Transverse-mode dynamics in vertical-cavity surface-emitting lasers with optical feedback

T ans e se-mode dynamics in e ical-ca i y su ace-emi ing lase s wi h op ical eedback M. S. To e,1C. Masolle ,2and Paul Mandel3 1Ins i u o de Fı ´sica ‘‘A oyo Seco,’’ UNCPBA Pin o 399 (7000) Tandil, A gen ina 2Ins i u o de Fı ´sica, Facul ad de Ciencias, Uni e sidad de la Repu ´blica, Igua 4225, Mon e ideo 11400, U uguay 3Uni e si e ´Lib e de B uxelles, Op ique Nonline ´ai e The ´o ique, Campus Plaine Code Pos ale 231, B-1050 B uxelles, Belgium 共Recei ed 22 Ma ch 2002, e ised manusc ip ecei ed 21 June 2002; published 26 No embe 2002兲 We s udy he ans e se-mode dynamics o e ical-ca i y su ace-emi ing lase s wi h weak op ical eed- back. We use a model ha akes in o accoun he spa ial dependence o he ans e se modes and o wo ca ie densi y p o iles, associa ed wi h con ined ca ie s in he quan um well egion o he lase and uncon ined ca ie s in he ba ie egion. Op ical eedback is included as in he Lang-Kobayashi model. We ind ha o adequa e pa ame e alues an iphase dynamics occu s. As he injec ion cu en a ies, he an iphase dynamics is des oyed h ough a sequence o pe iodic mixed s a es leading o in-phase dynamics. In hese mixed s a es he e a e ime in e als in which he modes a e in phase, ollowed by ime in e als in which hey a e in an iphase. We s udy he o igin o he an iphase dynamics, assessing he ole o he di e en spa ial p o iles. We show ha he compe i ion be ween he di e en p o iles leads o he obse ed an iphase beha io . DOI: 10.1103/PhysRe A.66.053817 PACS numbe 共s兲: 42.60.Mi, 42.55.Px, 42.65.S , 05.45.Pq I. INTRODUCTION The e ical-ca i y su ace-emi ing lase 共VCSEL兲is a ype o semiconduc o lase ha is eme ging as a key ele- men o high-speed in o ma ion p ocessing sys ems and op- ical communica ion ne wo ks 关1兴. The ad an ages o a VC- SEL o e a con en ional, edge-emi ing semiconduc o lase a e single-longi udinal-mode ope a ion, dense packing capa- bili y, low h eshold cu en , high modula ion bandwid h, na ow ci cula beam p o ile, and simple and e icien cou- pling o an op ical ibe . Nea h eshold, VCSELs ypically emi linea ly pola ized ligh in he undamen al ans e se mode. Howe e , i is o en obse ed ha he pola iza ion s a e selec ed a h eshold becomes uns able as he injec ion cu en is inc eased, and a swi ch o he o hogonal pola iza- ion s a e occu s 共see, e.g., Re . 关2兴and e e ences he ein兲. Fo high-powe ope a ion, high-o de ans e se modes a e exci ed and he VCSEL usually emi s mul iple ans e se modes. The complex pola iza ion and ans e se-mode be- ha io o VCSELs a e conside ed d awbacks om he iew- poin o mos applica ions, and ha e a ac ed b oad in e es , bo h heo e ically and expe imen ally 关3–20兴. I is well known ha op ical eedback om an ex e nal e lec o has impo an e ec s on he dynamics o VCSELs 关21–27兴. The e ec o op ical eedback depends on he amoun o powe ed back in o he lase ca i y, and on he ound ip ime o he ield in he ex e nal ca i y, which de e mines he eedback phase. Con olled op ical eedback migh s abilize he lase , educing i s linewid h, bu uncon- olled eedback 共una oidable in many applica ions兲migh des abilize he lase , inducing chao ic in ensi y luc ua ions and a b oad linewid h. One pa icula ly complex beha io is known as low- equency luc ua ions 共LFFs兲, and is cha ac- e ized by ab up andom in ensi y d opou s ollowed by g adual, de e minis ic eco e ies. In addi ion, in mul imode lase s op ical eedback migh induce a a ie y o complex egimes. Se e al au ho s ha e s udied heo e ically he dynamics o mul iple-longi udinal- mode con en ional 共edge-emi ing兲semiconduc o lase s wi h op ical eedback. Sukow e al. 关28兴s udied he e ec o eedback based on an ex ension o he single-mode Lang- Kobayashi 共LK兲model 关29兴, which inco po a es addi ional op ical modes ha a e coupled h ough he ca ie in e sion and h ough sel - and c oss-sa u a ion coe icien s. I was ound ha he s a is ics o he in ensi y luc ua ions in he LFF egime on a picosecond ime scale is essen ially inde- penden o he numbe o op ical modes in ol ed in he lase emission. Using a simila model, bu wi h a pa abolic gain p o ile, Rogis e e al. 