Cellula au oma on model o he simula ion o lase dynamics
J. L. Guisado,1,*F. Jime
´nez-Mo ales,1,† and J. M. Gue a2,‡
1Depa amen o de Fı
´sica de la Ma e ia Condensada, Uni e sidad de Se illa, P.O. Box 1065, 41080 Se illa, Spain
2Depa amen o de Op ica, Facul ad de CC. Fı
´sicas, Uni e sidad Complu ense de Mad id, 28040 Mad id, Spain
共Recei ed 28 No embe 2002; published 20 June 2003兲
The classical modeling app oach o lase s udy elies on he di e en ial equa ions. In his pape , a cellula
au oma on model is p oposed as an al e na i e o he simula ion o popula ion dynamics. E en hough he
model is simpli ied i cap u es he essence o lase phenomenology: 共i兲 he e is a h eshold pumping a e ha
depends in e sely on he decaying li e ime o he a oms and he pho ons; and 共ii兲depending on hese li e imes
and on he pumping a e, a cons an o an oscilla o y beha io can be obse ed. Mo e complex beha io s such
as spiking and pa e n o ma ion can also be s udied wi h he cellula au oma on model.
DOI: 10.1103/PhysRe E.67.066708 PACS numbe 共s兲: 07.05.Tp, 42.55.Ah, 42.60.J , 42.65.S
I. INTRODUCTION
In a lase sys em, he in e ac ions among simple a oms
and he adia ion hey p oduce can gi e ise o coope a i e
phenomena 关1兴. Howe e , he usual app oach o i s s udy is
based on e y de ailed mic oscopical equa ions, which some-
how mask he ac ion o such coope a i e p ope ies. In his
s udy, a simple cellula au oma on model is p esen ed, which
ep oduces much o he lase phenomenology, special a en-
ion being ocused on hese sel -o ganizing coope a i e e -
ec s. Ou model is in e es ing in ha i illus a es he eme -
gence o lase p ope ies as coope a i e phenomena based on
simple unde lying ules. I can also be use ul in calcula ing
lase ou pu in si ua ions which a e di icul o ea wi h he
adi ional app oach based on he esolu ion o de ailed di -
e en ial equa ions 关2,3兴. One example is when dealing wi h
complex bounda y condi ions o nume ical di icul ies in he
in eg a ion o he equa ions.
Cellula au oma a 关4,5兴a e a class o spa ially and em-
po ally disc e e ma hema ical sys ems, cha ac e ized by local
in e ac ion and synch onous dynamical e olu ion. They ha e
he abili y o gene a e a e y complex beha io om se s o
componen s ha in e ac locally wi h ela i ely simple ules.
They p o ide good models o a wide a ie y o physical
sys ems 关6,7兴exhibi ing coope a i e phenomena, such as
magne iza ion in solids 关8兴, eac ion-di usion p ocesses 关9兴,
luid dynamics o complex si ua ions 关10兴, g ow h phenom-
ena 关11兴, e c. As an example o an applica ion pa icula ly
close o lase dynamics, a cellula au oma on model has been
ecen ly applied o success ully s udy he exci ed-s a e dy-
namics o a omic oxygen, which play a p ominen ole in
c ea ing he au o a bo ealis 关12兴.
This pape is o ganized as ollows. Sec ion II p esen s a
b ie e iew o lase dynamics including he lase phenom-
enology ha ou model a emp s o ep oduce. Sec ion III
desc ibes he cellula au oma on model. The simula ions a e
ca ied ou and hei esul s a e p esen ed in Sec. IV. Finally,
he conclusions o his s udy a e explained in Sec. V.
II. LASER DYNAMICS
A lase is a de ice ha gene a es o ampli ies elec omag-
ne ic adia ion based on he s imula ed emission phenom-
enon. The basic componen s o a lase sys em a e 共1兲alase
medium—an app op ia e collec ion o a oms, molecules, ions
o a semiconduc o c ys al 共 hese elemen s will gene ally be
e e ed o as ‘‘a oms’’兲;共2兲apumping p ocess ha exci es
elec ons om hese a oms o uppe ene gy le els, due o
some ex e nal elec ical, op ical, o chemical ene gy sou ce;
共3兲op ical eedback elemen s ha e lec epea edly he a-
dia ion beam in o he lase medium 共in a lase oscilla o 兲,o
allow i o pass only once h ough i 共in a lase ampli ie 兲.
The wo king p inciple o lase is s imula ed emission, i.e.,
an exci ed a om can decay o a lowe s a e s imula ed by he
p esence o a pho on wi h ene gy equal o he di e ence
be ween he wo ene gy le els, emi ing a second pho on
wi h he same equency and p opaga ing in he same di ec-
ion. The p ocess o abso p ion has he same p obabili y, so
s imula ed emission domina es only when a popula ion in-
e sion is induced in he ma e ial by some pumping
mechanism.
