Different regimes of low-frequency fluctuations in vertical-cavity surface-emitting lasers

Di e en egimes o low- equency luc ua ions in e ical-ca i y su ace-emi ing lase s Ma c Sciamanna Se ice d’Elec omagne ´ isme e de Te ´le ´communica ions, Facul e ´Poly echnique de Mons, 31 Boule a d Dolez, B-7000 Mons, Belgium C is ina Masolle Ins 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 Neal B. Ab aham DePauw Uni e si y, G eencas le, Indiana 46135-0037 Fabien Rogis e , Pa ice Me ´g e , and Michel Blondel Se ice d’Elec omagne ´ isme e de Te ´le ´communica ions, Facul e ´Poly echnique de Mons, 31 Boule a d Dolez, B-7000 Mons, Belgium Recei ed July 11, 2002; e ised manusc ip ecei ed Augus 6, 2002 A nume ical s udy o e ical-ca i y su ace-emi ing lase s wi h op ical eedback demons a es he exis ence o wo di e en ypes o low- equency luc ua ions (LFFs). The compe i ion o wo equally dominan pola - iza ion modes cha ac e izes one ype o LFF, while he o he ype is cha ac e ized by powe d ops in a domi- nan pola iza ion mode and powe bu s s in he o hogonal dep essed mode. We cha ac e ize and compa e hese wo ypes o LFFs on he basis o hei pola iza ion p ope ies and hei dependency on he lase pa am- e e s. We show u he mo e ha a ansi ion is possible om one ype o LFF o he o he , depending on he alues o he linea aniso opies o he e ical-ca i y su ace-emi ing lase ca i y. © 2003 Op ical Socie y o Ame ica OCIS codes: 190.3100, 250.7260. 1. INTRODUCTION The e ical-ca i y su ace-emi ing lase (VCSEL) has ecei ed much a en ion since i s p oposal by K. Iga.1 The VCSEL shows up as a key de ice o op ical elecom- munica ion sys ems and all op ical signal handling. I exhibi s many ad an ages o e he con en ional edge- emi ing semiconduc o lase (EEL)2such as a single- longi udinal-mode ope a ion, low h eshold cu en , ci cu- la beam, low cos , and a dense packing capabili y. VCSELs a e also in e es ing om a undamen al iew- poin when conside ing hei ligh pola iza ion. The po- la iza ion o he emi ed ligh is no ixed and o en swi ches be ween wo o hogonal, linea ly pola ized (LP) s a es (xand y).3This pola iza ion swi ching is a d aw- back in pola iza ion-sensi i e applica ions bu on he o he hand may lead o new and in e es ing applica ions o op ical signal p ocessing.4 When using VCSELs in op ical elecommunica ion sys- ems, a small amoun o ligh is ine i ably ed back in o he VCSEL ca i y as he esul , o example, o pa asi ic e lec ions a he clea ed end o a ibe o om he disk in CD playe s. As wi h any semiconduc o lase , VCSELs a e sensi i e o op ical eedback. I was i s hough ha VCSELs may be ela i ely immune o op ical eed- back hanks o hei high ou pu -mi o e lec i i y, bu his e ec is o se by he sho VCSEL in aca i y ound ip. As a esul , he eedback a e, i.e., he ac ion o powe e lec ed back o he VCSEL pe uni o ime, is simila o ha o EELs o a gi en alue o ex e nal mi - o e lec i i y. Fi s s udies o VCSELs subjec o op i- cal eedback ha e iden i ied i e ypical dynamic egimes ha ha e p e iously been epo ed in EELs, such as linewid h na owing o b oadening, ex e nal-ca i y mode hopping, s able lase ou pu , chao ic dynamics, and lase es abiliza ion.5Howe e , he pola iza ion p ope - ies o VCSELs also lead o new and in e es ing phenom- ena when VCSEL a e coupled o an ex e nal ca i y. We can ci e, o example, pola iza ion-swi ching pheno- mena in pola ized op ical eedback,6,7 pola iza ion sel -modula ion in VCSELs coupled o op ical eedback h ough a qua e -wa e pla e,8–14 op ical- eedback- induced pola iza ion mode