Stability analysis of a laser with two modulated saturable absorbers Technology Journal of Applied Researchand

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Journal of Applied Research and Technology

www.jart.ccadet.unam.mx JournalofAppliedResearchandTechnology13(2015)576–581

Original

Stability analysis of a laser with two modulated saturable absorbers

Oscar Andres Naranjo-Montoya

CentrodeInvestigacionesenÓptica(CIO),LomadelBosque115,LomasdelCampestre,León,Guanajuato,CP37150,Mexico Received27April2015;accepted21September2015

Availableonline29November2015

Abstract

Thestabilityanalysisofamodelforalasersystemwithtwomodulatedsaturableabsorbers,eachonemodulatedatadifferentfrequency,is performed.Themodelisbasedonfourequationsdescribingthetemporalevolutionofthephotonflux,thepopulationinversionintheactivemedia, andthesaturationcoefficientsofeachsaturableabsorber.Thesystemdynamicsisdiscussedinordertofindstablesystemcontrolregions.

Keywords: Stabilityanalysis;Lasersystem;Modulatedsaturableabsorbers

1. Introduction

Thedynamicsofalasersystemwithtwosaturableabsorbers(seeFig.1)canbedescribedbyamodelbasedontheStatz–DeMars equations,whichoriginallyweredevelopedtodescribeoscillationsinaMaser(Statz&DeMars,1960).Thismodelhasundergone manymodificationstobeadopted forlasersystems (Tarassov,1985;Tang&Statz,1963;Thompson&Malacara,2001).Fora completephenomenologicaldescriptionofalaserwithtwosaturableabsorbers,onlyfourequationsareneeded:thephoton-flux equation,anequationforthepopulationinversiondensityintheactivemediumandtwosaturablepopulationinversionequations thatgivethesaturationcoefficientforeachsaturableabsorber(Wilson&Aboites,2013).Therefore,theStatz–DeMarsequations forathreelevellasersystemwithtwosaturableabsorberswithoutmodulationarewrittenasfollows:

dS

dt =ΓνσNS−ΓνL_α₁

L_mk_α₁S−ΓνL_α₂

L_mk_α₂S− 1 TS dN

dt =−β σ

ωNS+N₀−N τ dk_α₁

dt =−2σα1k_α₁S

ω +k0α₁−k_α₁ τ_α₁ dk_α₂

dt =−2σα₂k_α₂S

ω +k_0α₂ −k_α₂ τα₂

,

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whereSistheemittedphotondensity,Nisthepopulationinversionoftheactivemedium,kα₁ andkα₂ aretheresonantabsorptions ofthesaturableabsorbers1 and2 respectively,σα₁ andσα₂ arethesaturableabsorberscross-sections,andNα₁ andNα₂ arethe populationinversionsofthesaturableabsorbers(kα₁ =−σα₁N_α₁andk_α₂ =−σα₂N_α₂).Γ,ν,σandT stand,respectively,forcavity fillingcoefficient, opticalfrequency, active mediumcross-section andphoton lifetime inthe cavity; β isthe coefficient which accountsforthedifferenceinpopulationinversioncoursedbylasing;L_m,L_α₁ andL_α₂are,respectively,theactivemediumandthe saturableabsorberslengths;k_0α₁ andk_0α₂ arethelinearresonantsaturableabsorbersabsorptioncoefficientswithoutlasing;N₀is

E-mailaddress:[email protected]

PeerReviewundertheresponsibilityofUniversidadNacionalAutónomadeMéxico.

http://dx.doi.org/10.1016/j.jart.2015.10.010

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Active medium

Saturable absorber

1

Saturable absorber

2 Electro

optic modulator

Electro optic modulator

Fig.1.Three-levellasersystemwithtwosaturableabsorberswithelectro-opticmodulators.

thepopulationinversionintheactivemediumwithoutradiation;τ,τ_α₁ andτ_α₂ standforrelaxationtimeintheactivemediumand inthesaturableabsorbers,respectively;finally,ωisthephotonenergy(Aboites&Ramírez,1989).