A simpli ied bu ye ealis ic model o many eal lase s is
he ou -le el lase sys em shown in Fig. 1. The popula ion
dynamics o a lase 共 he a ia ion wi h ime in he numbe o
lase pho ons and in he popula ion in e sion, o numbe o
elec ons in he uppe lase le el minus he numbe o elec-
ons in he lowe lase le el兲is usually desc ibed as a sys-
em o coupled di e en ial equa ions called lase a e
equa ions.
The a e equa ions can be pu in o hei simples o ms
when he li e imes o E1and E3le el elec ons a e negli-
gible as compa ed o he li e ime o E2le el elec ons.
The e o e he E1le el elec on popula ion is N1⯝0, and
hus he popula ion in e sion is app oxima ely equal o he
uppe lase le el popula ion N( )⫽N2( )⫺N1( )⯝N2( );
and he abso p ion o lase pho ons by elec ons a le el E1is
negligible. Addi ionally, he pumping in o le el E2can be
desc ibed by a simple pumping a e R.
Unde hese assump ions, which a e qui e ealis ic, only
*P esen add ess: Cen o Uni e si a io de Me
´ ida, Uni e sidad de
Ex emadu a, 06800 Me
´ ida 共Badajoz兲, Spain. Elec onic add ess:
[email p o ec ed]
†Elec onic add ess: [email p o ec ed]
‡Elec onic add ess: [email p o ec ed]
PHYSICAL REVIEW E 67, 066708 共2003兲
1063-651X/2003/67共6兲/066708共8兲/$20.00 ©2003 The Ame ican Physical Socie y67 066708-1
le els E0and E2can be conside ed, and he a e equa ions
a e a sys em o wo coupled di e en ial equa ions 关2,3兴:
dn共 兲
d ⫽KN共 兲n共 兲⫺n共 兲
␶
c,共1兲
dN共 兲
d ⫽R⫺N共 兲
␶
a
⫺KN共 兲n共 兲.共2兲
He e Ris he pumping a e,
␶
ais he decay ime o he uppe
lase le el (E2),
␶
cis he decay ime o he lase pho ons in
he ca i y, and Kis a cons an called ‘‘coupling cons an .’’
The i s equa ion e lec s he a ia ion wi h ime o he num-
be o lase pho ons, n( ), which is ela ed o he lase beam
in ensi y. The second equa ion ep esen s he empo al a ia-
ion o he popula ion in e sion N( ). This is a se o wo
coupled nonlinea equa ions due o he p esence o he p od-
uc e m KN( )n( ).
Lase ope a ion does no gene ally p oduce a smoo h and
con inuous esponse, bu exhibi s di e en kinds o cha ac-
e is ic ansien o modula ion beha io s such as spiking,
elaxa ion oscilla ion, gain swi ching. Lase spiking, in pa -
icula , e e s o he pulses ha ypically occu du ing he
ini ial u n-on phase o many lase s. In hese cases, he lase
signal p esen s a sequence o sha p, la ge-ampli ude na ow
pulses o ‘‘spikes.’’ Each spike is ypically a ac ion o a
mic osecond wide, and hey a e sepa a ed by a ew mic o-
seconds. In some lase s, he spiking beha io jumps in an
e a ic way. In o he cases, o unde mo e s able condi ions,
i is possible o ob ain a mo e egula spiking beha io , in
which he ampli ude o he spikes g adually damps ou wi h
ime in o a elaxa ion p ocess. These a e some imes called
elaxa ion oscilla ions, and include lase s wi h la ge-
ampli ude oscilla ions when enough ime has elapsed o he
ampli ude o elax. This kind o beha io a ies o he di -
e en ypes o lase s. The spiking oscilla o y esponse is
cha ac e is ic o mos solid-s a e and semiconduc o lase s,
which ha e a subs an ially longe decay ime o he uppe
lase le el han he decay ime o he pho ons in he lase
ca i y:
␶
aⰇ
␶
c. Lase spiking and elaxa ion oscilla ions a e
no obse ed, on he o he hand, in mos gas lase s, which
usually ha e decay imes o he same o de o magni ude,
␶
a⬇
␶
c.
The s eady-s a e solu ion o he a e equa ions abo e he
h eshold 共i.e., wi h lase emission兲is
Ns⫽1
K
␶
c,共3兲
ns⫽
␶
c
冉
R⫺1
K
␶
a
␶
c
冊
.共4兲
In o de o his solu ion o ha e a physical meaning, he
numbe o pho ons, n, has o be g ea e han ze o. The mini-
mum alue o Rp oducing a lase emission in he ca i y,
known as h eshold pumping a e R is ound as
R ⫽1
K
␶
a
␶
c.共5兲
Fo he case when he spiking beha io in he lase has
elaxed o small-ampli ude luc ua ions, a linea ized small-
signal analysis o he a e equa ions abou he s eady-s a e
solu ion can be ca ied ou . Following wo di e en si ua-
ions a ise.