hopping, and gene alized mul is abili y.15 New applica ions o VCSELs o op ical elecommunica ions may be p oposed based on hese delay-induced dynamics. In pa icula , VCSELs wi h delayed eedback may gene a e ec o ial chaos in which no only he ield ampli ude and phase a e chao ic bu also he pola iza ion. Vec o ial chaos may be use ul o mul ichannel secu e communica ion sys ems.16 On he o he hand, high- equency oscilla ions e- sul ing om pola iza ion sel -modula ion may be Sciamanna e al. Vol. 20, No. 1/Janua y 2003/J. Op . Soc. Am. B 37 0740-3224/2003/010037-08$15.00 © 2003 Op ical Socie y o Ame ica used o op ical clock eco e y in all op ical egene a ion sys ems o as a s able sou ce o mic owa e oscilla ions.14 In his pape we s udy he eedback-induced, LFF e- gime. In his egime he lase in ensi y (obse ed on a nanosecond ime scale) exhibi s sudden andom d ops o- wa d ze o, eco e ing g adually o i s o iginal alue ( o a e iew see Re . 17). We show ha he coexis ence o wo pola iza ion modes allows VCSELs o p esen wo di e - en ypes o LFFs ha canno be esol ed by s udying he o al ou pu in ensi y bu ha can be clea ly dis in- guished by s udying he pola iza ion- esol ed modal in- ensi ies. Type I is cha ac e ized by compe i ion be ween he x- and y-pola iza ion modes wi h nea ly equal powe ; ype II is cha ac e ized by in ensi y d ops in he domi- nan pola iza ion mode and in ensi y bu s s in he de- p essed mode. Bo h LFF ypes a e ound in he same model, bu hei p esence depend on he lase and eed- back pa ame e s. We analyze he o igin o he wo LFF ypes and show a ou e o ype I LFF in ol ing ellip ically pola ized s a es in a bi u ca ion cascade. We also show ha a ansi ion om one ype o LFF o he o he migh be obse ed by modi ying he equency di e ence and/o he gain- o-loss a io o bo h pola iza ion modes. Ou e- sul s yield insigh in o pola iza ion dynamics o delayed VCSELs and mo i a e new expe imen s wi h VCSELs coupled o long ex e nal ca i ies. We analyze an example o ec o ial chaos ha appea s o be a d awback o pho- onic sys ems including VCSELs bu ha migh be use ul as a sou ce o chao ic communica ion sys ems. The plan o his pape is as ollows. In Sec ion 2 we p esen he model, which is an ex ension o he San Miguel–Feng–Moloney (SFM) model18 ha includes ex- e nal op ical eedback by aking in o accoun a single e- lec ion in he ex e nal ca i y. Sec ion 3 p esen s he wo ypes o LFF ha occu in he model and ha migh be clea ly dis inguished on he basis o hei pola iza ion p ope ies. Sec ion 4 demons a es ha a ying he bi- e ingence and/o he gain- o-loss a io o he pola iza- ion modes migh induce a ansi ion om one ype o LFF o he o he . Sec ion 5 analyzes he as pulsa ing dynamics unde lying he wo ypes o LFF, especially he phase cohe ence p ope ies o he modal in ensi ies. Sec- ion 6 shows ha he wo ypes o LFF a e also ound on an al e na i e model ha couples wo pola iza ion modes wi h di e en equencies and gains h ough nonlinea gain-sa u a ion coe icien s.19 Ou conclusions a e p e- sen ed in Sec ion 7. 