关30兴showed ha in he p esence o noise and in he LFF egime wo quali a i ely di e en be- ha io s on he picosecond ime scale a e possible: he longi- udinal modes can emi pulses in phase o oscilla e ou o phase, depending on he ope a ing pa ame e s. Vik o o and Mandel 关31兴s udied a mul imode ex ension o he LK model ha akes in o accoun he longi udinal ca ie g a ing asso- cia ed wi h a Fab y-Pe o con igu a ion and p edic ed he possibili y o an iphase dynamics. In ha model, he s eady s a e is des abilized ei he by a simple Hop bi u ca ion lead- ing o in-phase dynamics o he longi udinal modes, o by a degene a e Hop bi u ca ion leading o an iphase dynamics 关32兴. An iphase dynamics is an example o collec i e beha io in a sys em o globally coupled oscilla o s 关33兴. In lase s i esul s om he phase cohe ence o ime-dependen modal in ensi ies 关34,35兴. In he simples cases, i is cha ac e ized by he ac ha he o al in ensi y, which is he di ec sum o he modal in ensi ies o a e equa ion models, has many p ope ies o he single-mode in ensi y, while modal in ensi- ies display a mo e complex beha io . Se e al s udies o he ans e se-mode beha io o VCSELs ha e been based on a model o iginally p oposed by Valle, Sa ma, and Sho e 关3,4兴. The model includes spa ial p o iles o he ans e se op ical modes and o he ca ie densi y in he quan um well 共QW兲ac i e egion o he VCSEL. I also includes ca ie di usion. The model applies o weakly index-guided VCSELs, whe e he ans e se modal p o iles and modal equencies a e de e mined by he buil -in e ac i e index dis ibu ion, hus allowing a desc ip- PHYSICAL REVIEW A 66, 053817 共2002兲 1050-2947/2002/66共5兲/053817共9兲/$20.00 ©2002 The Ame ican Physical Socie y66 053817-1 ion in e ms o modal ampli udes. Fo a cylind ical VCSEL, he app op ia e ans e se modes a e he linea ly pola ized LPm,nmodes 关36兴. They ha e he p ope y LPm,n( , ␪ )⫽ ␾ m,n( )cos(m ␪ ) whe e ( , ␪ ) a e he pola coo dina es o he plane ans e se o he p opa- ga ion di ec ion. Se e al au ho s 关3,22,26,37,38兴ha e sim- pli ied he nume ical simula ions by assuming ha he azi- mu hal dependence o he modes wi h m⬎0 can be neglec ed, i.e., LPm,n( , ␪ )⯝ ␾ m,n( ), when hese modes a e degene a e. Howe e , he esul ing app oxima e modes a e no ca i y modes, excep i m⫽0. In his pape we use a model o VCSELs ha is an ex- ension o he model p oposed by Valle e al. 关3兴. Howe e , in o de o simpli y he calcula ions while keeping he model as comple e as possible, we assume ha only he i s h ee azimu hally symme ic modes LP0,1 ,LP 0,2 , and LP0,3 can be exci ed. Ca ie anspo e ec s a e included by conside ing wo ca ie densi ies, one o he ca ie s in he QW egion 共whe e he ca ie s a e in wo-dimensional quan um s a es兲, and one o he ca ie s in he ba ie egion 共whe e he ca - ie s a e in h ee-dimensional quan um s a es兲. The exchange o ca ie s be ween hese wo ese oi s 共ca ie cap u e in o he QWs and escape ou o he QWs兲is cha ac e ized by small bu ini e cap u e and escape imes. Ou app oach is he same as in he phenomenological s anda d a e equa ions o QW lase s 关39–41兴. Ex e nal op ical eedback is included as in he LK model, by conside ing a single e lec ion in he ex e nal ca i y. We show ha a weak op ical eedback may induce an- iphase dynamics o he ans e se modes, and we s udy how he an iphase dynamics is des abilized as he injec ion cu - en o he di usion coe icien a ies. We ind ha he an- iphase dynamics is des oyed h ough a sequence o pe iodic mixed s a es. In hese s a es, ime in e als in which he modes a e in phase al e na e wi h ime in e als in which hey a e in an iphase. We s udy he o igin o he an iphase beha io by conside ing equal and di e en spa ial p o iles o he ans e se modes. We show ha i is he compe i ion be ween he di e en p o iles ha leads o he obse ed an- iphase beha io . The e ec s o op ical eedback on he dynamics o VCSELs we e p e iously s udied by Law and Ag awal 关23,24兴, based on a model simila o ou s bu ha akes in o accoun se e al e lec ions in he ex e nal ca i y and does no conside