共a兲Nonspiking lase s: Fo lase s in which
␶
a⬇
␶
c, he
solu ions o n( ) and N( ) a e eal exponen ials. The sys em
is o e damped and he esponse o he lase o any pe u ba-
ion dies ou exponen ially owa ds he s eady s a e. The so-
lu ion is a s able node.
共b兲Relaxa ion oscilla ions: When
␶
aⰇ
␶
c he solu ions
ha e an exponen ially damped sinusoidal o m owa ds he
s eady-s a e alues. The sys em esponds o any pe u ba ion
exhibi ing elaxa ion oscilla ions. The solu ion is a s able o-
cus. The elaxa ion-oscilla ion equency, o en e e ed o as
spiking equency, is ound as
␻
sp⫽
冑
冉
1
2
␶
a
R
R
冊
2
⫺1
␶
a
␶
c
冉
R
R
⫺1
冊
.共6兲
The necessa y condi ion o he appea ance o he elax-
a ion oscilla ions is ha he quan i y inside he squa e oo be
posi i e, yielding
␶
a
␶
c
⬎
冉
R
R
冊
2
4
冉
R
R
⫺1
冊
.共7兲
III. CELLULAR AUTOMATA MODEL
Cellula au oma a 共CA兲a e ully disc e e dynamical sys-
ems, whe e he s a es a e chosen in a ini e se and dis ib-
u ed on a disc e e egula la ice, and he ime e olu ion is
un synch onously in all he si es o his la ice and each si e
FIG. 1. Schema ic iew o a ou -le el lase sys em.
GUISADO, JIME
´NEZ-MORALES, AND GUERRA PHYSICAL REVIEW E 67, 066708 共2003兲
066708-2
changes i s s a e acco ding o a local ule ha only depends
upon i s neighbo ing alues. Despi e he simplici y o hei
cons uc ion, CA a e ound o be capable o a di e se and
complex beha io and a e o en used as a p o o ype o he
analysis o he spon aneous eme gence o o de ed beha io
in spa ially ex ended sys ems ha a e locally coupled.
A. S a es o he cells
We conside a sys em composed o a squa e la ice o
Nc⫽L⫻Lcells wi h pe iodic bounda y condi ions. Two
a iables ai( ) and ci( ) a e associa ed o each node o his
la ice. The i s one, ai( ), ep esen s he s a e o he elec-
on in node ia a gi en ime . An elec on in he lase
g ound s a e akes he alue ai( )⫽0, while an elec on in
he uppe lase s a e akes he alue ai( )⫽1. A empo al
a iable a
˜
i( )苸
兵
0,1,2,...,
␶
a
其
is also in oduced in o de o
ake in o accoun he ini e ime
␶
a, o which an elec on
can emain in he uppe s a e. I he elec on is in he base
s a e, hen a
˜
i( )⫽0, o he wise a
˜
i( ⫹1)⫽a
˜
i( )⫹1 un il he
maximum alue
␶
ais eached and hen a
˜
i( ⫹1)⫽0.
The second a iable ci( )苸
兵
0,1,2,...,M
其
ep esen s he
numbe o pho ons in node ia ime . A la ge enough uppe
alue o Mis aken o a oid sa u a ion o he sys em. The e
is also ano he empo al a iable c
˜
i
j( )苸
兵
0,1,2,...,
␶
c
其
,
which measu es he amoun o ime since a pho on j
苸
兵
1,2,...,M
其
was c ea ed a node i.
␶
cis he li e ime o
each pho on. Fo a gi en pho on j,c
˜
i
j( ⫹1)⫽c
˜
i
j( )⫹1 un il
he li e ime
␶
cis eached and hen c
˜
i
j( ⫹1)⫽0.
B. Neighbo hood
The neighbo hood conside ed is he Moo e neighbo hood,
each cell ha ing nine neighbo s: he cell i sel , i s ou nea -
es neighbo s 共si ua ed in he posi ions no h, sou h, eas , and
wes 兲, and he ou nex neighbo s 共in he posi ions no heas ,
sou heas , no hwes , and sou hwes 兲.
C. T ansi ion ules
The ime e olu ion o he CA is gi en by a se o ules
which de e mines he s a e o any pa icula cell o he sys-
em a ime ⫹1, depending on he s a e o he cells included
in i s neighbo hood a ime . The e olu ion o he empo al
a iables a
˜
i( ) and c
˜
i
j( ) was desc ibed be o ehand. He e we
desc ibe only he e olu ion o ai( ) and ci( ).
共i兲I ai( )⫽0, hen
ai共 ⫹1兲⫽1 wi h p obabili y ␭.
共ii兲I ai( )⫽1 and ⌫i( )⫽兺neighbo sci( )⬎
␦
, hen
ci共 ⫹1兲⫽ci共 兲⫹1,
ai共 ⫹1兲⫽0,
whe e ⌫iis he numbe o pho ons included in he neighbo -
hood o he cell iand
␦
is a h eshold alue, which has been
aken o be 1 in ou simula ions.