Assumingthat thetwosaturableabsorbers haveequalrelaxationtimes(τα1 =τ_α₂ =τ_α),anddefiningthe nextadimensional parametersandvariables:t=t/τ,G=τ/t,δ=τ/τ_α,ρ₁=2σα1/βσ,ρ₂=2σα2/βσ,α=νσTNandα_α₁ =−νTk0α1L_α₁/L_m=

−νTσα1n_α₁/L_m, α_α₂ =−νTk0α2L_α₂/L_m=−νTσα2n_α₂/L_m; n(t)=νσTN(t), n_α₁(t)=−νTkα1(t)Lα1/L_m, n_α₁(t)=

−νTkα2(t)Lα2/L_mandn(t)=βBτS(t)/ν=βστS(t)/ω,theabovesystemcanberewrittenas:

dm

dt =Gm(n+nα₁ +nα₂−1) dn

dt =α−n(m+1) dn_α₁

dt =δαα₁−nα₁(ρ1m+δ) dnα₂

dt =δα_α₂−n_α₂(ρ2m+δ).

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Alltheparametersusedtodefinethesaturableabsorbersarefixed,exceptforα_α₁ andα_α₂,whichincludeameasureoftheactive centerabsorbentdensity;forthisreason,α_α₁ andα_α₂ areusedasthesaturableabsorberidentifyingparameters.Addinganexternal linearsinusoidalmodulation(e.g.usinganElectroOpticModulator(EOM))directlyintothesaturableabsorbersthroughtheirmain parameter(i.e.α_α₁ andα_α₂),thelasttwoaboveequationsmaybetransformedinto

dn_α₁ dt =δαα₁

1+cos(ωc1t) 2

−nα₁(ρ1m+δ) dn_α₂

dt =δαα₂

1+cos(ωc2t) 2

−nα₂(ρ2m+δ),

(3)

whereω_c₁ andω_c₂ standfortheexternalmodulationfrequenciesappliedtotheEOM.Thesefourdifferentialequationscomposethe workingsystem.ItmustbenotedthatinabsenceofamodulationfrequencyappliedtotheEOM,ω_c₁andω_c₂,thesystemreturnsto Eq.(2),i.e.rateequationsforalaserwithtwopassivesaturableabsorbers(Wilson,Aboites,Pisarchik,Pinto,&Barmenkov,2011).

2. LinearStabilityAnalysis

LinearStabilityAnalysisisusedtounderstandthesystemdynamics(Tabor,1989;Braun,1992).Theanalysisisbasedonthe lineardisturbanceequations;theseequationsarederivedfromtheoriginalequations(PintoRobledoetal.,2012).Asitiswellknown, themethodconsistsinlinearizingthedescribingequations,obtainingtheinitialstatecondition(i.e.whenthederivativesarezero), expandingthesystemabouttheinitialstatecondition,constructingtheJacobianmatrix,andfindingtheeigenvectorsandeigenvalues withadeterminantequaltozero.Thisgives,asaresult,thefixedpointsoftheequationsystem,whichmustbeanalyzedinorderto knowwhattypeofpointsthereare(i.e.fixed,source,saddle,etc.)(Wilson,Aboites,Pisarchik,Ruiz-Oliveras,&Taki,2011;Wilson, Aboites,Pisarchik,Pinto,&Taki,2011).TheequationsofinterestareEqs.(2)and(3);theseequationsarenon-autonomousdue

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totheexplicittime-dependencefoundinthecosineofthethirdandfourthexpressions.TobeabletoperformtheLinearStability Analysis,thosedependencesmustbeeliminated;todothat,variablechangesmustbeapplied,yielding,thenextequationsystem:

dm

dt =Gm(n+n_α₁+n_α₂ −1) dn

dt =α−n(m+1) dn_α₁

dt =δα_α₁

1+cos(x1) 2

−n_α₁(ρ1m+δ) dn_α₂

dt =δα_α₂

1+cos(x2) 2

−n_α₂(ρ2m+δ), dx₁

dt =ω_c₁ dx2

dt =ω_c₂

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NamingEq.(4)as:

f(m,n,n_α₁,n_α₂,x₁,x₂)= dm dt g(m,n,n_α₁,n_α₂,x1,x2)= dn

dt h(m,n,n_α₁,n_α₂,x₁,x₂)=dn_α₁

dt j(m,n,n_α₁,n_α₂,x1,x2)= dn_α₂

dt l(m,n,n_α₁,n_α₂,x₁,x₂)=dx1

dt q(m,n,n_α₁,n_α₂,x1,x2)=dx₂

dt ,

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and assuming that (m^∗,n^∗,n_α₁^∗,n_α₂^∗,x^∗₁,x^∗₂) is the steady state, that is, f(m^∗,n^∗,n^∗_α