ca ie cap u e and escape. In-phase and an iphase egimes we e ound in ha model bu ha aspec o he dy- namics was no he opic o hese pape s. He e we ocus on s udying in de ail he in-phase and an iphase beha io . This pape is o ganized as ollows. The model is desc ibed in Sec. II. Analy ical esul s o he s eady s a e a e p esen ed in Sec. III. Resul s o nume ical simula ions ha show dis inc dy- namical egimes o he ans e se modes a e p esen ed in Sec. IV. Finally, Sec. V con ains a summa y and he conclu- sions. II. THE MODEL We conside a cylind ically symme ic s uc u e, whose ac i e egion 共consis ing o se e al quan um wells兲is mod- eled as a single e ec i e quan um well o adius aand hick- ness dQW . Ba ie egions o hickness dblimi he QW e- gion. Two highly e lec ing mi o s sepa a ed by a dis ance L along he longi udinal zaxis de ine he lase ca i y. The in- jec ed cu en is azimu hally uni o m o e he ans e se a ea and a ies s epwise: j( )⫽jo o ⬍aand j( )⫽0 o he - wise. The emission beha io is de e mined by he buil -in index guiding in oduced by he ans e se e ac i e index s ep in he su ounding egion. The co e 共cladding兲 e ac i e index is aken o be nco e (nclad), i.e., he ans e se e ac- i e index p o ile is n( )⫽nco e o ⬍aand n( )⫽nclad o ⬎a. Fo his geome y he app op ia e ans e se modes a e he linea ly pola ized LPmn modes 关36兴, o which he ans e se a ia ion o he ield is gi en by ␺ mn共 , ␪ 兲⫽Jm共umn /a兲 Jm共umn兲cosm ␪ o ⬍a, ␺ mn共 , ␪ 兲⫽Km共wmn /a兲 Km共wmn兲cosm ␪ o ⬎a,共1兲 whe e Jmand Kma e Bessel unc ions o he i s and second kinds, espec i ely, umn⫽a关(nco ekmn)2⫺ ␤ 2兴1/2,wmn ⫽a关 ␤ 2⫺(ncladkmn)2兴1/2, ␤ L⫽q ␲ ,qis an in ege , and he wa e ec o kmn is ob ained om eigen alue equa ions. To simpli y he calcula ions we conside ha only h ee modes, ha ing azimu hal symme y, a e exci ed in he ange o pa- ame e s conside ed in his pape : ␺ 1共 兲⬅ ␺ 01共 , ␪ 兲⫽LP01 , ␺ 2共 兲⬅ ␺ 02共 , ␪ 兲⫽LP02 , and ␺ 3共 , ␪ 兲⬅ ␺ 03共 兲⫽LP03 .共2兲 The mode p o iles a e no malized such ha 兰 0 ⬁ 兩 ␺ i 兩 2( ) d ⫽1. Since he mode p o iles a e exponen ially small ou side he ac i e egion, his no maliza ion ha dly di e s om he physical no maliza ion 兰 0 a 兩 ␺ i 兩 2( ) d ⫽1. The equa ions o he slowly a ying complex ampli ude o he i h mode, ei( ), he densi y o ca ie s con ined in he QW egion, nw( , ), and he densi y o 共uncon ined兲ca ie s in he ba ie egion, nb( , ), a e 关3,29,38兴 dei d ⫽1⫹j ␣ 2 冉 gi⫺1 ␶ pi 冊 ei共 兲⫹kiei共 ⫺ ␶ 兲exp共⫺j ␻ i ␶ 兲, 共3兲 ⳵ nb ⳵ ⫽j共 兲 edb ⫺nb ␶ cap ⫹VQW Vb nw ␶ esc ⫺nb ␶ n ⫹Db 1 ⳵ ⳵ 冉 ⳵ nb ⳵ 冊 , 共4兲 ⳵ nw ⳵ ⫽Vb VQW nb ␶ cap ⫺nw ␶ esc ⫺nw ␶ n ⫺go共nw⫺n 兲兺 兩 ei 兩 2 兩 ␺ i 兩 2 ⫹Dw 1 ⳵ ⳵ 冉 ⳵ nw ⳵ 冊 .共5兲 In hese equa ions he modal ampli ude eiis no malized such ha 兩 ei 兩 2 兩 ␺ i 兩 2is he pho on densi y in he i h mode. The ca ie a iables a e a e aged along he longi udinal axis. TORRE, MASOLLER, AND MANDEL PHYSICAL REVIEW A 66, 053817 共2002兲 053817-2 The e o e, nw( , )关nb( , )兴 ep esen s he a e age ca ie densi y in he ans e se plane in he QW 共ba ie 兲 egion. I he QW egion consis s o se e al QWs, in e well ca ie anspo e ec s a e no conside ed, and nw ep esen s he a e age ca ie densi y in he QWs. The i s e m in he igh -hand side o Eq. 共3兲accoun s o op ical gain, losses, and phase-ampli ude coupling. He e, ␣ is he linewid h enhancemen ac o and giis he modal gain, gi共 兲⫽ 冕 0 ⬁go⌫i共nw⫺n 兲 兩 ␺ i 兩 2 d ,共6兲 whe e gois he gain coe icien , ⌫iis he con inemen ac o o he i h mode, and n is he anspa ency ca ie densi y. ␶ pi is he pho on li e ime o he i h mode. The second e m in he igh -hand side o Eq. 共3兲 akes in o accoun he ield e lec ed om he ex e nal ca i y. We conside a single e- lec ion, and he e o e he model is alid o weak and mod- e a e eedback le els. kiis he eedback coe icien o he i h mode: ki⫽(1⫺R2) 冑 R2Rex ␩ c/(R2 ␶ in)关24兴, whe e R2and Rex a e he ou pu and ex e nal mi o e lec i i ies, ␶ in is he soli a y lase ound- ip ime, and ␩ cis he coupling e - iciency. In gene al ␩ ccan be di e en o di e en ans- e se modes, bu in his s udy we ake ␩ c o be mode inde- penden . ␻ iis he op ical equency o he i h mode in he absence o eedback, and ␶ is he ex e nal-ca i y ound- ip ime. The e ms on he igh -hand side o Eq. 