共iii兲I ci( )⬎0 and he e is one pho on j o which
c
˜
i
j( )⫽
␶
c, hen
ci共 ⫹1兲⫽ci共 兲⫺1.
共i 兲I ai( )⫽1 and a
˜
i( )⫽
␶
a, hen
ai共 ⫹1兲⫽0.
These ansi ion ules ep esen he di e en physical p o-
cesses ha unc ion a he mic oscopical le el in a lase sys-
em. Rule 共i兲 ep esen s he pumping p ocess. Rule 共ii兲mod-
els he s imula ed emission—i he elec onic s a e o a cell
has a alue o ai( )⫽1 and he numbe o lase pho ons in
he neighbo hood is g ea e han a ce ain h eshold, hen a
he ime ⫹1 a new pho on will be c ea ed in ha cell and
he elec on will decay o he g ound le el. Rule 共iii兲 ep e-
sen s he pho on decay. Rule 共i 兲 ep esen s he elec on de-
cay in a way simila o ule 共iii兲—a e ime
␶
ao he exci ed
elec ons, hose elec ons will decay o he g ound le el. To
simpli y he model as much as possible, we conside his
decay o be en i ely non adia i e, i.e., spon aneous emission
is no aken in o accoun . Also, as in an ideal ou -le el lase
he popula ion o le el E1is negligible, s imula ed abso p-
ion has no been conside ed.
Addi ionally, in o de o ep esen he noise le el ob-
se ed in p ac ice 共 ypically o he o de o one noise pho on
pe ca i y mode兲, we in oduce a small con inuous noise
le el o andom pho ons in he lase mode a e e y ime s ep.
This is done by se ing ci( ⫹1)⫽ci( )⫹1 o a numbe o
cells smalle han 0.01% o he o al numbe o sys em cells,
whose posi ions a e andomly chosen. This noise 共along wi h
he popula ion in e sion induced by pumping兲is esponsible
o he ini ial s a up o he lase p ocess, and can p e en
he e en ual ex inc ion o his p ocess, in case he numbe o
lase pho ons d ops down o negligible alues.
IV. SIMULATIONS
The simula ions we e ca ied ou using la ices o 200
⫻200 and 300⫻300 cells. Th ee pa ame e s de e mine he
esponse o he sys em: he pumping p obabili y ␭, he li e-
ime o pho ons (
␶
c), and he li e ime o exci ed elec ons
(
␶
a). The pumping p obabili y ␭in oduced in he CA model
is linea ly ela ed o he cons an pumping a e Rincluded in
he a e equa ions, as explained in he Appendix. Thus he
lase a e equa ions’ p edic ion 关Eq. 共5兲兴 o he dependence
o h eshold pumping a e R wi h he li e imes
␶
aand
␶
ccan
be exp essed o he h eshold pumping p obabili y ␭ as
␭ ⫽1
K⬘
␶
a
␶
c
,共8兲
whe e K⬘is a cons an .
Figu e 2 shows he h eshold pumping p obabili y ␭ e -
sus he li e ime o he uppe lase le el (
␶
a) and he ca i y
li e ime
␶
c. Each poin has been ob ained in he ollowing
manne . A e a ansien ime, he sys em e ol es o 200
ime s eps and he a e age numbe o lase pho ons (n
¯
)is
CELLULAR AUTOMATON MODEL FOR THE SIMULATION . . . PHYSICAL REVIEW E 67, 066708 共2003兲
066708-3
eco ded. Fo each pai o alues
␶
cand
␶
a, he p ocedu e is
epea ed o alues o he pumping p obabili ies ␭ anging
om 0.0001 o 0.1. I he lase ac ion does no ini ia e in he
sys em, n
¯
is app oxima ely equal o he numbe o noise
pho ons (nnp) in oduced. When he lase ac ion s a s, pho-
ons a e p oduced by s imula ed emission 关 ia ule 共ii兲兴, and
he a e age numbe o pho ons is g ea e han nnp . Thus, ␭
can be compu ed as he minimum alue o he pumping
p obabili y o which he a e age numbe o pho ons is
highe han a gi en e e ence alue nmin . In ou simula ions,
we ha e aken nmin⫽1.25nnp . The numbe o noise pho ons
(nnp) is he numbe o noise pho ons in oduced in each ime
s ep mul iplied by he li e ime o each lase pho on.
Figu e 3 shows he h eshold pumping p obabili y ␭ e -
sus
␶
a
␶
con a loga i hmic scale. I can be seen ha all he
di e en cu es o Fig. 2 collapse in a unique s aigh line,
ha ing a slope close o ⫺1, which is in ag eemen wi h he
beha io p edic ed by he lase a e equa ions, i.e., Eq. 共8兲.
