameters are onventional for QD materials and geometry. They are speied inthe gure aption. The time t is normalized by the time interval t = 10 ps.A typial dynamis of the ML regime is shown in Fig.1. The intensity time traeshown in Fig. 1(a) displays the usual sequene of equidistant mode-loked pulses.The time dependene of the arrier densities Ng;q (t) in the wetting layer of the gainand absorber setions is shown in Fig. 1(d-e). They are similar to those of thesaturable gain and loss in a quantum well laser, for whih the gain setion is hara-terized by a very slow, almost linear reovery [8℄. On the ontrary, the reovery ofthe saturable gain G (t) and loss Q(t) of the QD setions appears to be muh faster[see Fig. 1(b-)℄. As an be seen in Fig. 1(b), after a fast reovery the saturablegain remains almost onstant between two onseutive pulses, G (t)  2ggLg. Theexistene of fast and slow reovery stages is a spei feature of quantum dots andis related to the ourrene of two ompeting mehanisms for the arrier transferbetween the wetting layers and the dots. This transfer is desribed in our model bythe funtion Fg (g; Ng) whih inludes two parts: nonlinear apture BNg (1  g)and linear esape Resg g. The nonlinear part of Fg (g; Ng) desribes intrinsi sat-uration introdued in Eq. (8) by the Pauli bloking fator. The strength of thissaturation depends on the arrier apture rate Bg: The linear part of Fg (g; Ng)desribes the arrier esape from the dots to the wetting layer. Together with thearrier relaxation rates g and  g, it indues the slow monotoni reovery as in thequantum well semiondutor lasers (Fig. 1(d)). The slow reovery is ontrolled bythe relaxation rates g +Resg and  g. Sine Bg  g +Resg , it is muh longer thanthe fast reovery stage. The fast gain reovery is known in optial ampliers basedon quantum dots, promising 1Tb/s speed for booster amplier [5℄. The fast reoveryof the saturable gain prevents the appearane of low frequeny self-pulsations whihdegrade the ML regime. It is diretly related to the strong damping of the relax-ation osillation frequeny in QD laser [2℄. Still, we have observed the low frequenyutuations in transients leading to stable ML.The dynamis of the variable Q(t) desribing the quantum dot saturable absorberalso shows two distint stages in the reovery proess (see Fig.1()). A similarfeature in the deay of QD absorber was observed experimentally in Ref. [11℄. Inthat paper fast reovery was attributed to the absorption saturation in the QD.Our modelling onrms this hypothesis as we nd a strong dependene of the fastreovery time on the rate Bq whih denes the saturation of the QD in our model.The nonlinear dynamis assoiated with the ML regime is rih and an inludehaos. An important dynamial feature we have observed is a bistability betweenthe ML regime and the non-lasing steady state regime. This bistability and theorresponding hysteresis loops in the harateristis were already experimentallyobserved and related to the nonlinear saturation of the QD absorption [4℄. Thissaturation ours due to the state lling and, therefore, is diretly linked to the fast3reovery stage.In Fig. 1, the round-trip net gain parameter G(t) = G(t)   Q(t) + ln is negativebetween pulses and beomes positive only during short time intervals when the pulseamplitude is large. Therefore aording to New's riterion [6℄, the regime shownin this Figure is stable with respet to small perturbations of the zero-intensitybakground. During the slow stage of the gain reovery, the net gain parameter G(t)is always negative.The omplete parameter and bifuration analysis of Eqs. (1)-(7) will be reportedelsewhere. We only briey mention