2. MODEL Ou nume ical simula ions a e based on he SFM equa ions18 ex ended o iso opic op ical eedback as E ˙x⫽ ␬ 共1⫹i ␣ 兲关共N⫺1兲Ex⫹inEy兴 ⫺共 ␥ a⫹i ␥ p兲Ex⫹ Ex共 ⫺ ␶ 兲exp共⫺i ␾ 兲 ⫹关 ␤ sp共N⫹n兲/2兴1/2 ␰ 1⫹关 ␤ sp共N⫺n兲/2兴1/2 ␰ 2, (1) E ˙y⫽ ␬ 共1⫹i ␣ 兲关共N⫺1兲Ey⫺inEx兴 ⫹共 ␥ a⫹i ␥ p兲Ey⫹ Ey共 ⫺ ␶ 兲exp共⫺i ␾ 兲 ⫺i关 ␤ sp共N⫹n兲/2兴1/2 ␰ 1⫹i关 ␤ sp共N⫺n兲/2兴1/2 ␰ 2, (2) N ˙⫽⫺ ␥ N关N⫺ ␮ ⫹N共 兩 Ex 兩 2⫹ 兩 Ey 兩 2兲 ⫹in共EyEx *⫺ExEy *兲兴, (3) n˙ ⫽⫺ ␥ sn⫺ ␥ N关n共 兩 Ex 兩 2⫹ 兩 Ey 兩 2兲 ⫹iN共EyEx *⫺ExEy *兲兴, (4) whe e Exand Eya e he slowly a ying ampli udes o he linea ly pola ized componen s o he elec ic ield, Nis he o al ca ie di e ence be ween conduc ion and alence bands, and nis he di e ence be ween he popula ion in- e sions o he spin-up and spin-down adia ion channels. ␬ is he ield decay a e, ␥ Nis he decay a e o N, ␥ sis he spin- lip elaxa ion a e, ␣ is he linewid h enhancemen ac o , ␮ is he no malized injec ion cu en ( ␮ ⫽1a h eshold), ␥ ais he linea dich oism, and ␥ pis he linea bi e ingence. The e ms p opo ional o Ex,y( ⫺ ␶ ) accoun o a de- layed e lec ion om he ex e nal mi o (mul iple e lec- ions a e neglec ed). ␶ is he ex e nal ca i y ound- ip ime, is he eedback a e, and ␾ ⫽ ␻ 0 ␶ is he eedback phase, whe e ␻ 0is he op ical equency o he pola iza- ion mode in he absence o bi e ingence and eedback. Spon aneous emission noise is aken in o accoun h ough he las wo e ms in Eqs. (1) and (2). ␤ sp is he spon aneous emission a e and ␰ 1, ␰ 2a e wo unco e- la ed whi e noises wi h ze o mean and uni a y a iance. 3. TWO DIFFERENT TYPES OF LFF REGIMES In ou simula ions we conside he ollowing pa ame e s ixed20: ␬ ⫽300 ns⫺1, ␥ N⫽1ns ⫺1, ␣ ⫽3, and ␾ ⫽6 ad. Figu es 1 and 2 display wo di e en ypes o LFFs. D ops o he o al in ensi y a e obse ed in bo h cases [Figs. 1(a) and 2(a)], bu he unde lying pola iza ion dynamics a e e y di e en . Type I is cha ac e ized by he compe i ion o he x- and y-pola iza ion modes wi h nea ly equal powe [Figs. 1(b) and (c)], while ype II is cha ac e ized by powe d ops in he dominan y-mode and powe bu s s in he dep essed x-mode [Figs. 2(b) and (c)]. Type I LFF was p e iously ound in simula ions o he SFM model,20 while ype II was obse ed expe imen ally and nume ically in Re . 21. I is wo h no ing ha he model o Re . 21 is also based on he SFM model bu di - e s om Eqs. (1)–(4) in wo espec s. Fi s Re . 21 in- cludes mul iple e lec ions in he ex e nal ca i y. The high e lec i i y o he ou pu mi o in VCSELs migh jus i y inclusion o mul iple ex e nal-ca i y ound ips e en o a low eedback a e, as i s sugges ed by Law and Ag awal.22 Inclusion o mul iple ound- ip e ms in he ield equa ion would mos p obably yield a be e quan i a i e ag eemen wi h expe imen s, since i adds co ec ion ac o s o he gain and equency o he s eady- s a e, ex e nal-ca i y modes.23,24 Howe e ou esul s show ha bo h ypes o LFF a e complemen a y and co- 38 J. Op . Soc. Am. B/Vol. 20, No. 1/Janua y 2003 Sciamanna e al. exis in he same model [Eqs. (1)–(4)] using a single ound- ip e m in he ex e nal ca i y as in he Lang– Kobayashi app oxima ion.25 As a i s conclusion, mul- iple eedback-delay e ms a e no esponsible o he ype II LFF epo ed expe imen ally in Re . 21. As we dem- ons a e below in Sec ion 4, whe he he VCSEL ope a es wi h ype I o ype II LFF is mainly dependen on he lin- ea ca i y aniso opies. A second di e ence be ween he model o Re . 21 and Eqs. (1)–(4) consis s in he inclusion o a sel -gain- sa u a ion coe icien in he model o Re . 21. Ou nu- me ical s udy indica es ha he gain-sa u a ion coe i- cien inc eases he damping o he as pulsa ing dynamics ha unde lie he slow LFF ime aces (see Sec- ion 5), bu is no esponsible o he de e mina ion o he ype o LFF dynamics. Figu es 1(d) and 2(d) show he deg ee o pola iza ion FP26 as a unc ion o ime a e a e aging on 1 ns o ype I o ype II LFF, espec i ely. FP is de ined on he basis o he S okes pa ame e s s0 o s3, which a e compu ed om s0⫽ 兩 Ex 兩 2⫹ 兩 Ey 兩 2, (5) s1⫽2Re 兵 Ex *Ey 其 , (6) s2⫽2Im 兵 Ex *Ey 其 , (7) s3⫽ 兩 Ex 兩 2⫺ 兩 Ey 兩 2. (8) FP is hen gi en by FP ⫽ 具 s1 典 2⫹ 具 s2 典 2⫹ 具 s3 典 2 具 s0 典 2, (9) in