To analyze he di e en dynamic beha io s o he sys em,
we allow i o e ol e o 500 ime s eps. Ini ially all he
elec ons a e in he g ound le el 关ai(0)⫽0兴and he e a e no
lase pho ons p esen 关ci(0)⫽0兴, excep a small ac ion
(0.01%) o noise pho ons. In each expe imen , he o al
numbe o lase pho ons, n( )⫽兺i⫽1
Ncci( ), and he o al
numbe o elec ons in he uppe lase s a e 共popula ion in-
e sion兲,N( )⫽兺i⫽1
Ncai( ) a e measu ed. In o de o classi y
he ange o alues o he pa ame e s o which oscilla ions
o a cons an egime in n( ), and N( ) appea , he Shannon
en opy So he dis ibu ion o alues aken by n( ) and
N( ) is calcula ed. This is ca ied ou by di iding he ange
o alues aken in 103in e als, and compu ing he e-
quency ia which he magni ude alue lies inside e e y
pa icula non oid in e al i. The Shannon en opy is ex-
p essed as
S共␭,
␶
c,
␶
a兲⫽⫺兺
i⫽1
m
ilog2 i,共9兲
whe e mis he numbe o non oid in e als. Figu e 4 shows
he con ou plo o he Shannon en opy o he alues dis i-
bu ion o he numbe o lase pho ons, Sc, o a ixed alue
o
␶
c⫽10. Simila plo s a e ob ained o o he alues o
␶
c.
The black line is he heo e ical s abili y cu e:
␶
a
␶
c
⫽
冉
R
R
冊
2
4
冉
R
R
⫺1
冊
,
which ollows om Eq. 共7兲. He e R/R is used ins ead o
␭/␭ , because Eq. 共A5兲 om he Appendix, whe e (␭/␭ )
⫽(R/R ), was aken in o accoun . The Shannon en opy
measu es he dispe sion in he dis ibu ion o he numbe o
lase pho ons 共o in he popula ion in e sion兲. I his numbe
is app oxima ely cons an , Swill end owa d ze o; i oscil-
la ions appea , S akes highe alues. The maximum alue
would esul om an equip obable dis ibu ion. Sis hus a
FIG. 2. Th eshold pumping p obabili y ␭ om he CA lase
model: 共a兲␭
dependence on he uppe lase le el li e ime
␶
a o
di e en alues o he ca i y li e ime
␶
c;共b兲␭
dependence on
␶
c
o di e en alues o
␶
a. Bo h igu es a e plo ed on a loga i hmic
scale.
␶
aand
␶
ca e measu ed in ime s eps.
FIG. 3. Dependence o he h eshold pumping p obabili y ␭
om he CA lase model e sus
␶
a
␶
c, plo ed on a loga i hmic
scale. The do s co espond o he esul s o he simula ions and he
con inuous line o he heo e ical p edic ion o he lase a e equa-
ions, i.e., Eq. 共8兲.
␶
aand
␶
ca e measu ed in ime s eps.
GUISADO, JIME
´NEZ-MORALES, AND GUERRA PHYSICAL REVIEW E 67, 066708 共2003兲
066708-4
good indica o o he p esence o oscilla ions in he sys em,
as i mus be low when he numbe o pho ons 共o he popu-
la ion in e sion兲is essen ially cons an and high when his
numbe oscilla es. Figu e 4 shows he p esence o oscilla-
ions abo e and o he igh o he heo e ical s abili y cu e
共da k zones兲, whe e
␶
a⬎
␶
cand R⬎R . On he o he hand, in
he b igh zones, whe e
␶
a⬇
␶
co R⭐R , he numbe o
lase pho ons does no oscilla e. This is in good ag eemen
wi h he heo e ical p edic ions o Eq. 共7兲, based on he lin-
ea iza ion o he lase a e equa ions, and wi h he beha io
obse ed in eal sys ems. Lase spiking is cha ac e is ic o
mos solid-s a e and semiconduc o lase s, which ha e a sub-
s an ially longe decay ime o he uppe lase le el han ha
o he pho ons in he lase ca i y. This is no gene ally ob-
se ed in gas lase s, whose decay imes usually ha e he
same o de o magni ude.
Figu es 5 and 6 show he empo al e olu ion o he num-
be o lase pho ons and he popula ion in e sion o wo
di e en alues o he sys em pa ame e s, cha ac e is ic o
he wo di e en egimes in which i can ope a e. In his
case, a la ice o 300⫻300 cells has been used in o de o
de ine he beha io mo e clea ly. In bo h igu es, he empo-
al e olu ion is shown a he le , and he e olu ion in a phase
space wi h he numbe o lase pho ons e sus he popula ion
in e sion is shown a he igh . Figu e 5 co esponds o he
pa ame e alues o which he Shannon en opy is low
共b igh zones in Fig. 4兲and, a e an ini ial gain swi ching
peak, he sys em shows a cons an beha io . In phase space,
a e an ini ial ansien he sys em goes o a ixed poin .