共4兲co espond, om le o igh , o 共i兲 he a e a which ca ie s a e injec ed in o he ba ie egion, 共ii兲 he a e a which ca ie s a e cap- u ed in o he QWs, 共iii兲 he a e a which ca ie s escape ou o he QWs, 共i 兲 he ca ie loss owing o a ious non adia- i e ecombina ion p ocesses, and 共 兲ca ie di usion ac oss he ba ie egion. The anspo e ec s a e included by a cap u e ime ␶ cap , an escape ime ␶ esc , and a di usion co- e icien Db. The ca ie loss is included by a ca ie li e ime ␶ n. Since he a iables nband nw e e o ca ie densi ies, he di e en sizes o he ba ie and QW egions mus be aken in o accoun . This is done by he a io Vb/VQW , whe e Vb⫽db ␲ a2is he olume o he ba ie egion, and VQW ⫽dQW ␲ a2is he olume o he QW egion. The e ms in he igh -hand side o Eq. 共5兲co espond, om le o igh , o 共i兲 he ca ie s cap u ed in o he QWs, 共ii兲 he ca ie s ha escape ou o he QWs, 共iii兲 he non a- dia i e ca ie loss, 共i 兲 he ca ie loss owing o s imula ed ecombina ion, and 共 兲ca ie di usion ac oss he QWs. Fo simplici y we conside he same non adia i e ecombina ion ime o he ca ie s in he QW egion and o he ca ie s in he ba ie egion 共 he e ec o di e en ecombina ion imes was s udied in 关38兴兲. III. STEADY-STATE SOLUTIONS The s a iona y solu ions o Eqs. 共3兲–共5兲a e ei共 兲⫽ei sexp关i共 ␻ i s⫺ ␻ i兲 兴, nw共 , 兲⫽nw s共 兲,nb共 , 兲⫽nb s共 兲, whe e ei s, ␻ i s,nw s( ), and nb s( ) sa is y gi s⫽ 冕 0 ⬁go⌫i共nw s⫺n 兲 兩 ␺ i 兩 2 d ⫽1/ ␶ pi⫺2kicos共 ␻ i s ␶ 兲, 共7兲 ␻ i s⫺ ␻ i⫽⫺ ␣ kicos共 ␻ i s ␶ 兲⫺kisin共 ␻ i s ␶ 兲,共8兲 nb s 冉 1 ␶ cap ⫹1 ␶ n 冊 ⫽j共 兲 edb ⫹VQW Vb nw s ␶ esc ⫹Db 1 ⳵ ⳵ 冉 ⳵ nb s ⳵ 冊 , 共9兲 nw s 冉 1 ␶ esc ⫹1 ␶ n 冊 ⫽Vb VQW nb s ␶ cap ⫺go共nw s⫺n 兲兺 兩 ei s 兩 2 兩 ␺ i 兩 2 ⫹Dw 1 ⳵ ⳵ 冉 ⳵ nw s ⳵ 冊 .共10兲 Equa ion 共7兲shows ha he s a iona y alues o he modal gains depend on he eedback le el bu no on he ca ie cap u e and escape imes. Equa ion 共8兲de e mines he op ical equencies o he ans e se modes in he p esence o eed- back, which a e also independen o ␶ cap and ␶ esc . In eg a - ing Eqs. 共9兲and 共10兲be ween ⫽0 and ⫽⬁gi es 共 ␥ cap⫹ ␥ n兲 冕 0 ⬁nb s d ⫽1 edb 冕 0 ⬁j共 兲 d ⫹VQW Vb ␥ esc 冕 0 ⬁nw s d ⫹Db 冕 0 ⬁1 ⳵ ⳵ 冉 ⳵ nb s ⳵ 冊 d ,共11兲 共 ␥ esc⫹ ␥ n兲 冕 0 ⬁nw s d ⫽Vb VQW ␥ cap 冕 0 ⬁nb s d ⫺兺 兩 ei s 兩 2 冕 0 ⬁go共nw s⫺n 兲 兩 ␺ i 兩 2 d ⫹Dw 冕 0 ⬁1 ⳵ ⳵ 冉 ⳵ nw s ⳵ 冊 d ,共12兲 whe e ␥ esc⫽1/ ␶ esc , ␥ cap⫽1/ ␶ cap , and ␥ n⫽1/ ␶ n. The num- be s o ca ie s in he ba ie and QW egions a e Nb共 兲⫽2 ␲ db 冕 0 ⬁nb共 , 兲 d , Nw共 兲⫽2 ␲ dQW 冕 0 ⬁nQW共 , 兲 d , and Eqs. 共11兲and 共12兲can be ew i en as 共 ␥ cap⫹ ␥ n兲Nb s⫽2 ␲ e 冕 0 ⬁j共 兲 d ⫹ ␥ escNw s ⫹Db2 ␲ db ⳵ nb s ⳵ 冏 ⫽0 ⫽⬁ ,共13兲 TRANSVERSE-MODE DYNAMICS IN VERTICAL-CAVITY . . . PHYSICAL REVIEW A 66, 053817 共2002兲 053817-3 共 ␥ esc⫹ ␥ n兲Nw s⫽ ␥ capNb s⫺2 ␲ dQW兺 兩 ei s 兩 2gi s/⌫i ⫹Dw2 ␲ dQW ⳵ nw s ⳵ 冏 ⫽0 ⫽⬁ ,共14兲 whe e Nb s⫽2 ␲ db 兰 0 ⬁nb s( ) d and Nw s ⫽2 ␲ dQW 兰 0 ⬁nQW s( ) d . Now i is clea why i is con enien o de ine he no maliza ion condi ion and all in eg als be- ween ⫽0 and ⫽⬁.nw( ⫽⬁)⫽nb( ⫽⬁)⫽0 and he di - usion e ms anish. Equa ions 共13兲and 共14兲can be simpli- ied o 共 ␥ cap⫹ ␥ n兲Nb s⫽J⫹ ␥ escNw s,共15兲 共 ␥ esc⫹ ␥ n兲Nw s⫽ ␥ capNb s⫺ ␥ pIT,共16兲 whe e J⫽2 ␲ 兰 0 ⬁j( ) d /eis he numbe o injec ed ca ie s pe uni ime, and IT⫽2 ␲ dQW兺 兩 ei s 兩 2is he o al numbe o pho ons in he QW egion. In Eq. 共16兲we ha e assumed ha gi s⬃1/(⌫i ␶ pi)⫽ ␥ p. This app oxima ion is alid o low eedback le els such ha 1/ ␶ piⰇki. F om Eqs. 共15兲and 共16兲we can elimina e Nb sand ob ain ␥ pIT⫽J 1⫹ ␥ n/ ␥ cap ⫺ ␥ nNw s 1⫹ ␥ n/ ␥ cap 冉 1⫹ ␥ n ␥ cap ⫹ ␥ esc ␥ cap 冊 . 共17兲 Two limi s a e in e es ing o analyze. When he ca ie s do no escape ou o he QW egion ( ␥ esc⬃0), we eco e he simple connec ion be ween he pho on numbe , he in- jec ed cu en , and he ca ie numbe in he QW egion, ␥ pIT⫽Je ⫺ ␥ nNw s, wi h a modi ied, ‘‘e ec i e’’ injec ed cu en Je ⫽J(1⫹ ␥ n/ ␥ cap)⫺1, which is sligh ly lowe han he ac ual injec ed cu en 共 ypically, ␶ nis o he o de o nanoseconds, and ␶ cap is o he o de o picoseconds; hus ␥ n/ ␥ capⰆ1). The ac o (1⫹ ␥ n/ ␥ cap)⫺1 ep esen s he loss o ca ie s 共due o non adia i e p ocesses兲du ing he cap u e ime. In o he wo ds, a ini e cap u e ime sligh ly diminishes he cu en densi y e ec i ely injec ed in o he QW egion. The o he limi co esponds o a la ge a io be ween he cap- u e and he escape imes, R⫽ ␶ cap / ␶ esc⫽ ␥ esc / ␥ cap . This leads o a la ge , nega i e con ibu ion o he las e m in Eq. 