which he S okes pa ame e s a e a e aged o e a de i- ni e ime span. In Figs. 1 and 2, a e aging is done on 1 ns, as men ioned abo e. FP anges be ween 0 o pola iza ion-incohe en ligh and 1 o pola iza ion- cohe en ligh . The wo LFF ypes may be dis inguished by he beha io o FP. In ype II LFF, FP is close o 1 in he ime span be ween wo d ops o he dominan mode. Analysis o he S okes pa ame e s con i ms ha he ligh is close o linea ly pola ized along y. Once he powe d op occu s FP dec eases s ongly as a esul o ime- dependen luc ua ions o he ligh pola iza ion, hen e- co e s again o 1 be o e he nex powe d op. By con- as , in ype I LFF, FP is always close o ze o: The ligh pola iza ion luc ua es s ongly wi h ime. 4. EFFECT OF BIREFRINGENCE AND GAIN ANISOTROPY: TRANSITION FROM ONE TYPE OF LFF TO THE OTHER The ype I LFF epo ed in Re . 20 occu s o in e nal pa- ame e s such ha bo h pola iza ion modes o he soli a y VCSEL a e uns able (which is a o ed by low ␥ s27). We ind ha ype I LFF may occu also in pa ame e egions whe e bo h modes a e s able (which is a o ed by la ge ␥ s) o whe e only one pola iza ion mode is s able. In his las case op ical eedback may u n on he uns able pola - iza ion mode, leading o ei he ype I o ype II LFF. Which o he wo LFF ypes occu s depends on ␥ aand ␥ p. The ansi ion be ween bo h ypes o LFF is illus a ed in Fig. 3. Figu e 3 shows ime aces o he o al in ensi y I o , he x-LP mode in ensi y Ixand he y-LP mode in en- si y Iy, a e a e aging on 1 ns, o di e en alues o ␥ a and ␥ pwhile he o he lase and eedback pa ame e s e- main ixed. Since he o al a e aged powe s o Figs. 1 and 2 a e signi ican ly di e en , one migh suspec ha ype II LFF can be obse ed a ound he h eshold o one pola iza ion a low injec ion, whe eas ype I LFF appea s a mode a e injec ion cu en wi h symme ic oscilla ion condi ions o he wo pola iza ion modes. Howe e ou nume ical simula ions in Fig. 3 show ha bo h ypes o LFF migh occu o he same injec ion cu en ␮ and ad- equa e alues o ␥ aand ␥ p. Type II LFF is a o ed by dich oism and bi e ingence. ␥ amodels he di e en gain- o-loss a io o he x- and y-pola ized modes ( ␥ a⬎0 gi es he y-mode a lowe h eshold). Fo pa ame e s such ha bo h modes com- pe e wi h nea ly equal mean powe ( ype I LFF), inc eas- Fig. 1. Type I LFF. (a) The o al in ensi y I o , (b), (c), he modal in ensi ies Ixand Iy, and (d) he deg ee o pola iza ion FP a e plo ed as a unc ion o ime a e a e aging on 1 ns. The b ack- e s indica e a e aged a iables. The pa ame e s a e ␥ a ⫽0.1 ns⫺1, ␥ p⫽10 ns⫺1, ␥ s⫽50 ns⫺1, ␮ ⫽1.8, ⫽60 ns⫺1, ␤ sp ⫽10⫺6ns⫺1, ␶ ⫽3ns. Fig. 2. Same as Fig. 1 bu showing ype II LFF. The pa am- e e s a e ␥ a⫽0.5 ns⫺1, ␥ p⫽2ns ⫺1, ␥ s⫽50 ns⫺1, ␮ ⫽1.1, ⫽40 ns⫺1, ␤ sp ⫽10⫺6ns⫺1, ␶ ⫽3ns. Sciamanna e al. Vol. 20, No. 1/Janua y 2003/J. Op . Soc. Am. B 39 ing ␥ amay cause he y-mode o be dominan and he x-mode o be dep essed, i.e., a ansi ion o ype II LFF. This ansi ion om ype I LFF o ype II LFF is shown in Figs. 3(a)–( ). Figu es 3(d)–( ) co espond o ype I LFF bu inc easing he dich oism leads o ype II LFF [Figs. 3(a)–(c)] in which he x-LP mode is dep essed. Va ying he bi e ingence pa ame e ␥ pmigh also lead o a ansi ion om one ype o LFF o he o he . When ␥ p⬎0 he x-mode has he lowe equency and may be a o ed because o he e ec o he sa u able dispe sion associa ed wi h he ␣ - ac o .27 Wi h eedback, o pa am- e e s such ha bo h modes compe e wi h equal powe , in- c easing ␥ pleads o a ansi ion o ype II LFF in which he x-mode is he dominan mode and he y-mode is he dep essed one. Such a ansi