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
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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
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