共17兲, and, he e o e, o a signi ican educ ion o he o al numbe o pho ons in he QW egion. We will show in he nex sec ion ha his a o s emission in he undamen al ans e se mode. A complemen a y way o unde s and he e ec o ca ie cap u e and escape is by conside ing he dependence o he h eshold cu en o he undamen al ans e se mode on he cap u e and escape imes. The h eshold cu en can be es i- ma ed om Eq. 共17兲as J h⫽ ␥ nNw s 冉 1⫹ ␥ n ␥ cap ⫹ ␥ esc ␥ cap 冊 .共18兲 Clea ly, an inc ease o ␥ esc inc eases he h eshold cu en , and, he e o e, o a ixed injec ion cu en he lase ope a es close o h eshold. IV. DYNAMICAL REGIMES We in eg a ed he model equa ions wi h he pa ame e s a⫽6 ␮ m, dQW⫽0.024 ␮ m共 h ee QWs each o hickness 0.08 ␮ m), db⫽1.2 ␮ m, index s ep⫽0.1, ␣ ⫽3, go ⫽ g ⳵ g/ ⳵ nwi h g⫽0.0715 ␮ m/ns and ⳵ g/ ⳵ n⫽5.95 ⫻10⫺8 ␮ m2,n ⫽1.33⫻106 ␮ m⫺3, ␶ cap⫽5 ps, ␶ esc ⫽25.5 ps, ␶ n⫽1.52 ns, ␶ ⫽1 ns, and Db⫽0.5 ␮ m2/ns. The ime in eg a ion s ep is ⌬ ⫽10⫺4ps and he space in eg a- ion s ep is ⌬ ⫽0.02 ␮ m. Fi s , we conside a degene a e si ua ion, in which all modes ha e he same con inemen ac o ⌫i⫽0.038, equency ( ␻ i ␶ ⫽0 ad), losses ( ␶ pi ⫽2.2 ps), and eedback le el (ki⫽k). The eedback le el, he di usion coe icien Dw, and he injec ion cu en I ⫽jo ␲ a2a e he ee pa ame e s o ou s udy. We show he exis ence o an an iphase dynamic egime o weak eedback and adequa e pa ame e alues. Nex , we s udy he e ec o ca ie di usion and modal p o iles on he an iphase egime. Finally, we show ha he an iphase egime is also obse ed in a mo e ealis ic si ua ion, in which he modes ha e di e - en op ical equencies. Wi hou eedback and close o h eshold, he single-mode s eady s a e is s able, while o la ge injec ion he ans e se mul imode s eady s a e is s able. The undamen al LP01 ans e se mode has he lowes h eshold and i is s able o low cu en . As he cu en inc eases he LP02 mode u ns on. FIG. 1. To al and modal in ensi ies as he injec ion cu en in- c eases. The di usion coe icien is Dw⫽0.5 ␮ m2/ns. 共a兲Wi hou eedback. 共b兲The eedback le el is k⫽1ns ⫺1. The hick 共 hin兲line shows he alue o he o al in ensi y 共injec ion cu en 兲. The modes a e ep esen ed as LP01 , dashed line; LP02 , do -dashed line; LP03 , do ed line. TORRE, MASOLLER, AND MANDEL PHYSICAL REVIEW A 66, 053817 共2002兲 053817-4 Fo e en la ge injec ion, he LP03 mode u ns on and he h ee modes coexis . Figu e 1共a兲displays he o al and modal in ensi ies in he absence o eedback, as he injec ion cu en 共 hin solid line兲g adually inc eases. The hick line co e- sponds o he o al powe , while he o he lines co espond o he modal powe s (LP01 dashed line; LP02 do -dashed line; LP03 do ed line兲. Weak eedback le els modi y his pic u e quan i a i ely bu no quali a i ely. Figu e 1共b兲co esponds o k⫽1ns ⫺1. Conside ing he in e nal ound ip ␶ in⫽0.045 ps, he cou- pling e iciency ␩ c⫽1, and he ou pu -mi o e lec i i y R2⫽0.995, his eedback le el co esponds o an ex e nal- mi o e lec i i y o Rex ⫽8.06⫻10⫺5, i.e., we wo k in he e y weak eedback egime. Figu e 1共b兲shows ha o his eedback he modal in ensi ies exhibi oscilla ions, and we ind dis inc dynamical egimes wi h inc easing cu en . To in es iga e in mo e de ail wha happens o di e en injec- ion cu en s, we plo in Fig. 2 he ime-a e aged alue o he o al and modal in ensi ies, as a unc ion o he injec ion cu en . In his igu e he injec ion cu en Iwas kep con- s an un il he s able egime was eached, and hen he a e - age alue o he o al and modal in ensi ies was calcula ed. Clea ly, wi h weak eedback he ans e se-mode dynamics is such ha he o al in ensi y looks single mode, i.e., i in- c eases linea ly wi h he injec ion cu en excep o he weak nonlinea esponse close o h eshold. To display an- o he ace o his in iguing cohe ence, we show in Fig. 3 he maximum and minimum alues o he o al 关Fig. 3共a兲兴 and he modal 关Fig. 