ion om ype I LFF o ype II LFF is shown in Figs. 3(d)–(i). Figu es 3(d)–( ) co e- spond o ype I LFF bu inc easing he bi e ingence leads o ype II LFF [Figs. 3(g)–(i)] in which he y-LP mode is dep essed. I is impo an o poin ou ha o la ge ␥ ao la ge ␥ p, o obse e bu s s o powe in he dep essed mode, spon aneous-emission-noise e ms ha e o be included in Eq. (1). In he absence o noise, o la ge enough ␥ ao ␥ p, he dep essed mode a e a ansien u ns o and e- mains o . S ochas ic p ocesses in he a e equa ions a e he e o e a minimal equi emen o obse e ype II LFF. Figu e 3 sugges s also ha ei he a ying ␥ ao ␥ pwould be su icien o obse e a ansi ion om ype I o ype II LFF, i.e., ha bo h linea aniso opies a e no needed o obse e he wo ypes o LFF. We ha e indeed e i ied nume ically ha keeping ␥ aequal o ze o and a ying ␥ p migh yield ei he ype I LFF (a low bi e ingence) o ype II LFF (a la ge bi e ingence). Fu he mo e i we keep ␥ pequal o ze o and a y ␥ a his migh yield ei he ype I LFF (a low dich oism) o ype II LFF (a la ge di- ch oism). An expe imen al con i ma ion o he in luence o he linea ca i y aniso opies on he ype o LFF dynamics de- se es echniques o induce and con ol bi e ingence and dich oism o he VCSEL ca i y. A leas wo echniques may be applied in ha sense. Fi s he so-called ho spo echnique makes use o an ex e nal collima ed beam and induces bi e ingence h ough elas o-op ic e ec s.28 Sec- ond Panajo o e al. epo ed con ol o VCSEL pola iza- ion and ca i y aniso opies h ough applica ion o s ain on he VCSEL moun ing.29 5. FAST POLARIZATION DYNAMICS Nex we analyze he beha io o he as -pulsing modal and o al in ensi ies du ing a powe d op. As indica ed in Re . 17 he LFF egime is he ime-a e aged en elope o a ain o pulses. An in e es ing ques ion ha de- se es in es iga ion is he analysis o he co ela ion be- ween he in ensi y pulses emi ed by bo h LP modes. Figu es 4(a)–(b) display he esul s o ype I LFF. Fig- u e 4(a) shows he modal in ensi ies Ix(black) and Iy (g ay) when a o al (and modal) powe d op occu s. I shows ha he modal in ensi ies exhibi an an iphase dy- namic be o e he powe d op bu a e he d op occu s hey synch onize and emi pulses in phase depa ing om he noise le el. Howe e we obse e ha when he x-LP mode in ensi y Ixinc eases, he y-LP mode in ensi y Iy dec eases, and ice e sa. The x- and y-LP modes ex- change ene gy on a slow ns ime scale, which co esponds o a pa ial an iphase beha io . Be o e he powe d op (a e he eco e y p ocess o he o al powe ), he o al in- ensi y will be nea ly cons an since he modes a e in an- iphase. Bu when a d op occu s he o al in ensi y will Fig. 3. T ansi ions om ype I o ype II LFF, and ice e sa, o di e en alues o ␥ aand ␥ p. The o he lase pa ame e s e- main ixed wi h he ollowing alues: ␮ ⫽1.2, ⫽40 GHz, ␥ s ⫽50 ns⫺1, ␶ ⫽3ns, ␤ sp ⫽10⫺6ns⫺1. (a), (d), and (g) show he ime ace o he o al in ensi y I o a e a e aging on 1 ns. (b), (e), and (h) show he a e aged x-LP mode in ensi y Ix. (c), ( ) and (i) show he a e aged y-LP mode in ensi y Iy.(a)–(c) co - espond o ype II LFF wi h ␥ a⫽1ns ⫺1, ␥ p⫽0.1 ns⫺1.