Figu e 6 co esponds o he pa ame e s alues o which he
Shannon en opy is high 共da k zones in Fig. 4兲and he sys-
em exhibi s oscilla ions. In his pa icula case, he sys em
shows damped oscilla ions which a e associa ed o a spi al
ajec o y owa ds a s eady s a e in phase space. Fo he al-
ues o he pa ame e s co esponding o Fig. 6, a h eshold
pumping p obabili y o ␭ ⫽0.0015 is ound. Acco ding o
Eq. 共A5兲, his means (R/R )⫽(␭/␭ )⫽(0.01/0.0015), which
leads o a spiking pe iod alue p edic ed heo e ically by Eq.
共6兲o Tsp
h⫽2
␲
/
␻
sp
h⫽137.3 ime s eps. The a e age spiking
pe iod o he oscilla ions esul ing om ou simula ions is
Tsp
sim⫽(127⫾10) ime s eps, which is in good ag eemen
wi h he heo e ical alue p edic ed by Eq. 共6兲.
The CA model is also e y use ul o obse ing he e o-
lu ion o spa io empo al pa e ns. Recen ly, quasi-
ins an aneous ans e se pa e ns in a b oad ape u e lase
ha e been measu ed o he i s ime 关13兴. In he case o a
CO2lase , a diso de ed dis ibu ion o spo s, non ep oduc-
ible om sho o sho , which yields a bounda y-de e mined
egula s uc u e when in eg a ed in p og essi ely longe
ime windows, has been ound.
Figu e 7 shows some snapsho s o he ins an aneous ans-
e se pa e ns shown by he lase pho on popula ion in he
la ice. Black colo ep esen s a cell wi h no pho on p esen ,
g ay ep esen s cells wi h one pho on, and whi e ep esen s
cells wi h wo o mo e pho ons. Figu es 7共a兲and 7共b兲co e-
FIG. 4. Con ou plo o he Shannon en opy o he dis ibu ion
o he numbe o lase pho ons o a ixed alue o
␶
c⫽10. Low
alues o Sc共b igh zones兲indica e ha he esponse o he sys em
is nonoscilla o y, while high alues 共da k zones兲indica e an oscil-
la o y esponse. The black line is he heo e ical s abili y cu e.
FIG. 5. E olu ion o he sys em o he pa ame e s: ␭⫽0.1,
␶
c⫽8,
␶
a⫽30, whe e
␶
cand
␶
aa e measu ed in ime s eps. La ice: 300
⫻300 cells. 共a兲Numbe o lase pho ons and popula ion in e sion e sus ime. 共b兲E olu ion in a phase space o med by he numbe o lase
pho ons e sus he popula ion in e sion.
CELLULAR AUTOMATON MODEL FOR THE SIMULATION . . . PHYSICAL REVIEW E 67, 066708 共2003兲
066708-5

spond o he oscilla o y beha io a e a ansien ime. In
his case, he sys em shows spa ial s uc u es ha e ol e in
ime esembling wa e on s ha in e ac des uc i ely wi h
each o he . The ypical scale o hese pa e ns is dependen
on he alue o he pa ame e s, as can be seen by compa ing
Figs. 7共a兲and 7共b兲. Figu es 7共c兲and 7共d兲co espond o he
nonoscilla o y beha io and bo h igu es ha e he same pa-
ame e s alues. Figu e 7共c兲was aken a ime ⫽50 ime
FIG. 6. E olu ion o he sys em o he pa ame e s: ␭⫽0.01,
␶
c⫽14,
␶
a⫽160, whe e
␶
cand
␶
aa e measu ed in ime s eps. La ice:
300⫻300 cells. 共a兲Numbe o lase pho ons and popula ion in e sion e sus ime. 共b兲E olu ion in phase space.
FIG. 7. Ins an aneous ans-
e se spa io empo al pa e ns
shown by he lase pho on popula-
ion in he sys em. 共a兲and 共b兲co -
espond o oscilla o y beha io ,
whe eas 共c兲and 共d兲co espond o
nonoscilla o y beha io . The al-
ues o he pa ame e s a e 共a兲␭
⫽0.01,
␶
c⫽14,
␶
a⫽160; 共b兲␭
⫽0.03,
␶
c⫽6,
␶
a⫽150; 共c兲␭
⫽0.1,
␶
c⫽8,
␶
a⫽30, ⫽50; 共d兲
␭⫽0.1,
␶
c⫽8,
␶
a⫽30, ⫽200.
␶
c,
␶
a, and a e measu ed in ime
s eps.
GUISADO, JIME
´NEZ-MORALES, AND GUERRA PHYSICAL REVIEW E 67, 066708 共2003兲
066708-6
s eps, whe eas Fig. 7共d兲a ime ⫽200 ime s eps. In his
case, i can be obse ed ha he lase p ocess is ac i a ed
ini ially wi h expanding a alanches and hen eaches a s a-
iona y s a e a which he lase pho ons dis ibu ion is
homogeneous.