3共b兲兴 in ensi ies whose ime a e age is shown in Fig. 2. Figu e 3 e eals a complex unde lying ans e se-mode beha io , which is ypical o an iphase dy- namics in globally coupled nonlinea oscilla o s. A. An iphase dynamics Fo I⭐1.7 mA he VCSEL is single mode. The in ensi y o he LP01 mode exhibi s undamped elaxa ion oscilla ions, whose ampli ude dec eases o inc easing I. Figu e 3 indi- ca es ha o I⫽1.7 mA he e is single-mode s eady-s a e ope a ion in ol ing only he LP01 mode. Fo Isligh ly la ge , he LP02 mode eme ges, des abilizing he s eady-s a e LP01 mode. In he in e al 1.7⬍I⬍2.2 mA, he wo modes oscil- la e in phase. As he cu en is inc eased, a wo-mode s eady s a e is eached o I⫽2.3 mA. As o he case I⫽1.7 mA, inc easing Ides abilizes he s eady-s a e ope a ion ia he eme gence o a new mode. Fo I⫽2.4 mA he LP03 mode eme ges and o I⬎2.4 mA we obse e di e en egimes o h ee-mode ope a ion. Fi s , o 2.4⬍I⬍2.7 mA he e a e oscilla ions o he o al in ensi y, since he maximum and minimum alues di e 关Fig. 3共a兲兴. Fo I⬎2.7 mA he maxi- mum and minimum alues o he o al in ensi y a e nea ly equal, and Fig. 3共b兲shows ha wo dis inc dynamical e- gimes ac ually occu . Fo 2.7⬍I⬍3.7 mA he e a e oscilla- ions o he modal in ensi ies which nea ly compensa e in he o al in ensi y, while o I⭓3.7 mA each ans e se mode is in s eady s a e. Figu es 3共a兲and 3共b兲sugges ha in he in e al 2.7⬍I ⬍3.7 mA an iphase dynamics occu s, since he modal in en- si ies oscilla e while he o al in ensi y emains nea ly con- s an . In o de o s udy he phase ela ions among he modes in he di e en dynamical egimes, Fig. 4 shows o inc eas- ing alues o he injec ion cu en he o al and modal in en- FIG. 2. To al and modal a e aged in ensi ies as a unc ion o he injec ion cu en . All pa ame e s a e as in Fig. 1共b兲. The hick line shows he alue o he o al in ensi y. The modal in ensi ies a e ep esen ed as LP01 , dashed line; LP02 , do -dashed line; LP03 , do - ed line. FIG. 3. Maximum and minimum alues o he 共a兲 o al and 共b兲 modal in ensi ies as a unc ion o he injec ion cu en . All pa am- e e s a e as in Fig. 1共b兲. The modal in ensi ies a e ep esen ed as LP01 , dashed line; LP02 , do -dashed line; LP03 , do ed line. TRANSVERSE-MODE DYNAMICS IN VERTICAL-CAVITY . . . PHYSICAL REVIEW A 66, 053817 共2002兲 053817-5 si ies. As be o e, he hick line shows he o al powe , while he hin lines show he modal powe s (LP01 , dashed line; LP02 , do -dashed line; LP03 , do ed line兲. Fo low cu en he e is he single-mode pe iodic egime shown in Fig. 4共a兲. Figu e 4共b兲displays a wo-mode in-phase egime wi h e y di e en oscilla ion ampli udes. Figu e 4共c兲co esponds o a wo-mode s eady-s a e egime. The eme gence o he hi d mode leads o in-phase oscilla ions o he o he wo modes, Fig. 4共d兲. The ansi ion o an iphase oscilla ions, Fig. 4共 兲,is h ough a mixed s a e displayed in Fig. 4共e兲. No e ha in Fig. 4共e兲 he e a e ime in e als in which he LP01 and LP02 mode pulses a e in phase ollowed by ime in e als in which hey a e in an iphase. As a whole, his egime is pe iodic. Fo e en la ge injec ions a h ee-mode s eady s a e is eached 关Fig. 4共g兲兴. An iphase beha io was also ound by Valle 关42兴, in he compe i ion o he LP11 c关wi h a cos2( ␪ ) dependence in ensi y p o ile兴, and he LP11 s关wi h a sin2( ␪ ) dependence in ensi y p o ile兴, when he VCSEL is subjec ed o injec ion cu en modula ion. Mo eo e , p e ious s udies by Law and Ag awal 关23,24兴o he dynamics o VCSELs wi h eedback 共based on a model simila o ou s, bu ha includes se e al e lec ions in he ex e nal ca i y and does no ake in o accoun ca ie cap u e and escape兲 e ealed he exis ence o in-phase and an iphase beha io . In 关23,24兴 he au ho s conside ed high e lec i i y, he compe i ion o only wo ans e se modes, and di e en con ac geome ies o he injec ion cu en . Fo ins ance, wi h a disk-con ac geome y, pe iodic in-phase and an iphase beha io s o he LP01 and LP11 modes we e ound 共see Figs. 5b and 5c o 关23兴兲 o di e en alues o he eed- back pa ame e . Thus, an iphase dynamics seems o be a o- bus gene al ea u e independen o he de ails o he model. As discussed in Sec. III, he e ec o ca ie cap u e and escape is o modi y he cu en e ec i ely injec ed in o he lase ca i y. Ou nume ical simula ions e i y ha inc easing ␶ esc is indeed equi alen o inc easing he injec ion cu en , and a ansi ion om in-phase o an iphase beha io can be obse ed. In he ollowing sec ions we s udy he in luence o ca ie di usion, modal p o iles, and di e en op ical e- quencies on he an iphase beha io . B. In luence o he ca ie di usion Va ying he di usion coe icien also changes he h esh- old cu en , and he e o e dec easing Dwhas an e ec simila FIG. 4. Dynamic egimes o inc easing injec ion cu en . 