(d)–( ) co espond o ype I LFF wi h ␥ a⫽0.1 ns⫺1, ␥ p⫽0.1 ns⫺1. (g)–(i) co espond o ype II LFF wi h ␥ a⫽0.1 ns⫺1, ␥ p ⫽15 ns⫺1. Fig. 4. Fas -pulsa ing dynamics o o al and modal in ensi ies o ype I LFF. (a) shows he una e aged Ix(black cu e) and Iy (g ay cu e), and co esponds o an enla gemen o he LFF ime ace o Fig. 1. Bo h modes a e in pa ial an iphase be o e he powe d op while exhibi ing in-phase pulsa ing ou pu jus a e he d op, hence du ing he eco e y p ocess. (b) and (c) show he PDFs o he o al (hea y black cu e) and modal (x, hing ay cu e; y, dashed g ay cu e) in ensi ies o a ime span o 30 ns be o e (b) and a e (c) he powe d op. 40 J. Op . Soc. Am. B/Vol. 20, No. 1/Janua y 2003 Sciamanna e al. display a ain o pulses depa ing om he noise le el simila o wha occu s in he modal in ensi ies, since modes emi in phase. We gain insigh in o he co ela ion p ope ies o he LP-mode in ensi ies in he LFF egime by compu ing, as shown in Figs. 4(b) and (c), he p obabili y dis ibu ion unc ions (PDFs) o he o al (hea y black cu e), x-LP mode in ensi y ( hin g ay cu e), and y-LP mode in ensi y (dashed g ay cu e) o a 30-ns ime in e al be o e (b) and a e (c) he powe d op. Fo cla i y he in ensi ies I a e no malized by hei co esponding mean alues 具I典. In bo h cases (b) and (c) he PDFs o he modal in ensi ies mono onically decay om he noise le el, indica ing emis- sion o pulses wi h he mos p obable in ensi y close o he noise le el. Be o e a powe d op, he PDF o he o al in ensi y displays a peak a ound i s mean in ensi y alue. In ha case he o al in ensi y exhibi s small luc ua ions a ound i s mean alue, since LP modes a e in pa ial an- iphase be o e a powe d op. In con as he PDF o he o al in ensi y mono onically decays o a ime in e al jus a e he powe d op, i.e., du ing he powe eco e y p ocess, since he wo LP modes emi pulses in phase and he co esponding o al in ensi y exhibi s pulses in phase wi h he LP modes. Coexis ence o in-phase and pa ial-an iphase dynam- ics is epo ed he e o he i s ime in a pola iza ion- uns able semiconduc o lase . This opic has a ac ed much in e es in connec ion wi h delayed lase sys ems because o i s po en ial o con ibu e o he unde s anding o he in luence o a mul imode beha io on he o al in- ensi y dynamics. Ou esul s sugges ha a combina- ion o wo ypes o phase cohe ence in he modal in ensi- ies is a gene al ea u e o delayed-mul imode lase sys ems, including sys ems wi h wo pola iza ion modes,12,13 wi h mul iple ans e se modes,30 o EELs wi h many longi udinal modes,31,32 e en i he physics un- de lying he exci a ion o modes is di e en . Figu es 5(a)–( ) display he esul s o ype II LFF. Be o e he d op he dep essed pola iza ion is o , and i ises wi h an in-phase pulsing beha io a e he d op o he dominan mode. The pa ial synch oniza ion o pulses in he pola ized modes a e a powe d op is he e- o e a common ea u e o bo h ypes o LFFs. In Figs. 5(a)–( ) he powe bu s in he dep essed mode is delayed wi h espec o he powe d op in he dominan mode. Howe e we ha e also obse ed in he same ime ace o LFF bu s s o he dep essed mode which a e simul a- neous wi h d ops o he dominan mode. Ou esul s sugges ha he delay be ween he d op in he dominan LP mode and he bu s o he o he dep essed LP mode is a andom a iable wi h a nonze o mean alue. The s a- is ical p ope ies o ha a iable delay is a subjec o u- u e wo k. Since spon aneous-emission noise plays an impo an ole in he dynamics o he dep essed mode, we specula e ha he bu s s a e due o he complex in e play o noise and ca ie dynamics: The inc ease in he ca - ie popula ion ha ollows he d op o he dominan mode bene i s he dep essed mode which hen ises wi h a bu s om he noise le el. In ag eemen wi h Re . 