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. Ma-
solle is pa ially suppo ed by PEDECIBA (P oyec o de
Desa ollo de Ciencias Basicas, U uguay) and CSIC
(Comision Sec o ial de In es igacion Cien i ica, U u-
guay).
M. Siamanna may be eached by e-mail a Sciamanna
@ elecom. pms.ac.be.
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