1,n^∗_α

2,x^∗₁,x^∗₂)=0, g(m^∗,n^∗,n^∗_α

1,n^∗_α

2,x^∗₁,x^∗₂)=0, h(m^∗,n^∗,n^∗_α

1,n^∗_α

2,x^∗₁,x^∗₂)=0, j(m^∗,n^∗,n^∗_α

1,n^∗_α

2,x^∗₁,x^∗₂)=0, l(m^∗,n^∗,n^∗_α

1,n^∗_α

2,x^∗₁,x^∗₂)=0, and q(m^∗,n^∗,n^∗_α

1,n^∗_α

2,x^∗₁,x^∗₂)=0, then in order to find whether the steady state is stable or unstable, a small perturbation (representedbythesubscriptp)mustbeaddedtoit,

f =f^∗+f_p g=g^∗+g_p h=h^∗+h_p j=j^∗+jp

l=l^∗+lp

q=q^∗+qp,

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With(fp,g_p,h_p,j_p,l_p,q_p)1

Now,themainquestionforpracticalpurposesiswhetherwilltheperturbationsgrow(steadystateunstable)ordecay(steadystate stable).Tobeabletoobservewhethertheperturbationsgrowordecay,theperturbationderivativesmustbefound(Wilson,Aboites, Pisarchik,etal.,2011).

df_p

dt =f(m^∗,n^∗,n^∗_a

1,n^∗_a

2,x^∗₁,x^∗₂)+ δ

δmf(m^∗,n^∗,n^∗_a

1,n^∗_a

2,x^∗₁,x^∗₂)fp+ δ

δnf(m^∗,n^∗,n^∗_a

1,n^∗_a

2,x^∗₁,x^∗₂)gp

+ δ

δn_a₁f(m^∗,n^∗,n^∗_a₁,n^∗_a₂,x₁^∗,x^∗₂)hp+ δ

δn_a₂f(m^∗,n^∗,n^∗_a₁,n^∗_a₂,x^∗₁,x^∗₂)jp+ δ

δx₁f(m^∗,n^∗,n^∗_a₁,n^∗_a₂,x^∗₁,x^∗₂)lp

+ δ δx2

f(m^∗,n^∗,n^∗_a

1,n^∗_a

2,x^∗₁,x^∗₂)qp+highorderterms. (7)

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FollowingtheTaylorseriesexpansionshowninEq.(7)thederivativesforeachperturbationare:

df_p dt = δ

δmf ·fp+ δ

δnf·gp+ δ

δn_α₁f·hp+ δ

δn_α₂f·jp+ δ

δx₁f·lp+ δ δx₂f·qp

dgp

dt = δ

δmg·f_p+ δ

δng·g_p+ δ

δn_α₁g·h_p+ δ

δn_α₂g·j_p+ δ δx1

g·l_p+ δ δx2

g·q_p dh_p

dt = δ

δmh·fp+ δ

δnh·gp+ δ

δn_α₁h·hp+ δ

δn_α₂h·jp+ δ

δx₁h·lp+ δ δx₂h·qp

djp

dt = δ

δmj·f_p+ δ

δnj·g_p+ δ

δn_α₁j·h_p+ δ

δn_α₂j·j_p+ δ δx1

j·l_p+ δ δx2

j·q_p dl_p

dt = δ

δml·fp+ δ

δnl·gp+ δ

δn_α₁l·hp+ δ

δn_α₂l·jp+ δ

δx₁l·lp+ δ δx₂l·qp

dqp

dt = δ

δmq·f_p+ δ

δnq·g_p+ δ

δn_α₁q·h_p+ δ

δn_α₂q·j_p+ δ δx1

q·l_p+ δ δx2

q·q_p.