20 we ind nume ically ha spon aneous-emission noise is no needed o obse e ype I LFF, in con as wi h ype II LFF. We he e o e p o- pose a i s de e minis ic explana ion o ype I LFF. We analyze he ou e o LFF by compu ing bi u ca ion dia- g ams o he x- and y-linea ly pola ized in ensi ies as a unc ion o he eedback a e ; see Fig. 6. Figu e 7 shows ypical dynamics o di e en alues o he eedback a e and illus a es he bi u ca ion s udy o Fig. 6. Ini ially o small eedback a es in Fig. 6, he ou e o LFF in- ol es emission o bo h x- and y-pola ized modes. We cal- cula e he op ical spec um o he x- and y-pola ized ields, inding ha hey ha e he same op ical equency, i.e., i is an ellip ically pola ized s eady s a e; see Fig. 7(a). Fo inc easing eedback Fig. 6 shows a bi u ca ion cascade al e na ing be ween ellip ically pola ized s eady s a es and egions o ime-pe iodic [see Fig. 7(b)] o e en chao ic dynamics [see Fig. 7(c)]. A s ill la ge eedback a es, windows o chaos b oaden and lead o ype I LFF, as shown in Figs. 7(d)–( ). This ou e o LFF in VCSELs Fig. 5. Fas -pulsa ing dynamics o (a), (d) I o ,(b),(e)Ix, and (c), ( ) Iy o ype II LFF. The una e aged in ensi ies a e plo ed as a unc ion o ime in he ime in e al co esponding o one powe d op o Fig. 2. (d)–( ) a e enla gemen s o (a)–(c), espec- i ely, o cla i y he in-phase pulsa ing dynamics o he LP modes a e he powe d op. Fig. 6. Bi u ca ion diag am o (a) Ixand (b) Iyas a unc ion o he eedback a e showing one possible ou e o ype I LFF. The pa ame e s a e ␥ a⫽0.5 ns⫺1, ␥ p⫽8ns ⫺1, ␥ s⫽10 ns⫺1, ␮ ⫽1.03, ␤ sp ⫽0, ␶ ⫽1ns. Sciamanna e al. Vol. 20, No. 1/Janua y 2003/J. Op . Soc. Am. B 41 di e s om ha obse ed in con en ional pola iza ion- s able EELs.33 As p e iously epo ed20 he s eady s a es o he delayed VCSEL sys em co espond o ex e nal-ca i y modes ha a e ei he linea ly pola ized along xo yo ellip ically pola ized. Ou esul s indica e ha ellip ically pola ized ex e nal-ca i y modes appea in he bi u ca ion ou e o LFF, and he e o e mo i a e u - he heo e ical s udies on linea -s abili y analysis o ex e nal-ca i y modes in he delayed-VCSEL p oblem. 6. ALTERNATIVE MODEL Se e al di e en models ha e been used o s udy pola - iza ion dynamics in single- ans e se-mode VCSELs. A ypical al e na i e app oach o he SFM model o soli- a y VCSELs is o conside a wo-mode- a e-equa ions model including sel - and c oss-sa u a ion coe icien s.19 The gain-sa u a ion coe icien s model in a phenomeno- logical way di e en coupling p ocesses be ween pola iza- ion modes such as hose induced by spa ial hole bu ning ha is he esul o ca ie di usion. The pola iza ion s abili y is de e mined by he alues o he sel - and c oss- sa u a ion coe icien s. Such a model can be ex ended o op ical eedback by use o he Lang–Kobayashi app oach yielding a delayed- eedback e m in he ield equa ions as in Eqs. (1) and (2). The model has been success ully used o desc ibe o example pola iza ion sel -modula ion in VCSELs coupled o a pola iza ion- o a ing op ical eedback10,12–14 and o s udy op ical- eedback-induced po- la iza ion swi chings.7,34 The model equa ions a e w i en as E ˙x⫽ ␬ 共1⫺i ␣ 兲共NFx⫺1兲Ex ⫺共 ␥ a⫹i ␥ p兲Ex⫹ Fx共 ⫺ ␶ 兲exp共⫺i ␾ 兲 ⫹共 ␤ sp N/2兲1/2 ␰ 1⫹共 ␤ sp N/2兲1/2 ␰ 2, (10) E ˙y⫽ ␬ 共1⫹i ␣ 兲共NFy⫺1兲Ey ⫹共 ␥ a⫹i ␥ p兲Ey⫹ Ey共 ⫺ ␶ 兲exp共⫺i ␾ 兲 ⫺i共 ␤ sp N/2兲1/2 ␰ 1⫹i共 ␤ sp N/2兲1/2 ␰ 2,(11) N ˙⫽⫺ ␥ N关N⫺ ␮ ⫹N共 兩 Ex 兩 2⫹ 兩 Ey 兩 2兲兴, (12) wi h Fx,y⫽1⫺ ⑀ xx,yx 兩 Ex 兩 2⫺ ⑀ xy,yy 兩 Ey 兩 2. (13) ⑀ xx , ⑀ yy a e sel -gain-sa u a ion coe icien s and ⑀ xy , ⑀ yx a e c oss-gain-sa u a ion coe icien s. Linea ca i y aniso opies ␥ aand ␥ pa e included in he model since Sec ion 4 has shown hey we e impo an in obse ing bo h ypes o LFF. Excep o he inclusion o ␥ aand ␥ p, Eqs. (10)–(13) a e iden ical o hose o he model used in Re s. 12–14 i one conside s he di e en no maliza ion o he ield and he ca ie a iable.27 Masolle