V. DISCUSSION
Depending on he alues o he h ee pa ame e s
␶
a,
␶
c,␭
o he sys em, ou simula ions show ha wo dis inc cha ac-
e is ic beha io s appea . The Shannon en opy o he lase
pho ons dis ibu ion and o he popula ion in e sion allows
us o ob ain he pa ame e anges in which each beha io
akes place.
A high Shannon en opy 共da k zones in Fig. 4兲indica es a
g ea e dispe sion in he popula ions han in o he a eas o
he pa ame e s space. Fo hese alues o he pa ame e s, he
CA model ep oduces he ypical kind o beha io known as
lase spiking— he empo al e olu ion o he popula ions o
lase pho ons and in e sion elec ons oscilla e in a co ela ed
way, as shown in Fig. 6共a兲. A se ies o sha p, na ow
‘‘spikes’’ can be obse ed, whose ampli ude dec eases wi h
ime. In he phase space desc ip ion 关Fig. 6共b兲兴, he sys em
ollows a spi al ajec o y which con e ges owa ds a s eady-
s a e limi poin .
On he o he hand, a low Shannon en opy 共b igh zones
in Fig. 4兲indica es a smalle dispe sion in he popula ions.
Fo hese pa ame e alues, he beha io shown by he CA
model co esponds o a cons an egime—a e an ini ial
ansien , he o al numbe o lase pho ons and in e sion
elec ons emain app oxima ely cons an , as is shown in Fig.
5共a兲. In he phase space desc ip ion, Fig. 5共b兲, a e a sho
ansien he sys em eaches a ixed poin .
Ou model is success ul in showing ei he one o he o he
o hese wo egimes, depending on he cha ac e is ic decay
imes in ol ed. Lase spiking appea s gene ally in si ua ions
in which he li e ime o he elec ons in he exci ed s a e is
subs an ially longe han he li e ime o he lase pho ons,
whe eas he cons an egime mos ly appea s when bo h li e-
imes a e o he same o de o magni ude.
The CA model can be applied du ing he in e es ing pe-
iod o s ong spiking, o which he e is no simple analy ic
solu ion o he lase a e equa ions and he adi ional ap-
p oach elies on he nume ical in eg a ion o he sys em o
di e en ial equa ions.
In Fig. 7, we can obse e ha he kind o spa io empo al
ans e se pa e ns shown by he lase pho ons popula ion
depend on he popula ion dynamics egime. Fo he lase
spiking egime, spa io empo al s uc u es appea in he o m
o p opaga ing wa e on s, Figs. 7共a兲and 7共b兲. The cha ac-
e is ic scale o hese s uc u es depends on he h ee pa am-
e e s o he sys em. I is ema kable ha ou a he simple
and gene ic lase model p oduces spa io empo al s uc u es
simila o hose obse ed expe imen ally and wi h he nu-
me ical in eg a ion o e y de ailed Maxwell-Bloch equa-
ions models 关13,14兴.
On he o he hand, when he sys em is in he cons an
egime, he e is a homogeneous dis ibu ion o lase pho ons,
Fig. 7共d兲. This is in ag eemen wi h he app oxima ely con-
s an alues shown in he o al numbe o lase pho ons. The
lase p ocess is ini ially ac i a ed wi h expanding a a-
lanches, as shown in Fig. 7共c兲. When he a alanches ha e
sp ead h oughou he whole sys em, he cons an s eady
s a e is eached.
In conclusion, we ha e p esen ed he e a CA model o he
simula ion o he popula ion dynamics in a lase ca i y.
While simpli ied, he model is able o cap u e he essen ial
ea u es and phenomenology encoun e ed in lase s such as
elaxa ion oscilla ions, spiking beha io , and pa e n o ma-
ion. The CA app oach o lase s is a e y p omising ool ha
can supplemen he adi ional app oach based on he solu-
ion o se s o coupled di e en ial equa ions. I s pe o mance
is e y good, so ha he simula ion can be ca ied ou on
la ge la ices. The ull pa allelism o he CA can be imple-
men ed on special pu pose compu e s o in compu e s wi h
ull pa allel a chi ec u es, and he e o e an impo an sa ing
in compu e ime can be ob ained. A CA model can ep esen
an ad an age in cases in which he sys em o di e en ial
equa ions ha e con e gence p oblems, o example, s i di -
e en ial equa ions. The Maxwell-Bloch equa ions ha de-
sc ibe many lase sys ems a e o his ype. Ou model is ee
om his kind o con e gence p oblems, so i can be e y
use ul o his kind o lase s. Howe e , wo k mus be done in
o de o es his po en ial applica ion, including a quan i a-
i e compa ison o he compu ing imes o he CA model
e sus he nume ical in eg a ion o he ull se o Maxwell-
Bloch equa ions.
In he cu en s a e o ou wo k, we see no obs acle o ine
une he p oposed CA model in o de o mee he cha ac e -
is ics o any pa icula lase sys em. This mus be add essed
in he nea u u e. Addi ionally, we hope ha simila models
can be used o s udy o he p oblems o cu en in e es , such
as chao ic lase dynamics 关15兴o he dynamics o a wo-
pho on lase 关16兴.