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ThesystempresentedinEq.(8)canberewritteninthematrixform:

⎛

⎜⎜

⎜⎝ f_p g_p h_p j_p l_p q_p

⎞

⎟⎟

⎟⎠

=

⎛

⎜⎜

⎝ δ δmf δ

δnf δ δna₁

f δ

δna2

f δ

δx1

f δ

δx2

f δ

δmg δ δng δ

δn_a₁g δ

δn_a2g δ δx₁g δ

δx₂g δ

δmh δ δnh δ

δn_a₁h δ

δn_a2h δ δx₁h δ

δx₂h δ

δmj δ δnj δ

δna₁

j δ

δna2

j δ

δx1

j δ

δx2

j δ

δml δ δnl δ

δn_a₁l δ

δn_a2l δ δx₁l δ

δx₂l δ

δmq δ δnq δ

δn_a₁q δ

δn_a2q δ δx₁q δ

δx₂q

⎞

⎟⎟

⎠

⎛

⎜⎜

⎜⎝ f_p gp

hp

jp

l_p q_p

⎞

⎟⎟

⎟⎠

=j

⎛

⎜⎜

⎜⎝ f_p gp

hp

jp

l_p q_p

⎞

⎟⎟

⎟⎠

, (9)

wherejdenotestheJacobianmatrixoftheoriginalsystematthesteadystate.SubstitutingthevaluesintheJacobianfunction,the nextmatrixisobtained:

⎛

⎜⎜

⎝

G(n+n_α₁+n_α₂−1) Gm Gm Gm 0 0

−n −m−1 0 0 0 0

−nα1ρ₁ 0 −ρ1m−δ 0 −δαα₁

2 sin(x1) 0

−nα₂ρ2 0 0 −ρ2m−δ 0 −δα_α₂

2 sin(x2)

0 0 0 0 0 0

⎞

⎟⎟

⎠

. (10)

Thenextstepistofindtheeigenvectorsandeigenvaluesofthesystem(j−λI)∂p=0;so,thedeterminantwouldbe:

|j−λI|=0. (11)

Thematrixfortheformerequationis:

⎛

⎜⎜

⎝

G(n+n_α₁+n_α₂−1)−λ Gm Gm Gm 0 0

−n −m−1−λ 0 0 0 0

−nα1ρ1 0 −ρ1m−δ−λ 0 −δα_α₁

2 sin(x1) 0

−nα2ρ₂ 0 0 −ρ2m−δ−λ 0 −δαα₂

2 sin(x2)

0 0 0 0 −λ 0

0 0 0 0 0 −λ

⎞

⎟⎟

⎠

=0.

(12)

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0 1

1 2 3

3 2

1

4

Stable state Unstable state

5 6

2 3 4

α

αα²

α^α¹

Fig.2.Stabilityconditiongivenbytherelationbetweenαα₁,αα₂andα.

Thesolutionsfortheperturbedsteady-stateoftheoriginalsystemare:

m^s=0 n^s =α x^s₁=0 x^s₂=0 n^s_α

1 =α_α₁ n^s_α

2 =α_α₂.

(13)

SubstitutingtheformersolutionsinEq.(12),thematrixtransformsinto:

⎛

⎜⎜

⎜⎝

G(α+α_α₁+α_α₂ −1)−λ 0 0 0 0 0

−α −1−λ 0 0 0 0

−αα1ρ₁ 0 −δ−λ 0 0 0

−αα2ρ₂ 0 0 −δ−λ 0 0

0 0 0 0 −λ 0

0 0 0 0 0 −λ

⎞

⎟⎟

⎟⎠

=0. (14)

Matrix(14)hasthefollowingcharacteristicequation:

[G(α+αα₁+αα₂−1)−λ][−1−λ][2(−δ−λ)][−2λ]=0, (15) whereλ₁=G(α+α_α₁ +α_α₂−1),λ₂=−1,λ₃=−δ,λ₄=−δ,λ₅=0andλ₆=0areeigenvalueswhichareallreal;λ₂,λ₃and λ₄are alwaysnegative(i.e.theperturbationwilldecay),andλ₅ andλ₆ arecriticallystable.Therefore,thestabilityconditionis definedonlybythesignofλ₁,i.e.thefixedpointisasourcewhen(αα1 +α_α₂+α)>1,asshowninFigure2.

3. Conclusions

Thestabilityanalysisofthemodelforalasersystemwithtwomodulatedsaturableabsorbersisperformed.Themodelisbasedon fourequationsdescribingthetemporalevolutionofthephotonflux,thepopulationinversionintheactivemedia,andthesaturation coefficientsofthetwosaturableabsorbers.Thestabilityconditionofthesystemsisfoundtodependonlyonthesignofthefirst eigenvalueofthesystemλ.Asitisgraphicallyshown,thestabilityconditionsaregivenbytherelationbetweenα,αα₁andαα₂.

Conﬂictofinterest

Theauthorshavenoconflictsofinteresttodeclare.

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