and Ab aham ha e analyzed such a wo-mode model in he amewo k o he LFF egime.20 They ha e ound ha ype I LFF can be obse ed p o ided ha non- ze o gain-sa u a ion coe icien s a e included in he a e equa ions. As can be expec ed om ou p e ious analy- sis, we now show ha such a model could also ep oduce ype II LFF depending on he alues o he linea ca i y aniso opies. Fi s ou nume ical simula ions indica e ha o obse e LFF, nonze o alues o he c oss-gain- sa u a ion coe icien s should be included in Eqs. (10)– (13), bu he sel -gain sa u a ion coe icien s may be omi - ed. Ou esul s he e o e subs an ia e p e ious nume ical s udies on ha delayed-VCSEL model20 by dis- c imina ing be ween he in luence o he c oss-gain sa u- a ion coe icien and ha o he sel -gain sa u a ion coe - icien on inducing LFF. Figu e 8 illus a es ype II LFF ob ained wi h he model o Eqs. (10)–(13) o he same lase and eedback pa ame e s as in Fig. 2 and wi h ⑀ xx ⫽ ⑀ yy ⫽0 and ⑀ xy ⫽ ⑀ yx ⫽0.01. I shows ha he y-LP mode in ensi y ex- hibi s powe d ops simila o he dynamics o he o al in- ensi y, and ha he x-LP mode is s ongly dep essed bu Fig. 7. Time aces o Ix(black cu es) and Iy(g ay cu es) o (a) ⫽23 ns⫺1,(b) ⫽23.2 ns⫺1, (c) ⫽34 ns⫺1.(d)–( ) show I o ,Ix, and Iy, espec i ely, a e a e aging o e 1 ns o ⫽60 ns⫺1. The igu e shows ype I LFF occu ing a e he bi- u ca ion cascade o Fig. 6. The o he pa ame e s a e as in Fig. 6. Fig. 8. Time aces o (a) he o al in ensi y I o and (b), (c) he modal in ensi ies Ixand Iy, espec i ely, a e a e aging on 1 ns, showing ype II LFF wi h he model o a e Eqs. (10)–(13). The lase and eedback pa ame e s a e he same as in Fig. 1 wi h ⑀ xx ⫽ ⑀ yy ⫽0and ⑀ xy ⫽ ⑀ yx ⫽0.01. 42 J. Op . Soc. Am. B/Vol. 20, No. 1/Janua y 2003 Sciamanna e al. exhibi s powe bu s s consequen o he d ops in he main mode. Fu he mo e we a i e a he same conclusion conce ning he in luence o he linea aniso opies as ha o Sec ion 4: Type II LFF is a o ed by la ge ␥ aand/o la ge ␥ p. Ou esul s, combined wi h hose o Re . 20, show ha bo h ypes o LFF dynamics a e ound wi h he a e Eqs. (10)–(13). Ou conclusions also ma ch hose o Sec ion 4 conce ning he in luence o he linea ca i y aniso opies on he ype o LFF dynamics. 7. CONCLUSION In summa y nume ical solu ions o a model o VCSELs wi h op ical eedback show ha VCSELs migh exhibi wo quali a i ely di e en egimes o LFFs ha can be clea ly dis inguished by he pola iza ion beha io . We ha e shown ha he deg ee o pola iza ion FP allows a clea disc imina ion be ween he wo ypes o LFF dy- namics. Tha app oach leads o a new insigh no sug- ges ed by he p e ious app oach o Re s. 20, 21 ha could be es ed expe imen ally. The wo ypes o LFF dynam- ics exis in a wide ange o pa ame e s and o wo di e - en physical models. Fu he mo e we ha e shown ha a ansi ion om one ype o LFF o he o he migh be ob- se ed by modi ying he equency spli ing and/o he gain- o-loss a io o he modes. Since only ype II LFF has been expe imen ally epo ed in VCSELs, ou esul s mo i a e new expe imen s on VCSELs coupled o long ex- e nal ca i ies. In ligh o ou esul s special a en ion may be gi en o he in luence o he linea ca i y aniso o- pies. Ou esul s also analyze an example o ec o ial chaos in lase s which migh be o use o chao ic commu- nica ion sys ems. ACKNOWLEDGMENTS M. Sciamanna is a Resea ch Fellow o he Fonds Na ional de la Reche che Scien i ique (FNRS, Belgium). Ou e- sea ch is also suppo ed by he In e uni e si y A ac ion Pole (IAP) p og am o he Belgian go e nmen . C. 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