the main tendenies of the ML regime. In-reasing the apture rates Bg;q leads to shorter pulses whih is dierent from theresults obtained for atively modeloked lasers [12℄. The esape rates Resg;q do notsigniantly inuene the pulse duration if Resg;q  Bg;q. The trends for the otherparameters are similar to the quantum well lasers.In onlusion , we proposed a model based on delay dierential equations for passiveML in quantum dot lasers. The model predits the appearane of ML pulses with noaompanying self-pulsations. The gain reovery is spei to quantum dot lasers.It explains the absene of the low frequeny self-pulsations and bistability betweenthe ML regime and lasing-o state.The authors aknowledge support of the Fonds National de la Reherhe Sientique(Belgium), the Interuniversity Attration Pole Programme - Belgian Siene Poliy,the State of Berlin, and the EU under grant TOB B1-1911. E.A.V. would like tothank the Isaa Newton Institute (Cambridge, UK) for its hospitality.Referenes[1℄ D. Bimberg, M. Grundmann, and N. N. Ledentsov , Quantum dot heterostru-tures (J. Wiley, 1999).[2℄ D. O'Brien, S.P. Hegarty, G. Huyet, and A.V. Uskov, \Sensitivity of quantum-dot semiondutor lasers to optial feedbak,"Opt. Lett. 29, 1072-1074 (2004).[3℄ M. Kuntz, G. Fiol, M. Lammlin, D. Bimberg, M.G. Thompson, K.T. Tan,C. Marinelli, R.V. Penty, I.H. White, V.M. Ustinov, A.E. Zhukov, Yu.M.Shernyakov, A.R. Kovsh, \35 GHz mode-loking of 1.3 m quantum dotlasers,"Opt. Lett. 29, 1072-1074 (2004).[4℄ X. Huang, A. Stintz, H. Li, L.F. Lester, J. Cheng, K.J. Malloy, \Passive mode-loking in 1.3 m two-setion InAs quantum dot lasers,"Appl. Phys. Lett. 78,2825-2827 (2001).[5℄ D. Bimberg, \Quantum dots for lasers, apliers and omputing,"J. Phys. D:Appl. Phys. 38, 2055-2058 (2005). 4[6℄ G.H.C. New, \Pulse evolution in mode-loked quasi-ontinuous lasers,"IEEE J.Quantum Eletron. 10, 115-124 (1974).[7℄ A.G. Vladimirov, D. Turaev, and G. Kozyre, \Delay dierential equations formode-loked semiondutor lasers,"Opt. Lett. 29, 1221-1223 (2004).[8℄ A.G. Vladimirov and D. Turaev, \Model for passive mode loking in semion-dutor lasers,"Phys. Rev. A 72 033808 (2005).[9℄ A.V. Uskov, Y. Bouher, J. LeBihan, and J. MInerney, \Theory of a self-assembled quantum-dot semiondutor laser with Auger arrier apture: Quan-tum eÆieny and nonlinear gain,"Appl. Phys. Lett. 73, 1499-1501 (1998).[10℄ E.U. Rafailov, M.A. Cataluna, W. Sibbett, N.D. Il'inskaya, Yu.M. Zadiranov,A.E. Zhukov, V.M. Ustinov, D.A. Livshits, A.R. Kovsh, and N.N. Ledentsov,\High-power pioseond and femtoseond pulse generation from a two-setionmode-loked quantum-dot laser,"Appl. Phys. Lett. 87, 081107 (2005).[11℄ E.U.Rafailov, S.J. White, A.A. Lagatsky, A. Miller, W. Sibbett, D.A. Livshits,A.E. Zhukov, and V.M. Ustinov, \Fast quantum dot saturable absorber forpassive mode-loking of solid-state lasers,"IEEE Photonis Tehnology Lett.16 2439-2441 (2004).[12℄ C. Xing and E.A. Avrutin, \Multimode spetra and ative mode loking po-tential of quantum dot lasers,"J. Appl. Phys. 97, 104301 (2005).5Figure 1: Time evolution of the eletri eld intensity jA (t)j2 ; the saturable gain G (t)and the saturable loss Q (t) in the dots, the arrier densities Ng;q (t) in the wetting layer ina mode-loked quantum dot laser with absorber. The reovery of G (t) and Q (t) onsistsof two stages: fast and slow. The parameters are:  = 0:2; T = 5;  = 10; g = q =  g =0:01;  q = 1;Ng0 = 5;Nq0 = 6; ggLg = 1:6; gqLq = 3; s = 10:6

A model for mode-locking in quantum dot lasers

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A model for mode-locking in quantum dot lasers

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