共a兲I ⫽1.7 mA; 共b兲I⫽2.1 mA; 共c兲I⫽2.3 mA; 共d兲I⫽2.55 mA; 共e兲I ⫽2.67 mA; 共 兲I⫽3.3 mA; 共g兲I⫽3.7 mA. All o he pa ame e s a e as in Fig. 1共b兲. FIG. 5. Dynamic egimes o dec easing ca ie di usion. k ⫽1ns ⫺1,I⫽2.8 mA. 共a兲Dw⫽3.0 ␮ m2/ns; 共b兲Dw ⫽2.0 ␮ m2/ns; 共c兲Dw⫽1.5 ␮ m2/ns; 共d兲Dw⫽0.5 ␮ m2/ns; 共e兲 Dw⫽0.01 ␮ m2/ns. TORRE, MASOLLER, AND MANDEL PHYSICAL REVIEW A 66, 053817 共2002兲 053817-6 o inc easing he injec ion. Figu e 5 shows he ans e se- mode dynamics o i e dec easing alues o he di usion coe icien . Fo la ge di usion 关Fig. 5共a兲兴 he lase ope a es on he undamen al ans e se mode in a pe iodic egime. As he di usion coe icien dec eases, we obse e a ansi ion o an iphase oscilla ions in ol ing he LP01 and LP02 modes 关Figs. 5共c兲and 5共d兲兴. As he di usion coe icien is u he dec eased, we ind a mo e complex ype o an iphase egime: an iphase oscilla ions in ol ing he LP02 and LP03 modes, while he LP01 mode exhibi s small oscilla ions ha a e in phase, al e na ely wi h he LP02 and he LP03 modes 关Fig. 5共e兲兴. The esul s ob ained o di e en alues o he di u- sion coe icien clea ly show ha in his model i is he popu- la ion g a ing due o he Fab y-Pe o con igu a ion ha in- duces he mul imode beha io , as occu s in o he ypes o homogeneously b oadened lase s. C. T ans e se p o ile compe i ion In o de o unde s and he o igin o an iphase dynamics in his model, we analyze he compe i ion among he h ee ans e se modes. Fo ha pu pose, we shall eplace he LP p o iles used up o he e 关Eq. 共1兲兴 by o he p o iles. Figu e 6 shows he e ec o di e en p o iles. Figu e 6共a兲 is used as a e e ence: he ans e se p o iles a e he LP01 , LP02 , and LP03 modes, and an an iphase oscilla ion is ob- se ed be ween he LP01 and LP02 modes. Nex , we assume ha modes 2 and 3 ha e uni o m p o iles wi hin he ac i e egion 关 兩 ␺ i( ) 兩 2⫽ci o ⬍aand 兩 ␺ i( ) 兩 2⫽0 o he wise兴 while he hi d mode is he undamen al ans e se mode LP01 . This case is shown in Fig. 6共b兲, whe e we obse e ha he wo modes wi h equal p o iles a e iden ical 共do -dashed line兲, while he hi d mode 共dashed line兲oscilla es such ha la ge 共small兲peaks in he iden ical modes co espond o small 共la ge兲peaks in he hi d mode. Las , we conside he case in which all modes ha e uni o m p o iles. In his case, all h ee modes a e iden ical and oscilla e in phase, and he o al in ensi y is exac ly h ee imes he in ensi y o any mode 关Fig. 6共c兲兴. The ini ial condi ions a e he same in Figs. 6共a兲,6共b兲, and 6共c兲. The ini ial condi ions a e aken all h ough he pape FIG. 6. E ec o di e en ans e se mode p o iles. k ⫽1ns ⫺1,I⫽2.8 mA, Dw⫽0.5 ␮ m2/ns. 共a兲The ans e se modes a e LP01 ,LP 02 , and LP03 .共b兲Two ans e se modes ha e he same uni o m p o iles, and he hi d mode is he LP01 mode. 共c兲The h ee ans e se modes ha e he same uni o m p o iles. FIG. 7. Dynamic egimes o inc easing injec ion cu en , when he ans e se modes ha e di e en op ical equencies. 共a兲I ⫽1.7 mA; 共b兲I⫽2.1 mA; 共c兲I⫽2.3 mA; 共d兲I⫽2.55 mA; 共e兲I ⫽2.67 mA; 共 兲I⫽3.3 mA; 共g兲I⫽3.7 mA. ␭1⫽851.8 nm, ␭2 ⫽852.0 nm, and ␭3⫽852.2 nm. All o he pa ame e s a e as in Fig. 4. TRANSVERSE-MODE DYNAMICS IN VERTICAL-CAVITY . . . PHYSICAL REVIEW A 66, 053817 共2002兲 053817-7 wi h he lase o , i.e., he modal ampli udes a e a he noise le el and he ca ie densi ies a e a he anspa ency alue. The di e en dynamic egimes shown in Figs. 6共a兲,6共b兲, and 6共c兲a e hus a consequence o he ans e se-mode p o iles conside ed. F om hese esul s i is clea ha in his model he an- iphase dynamics has i s o igin in he mode coupling and compe i ion 共 ou h e m兲in Eq. 