ACKNOWLEDGMENT
J. L. Guisado would like o hank Ramon Risco o help-
ul sugges ions and discussions.
APPENDIX: RELATIONSHIP BETWEEN THE PUMPING
RATE RAND THE PUMPING PROBABILITY ␭
Le us see how he pumping a e R, included in he lase
a e equa ions, is ela ed o he pumping p obabili y ␭, in o-
duced in he CA model. Ris de ined as he numbe o elec-
ons ha a e exci ed om he g ound s a e E0 o he s a e E3
pe uni ime, conside ed as a cons an in he a e equa ions:
R⫽dn3
d ⇒n3⫽R .共A1兲
In he CA model, pumping is in oduced as a p obabilis ic
p ocess, whe e ␭is he pumping p obabili y pe uni ime o
an elec on om s a e E0 o E3. This means ha i he e a e
n0elec ons in E0, hen he numbe o exci a ions in a ime
in e al d will be n␭d . So he popula ion o s a e E0de-
c eases as
CELLULAR AUTOMATON MODEL FOR THE SIMULATION . . . PHYSICAL REVIEW E 67, 066708 共2003兲
066708-7
dn0⫽⫺␭d n0⇒n0⫽n0ie⫺␭ .共A2兲
I we assume ha n3is ini ially unpopula ed, hen n0i
⫽n0⫹n3and he popula ion o E3will inc ease wi h pump-
ing as
n3⫽n0i⫺n0⫽n0i共1⫺e⫺␭ 兲.共A3兲
Then, he pumping a e Rwill be
R⫽dn3
d ⫽␭n0ie⫺␭ .共A4兲
In gene al, i he only p ocess in he sys em is he pump-
ing, Rwill dec ease exponen ially wi h ime as he popula-
ion o elec ons in E0dec eases. Bu he elec ons pumped
o le el E3decay down again o le el E0 epopula ing he
g ound le el, and 共as long as he pumping p obabili y is
small兲a e a ansien pe iod, Rs abilizes o a cons an
alue:
R⫽␭n0c⯝cons , 共A5兲
whe e n0cs ands o he app oxima ely cons an alue o-
wa d which he popula ion o le el E0 ends. So a e a an-
sien pe iod, he e is a linea ela ionship be ween he pump-
ing p obabili y conside ed in ou model and he cons an
pumping a e conside ed in he lase a e equa ions. This
means ha he pumping p obabili ies om he CA model
mus ollow Eq. 共5兲wi h he excep ion o he alue o he
cons an K:
␭ ⫽1
K⬘
␶
a
␶
c
.共A6兲
关1兴H. Haken, Syne ge ics: An In oduc ion. Nonequilib ium Phase
T ansi ions and Sel -o ganiza ion in Physics, Chemis y and
Biology 共Sp inge -Ve lag, Be lin, 1983兲.
关2兴A. Siegman, Lase s 共Uni e si y Science Books, Mill Valley,
CA 1986兲.
关3兴O. S el o, P inciples o Lase s 共Plenum P ess, New Yo k,
1989兲.
关4兴S. Wol am, Physica D 10,1共1984兲.
关5兴S. Wol am, Cellula Au oma a and Complexi y 共Addison-
Wesley, New Yo k, 1994兲.
关6兴B. Chopa d and M. D oz, Cellula Au oma a Modeling o
Physical Sys ems 共Camb idge Uni e si y P ess, Camb idge,
1998兲.
关7兴T. To oli and N. Ma golus, Cellula Au oma a Machines: A
New En i onmen o Modelling 共MIT, Camb idge, MA,
1987兲.
关8兴G. Vichniac, Physica D 10,96共1984兲.
关9兴B. Chopa d, P. Lu hi, and M. D oz, Phys. Re . Le . 72, 1384
共1994兲.
关10兴A.K. Guns ensen, D.H. Ro hman, S. Zaleski, and G. Zane i,
Phys. Re . A 43, 4320 共1991兲.
关11兴J. G a ne and D. G i ea h, Ad . Appl. Ma h. 21, 241 共1998兲.
关12兴P. Seybold, L. Kie , and C.-K. Cheng, In . J. Quan um Chem.
75, 751 共1999兲.
关13兴F. Encinas-Sanz, I. Ley a, and J.M. Gue a, Phys. Re . A 62,
043821 共2000兲.
关14兴F. Encinas-Sanz, I. Ley a, and J.M. Gue a, Phys. Re . Le .
84, 883 共2000兲.
关15兴G. Huye and J. T edicce, Physica D 96, 209 共1996兲.
关16兴O. P is e , W.J. B own, M.D. S enne , and D.J. Gau hie , Phys.
Re . Le . 86, 4512 共2001兲.
GUISADO, JIME
´NEZ-MORALES, AND GUERRA PHYSICAL REVIEW E 67, 066708 共2003兲
066708-8
View publica ion s a sView publica ion s a s