共5兲. The compe i ion among he ans e se modes wi h di e en spa ial p o iles o ‘‘bu n holes’’ in he same ese oi o ca ie s 共i.e., he ca ie s in he QWs兲leads o he obse ed an iphase dynamics. Equi a- len ly, one can in e p e his mechanism as a ans e se ca - ie g a ing induced by he di e en weigh s 兩 ␺ i 兩 2o he lasing modes. This beha io has also been ound ecen ly in he dynamics o he longi udinal modes o an edge-emi ing lase wi h op ical eedback 关31,32兴and i is he same mecha- nism ha leads o an iphase dynamics. D. In luence o di e en op ical equencies and di e en eedback le els In he p eceding subsec ions, he h ee ans e se modes ha e he same op ical equencies, and one migh ques ion i he an iphase egime ound is no a singula p ope y o he degene a e equa ions. Figu e 7 shows esul s in which he modes ha e di e en op ical equencies: ␭1⫽851.8 nm, ␭2⫽852.0 nm, and ␭3⫽852.2 nm, all o he pa ame e s be- ing as in Fig. 4. The an iphase egime pe sis s, and his con- i ms ha , gi en he smallness o he in e mode equency sepa a ion wi h espec o he op ical equency, in e mode equency di e ences may be ea ed by a pe u ba ion heo y o which Eqs. 共3兲–共5兲a e he ze o o de app oxima- ion. The an iphase egime is also obus wi h espec o sligh ly di e en eedback le els. Figu e 8 shows esul s o k1⫽k2⫽1ns ⫺1,k3⫽2ns ⫺1, all o he pa ame e s being as in Fig. 4共 兲. Clea ly he an iphase egime su i es. V. SUMMARY AND CONCLUSIONS The ans e se-mode dynamics o an index-guided e ical-ca i y su ace-emi ing lase wi h weak op ical eed- back was analyzed using a model ha akes in o accoun h ee ans e se modes and wo ca ie densi y p o iles, as- socia ed wi h con ined ca ie s in he quan um well egion o he lase . We ound an iphase dynamics and s udied he de- s abiliza ion o his egime when he injec ion cu en o he ca ie di usion a ies. We ound ha he an iphase dynam- ics is des abilized hough a sequence o mixed s a es, in which ime in e als in which he mode pulses a e in phase al e na e wi h ime in e als in which hey a e in an iphase. We ha e also shown ha in his model he an iphase dynam- ics is due o he ans e se p o iles o he op ical modes, which compe e o he same ese oi o ca ie s. An iphase dynamics is usually s udied in he ame o modal a e equa ions 关30–32兴, whe e he ca ie densi y is expanded ei he in Fou ie se ies o in modal se ies. These se ies ha e o be unca ed and he app oxima ion induced by his unca ion is di icul o assess 关43兴. The powe o he model s udied in his pape is ha no such expansion has been in oduced and he ca ie densi y dynamical equa ion is comple e, including di usion. As a esul , he e is no am- bigui y as o he o igin o he an iphase dynamics. This is especially clea om he analysis o Sec. IVC, whe e an- iphase dynamics was clea ly a ibu ed o he ans e se g a ing o he ca ie s. The nega i e aspec o his model is ha li le can be concluded analy ically abou he ime- dependen egimes o he model equa ions 共3兲–共5兲. Nume i- cal simula ions a e essen ial o unde s and he dynamical e- gimes. ACKNOWLEDGMENTS M.S.T. was suppo ed in pa by a g an om Sec e a ı ´a de Ciencia y Te ´cnica 共UNCPBA-A gen ina兲, C.M. was sup- po ed in pa by P oyec o de Desa ollo de Ciencias Basicas 共PEDECIBA兲and Comision Sec o ial de In es igacion Cien- i ica 共U uguay兲, and P.M. was suppo ed by he Fonds Na- ional de la Reche che Scien i ique and he In e uni e si y A ac ion Pole p og am o he Belgian go e nmen . 关1兴Ve ical-Ca i y Su ace-Emi ing Lase s, edi ed by K.D. Choque e and D. G. Deppe, SPIE P oc. Vol. 3003 共SPIE, Bell- ingham, WA, 1997兲. 关2兴M. San Miguel, in Semiconduc o Quan um Op oelec onics, edi ed by A. Mille , M. Eb ahimzadeh, and D.M. Finlayson 共Ins i u e o Physics, B is ol, 1999兲, p. 339. 关3兴A. Valle, J. Sa ma, and K.A. Sho e, Op . Commun. 115, 297 共1995兲. 关4兴A. Valle, J. Sa ma, and K.A. Sho e, IEEE J. Quan um Elec on. 31, 1423 共1995兲. 关5兴M. San Miguel, Q. Feng, and J.V. Moloney, Phys. Re . A 52, 1728 共1995兲. 关6兴J. Ma in-Regalado, S. Balle, M. San Miguel, A. Valle, and L. 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