ReportoftheFirstRefereeInthismanuscript,observationofsingle-modeinstabilitiesisreportedinquantumcascadelasers(QCLs)forwhichatheoryisdeveloped.Incoherentopticalgainisfoundtoco-existwithcoherentparametricgain.Dependingontheratioofthetwotypesofgain,emissioncanbeeitheramplitudemodulated(AM)orfrequencymodulated(FM).Thisratiocanbetunedviathelaserparametersandtheexternalpumpingrate,therebyallowingsomecontrolandtheabilitytoengineeradeviceforaspecifictask,e.g.,ultrashortpulsegenerationinthemid-infraredregime.Thismanuscriptisamasterpieceofinformativecontentandwasapleasuretoread.Beyondthewellpresentedfigures,concisetheoreticaltreatment,andthoroughdiscussionofresults,thecontentisnovelandinteresting.Moreover,thesupplementalmaterialenablesanyonetochecktheanalysis.Numerousimportantresultsappearinthispaper;onlyafewofthemarehighlightedhere.*Experimentalresultsareclean,makingtheobservationofasingle-modeinstabilityinQCLsveryclearandinteresting.Allfeaturesoftheseresults,includingspectralhysteresis,theonsetof"harmonic"then"dense"states,variationofsidebandseparation,andmanyotherswerediscussedinthecontextofthetheorypresented.Explanationsaremostlyqualitative,butconvincing,consideringthefactthatdirectmeasurementsofAMorFMcharacterarenotpossible[p.6].*Inthetheoreticalanalysis,shortcomingsduetoapproximationsarepresentedwhereappropriate.Theapproximationsusedaresuitableforanintroductiontothistheory.However,welookforwardtofutureworkinthecaseofbi-orthogonalspatialmodes.*ProgressismadeinunderstandingtheinstabilitythresholdandtheapproachverifiedinthereproductionoftheRNGHthreshold.Alower-than-might-be-expectedinstabilitythresholdisexplainedbyanunclampedgainprovidedtothesidebandsbyaspatialpopulationgrating.Theoreticalpredictionsaremuchlowerthanthoseobservedexperimentally,buttheauthors'planforfutureworkistheobviouspathtowardimprovement.Thus,betterandmoreprecisepredictionsaredelayeduntilthetheorycanbefurtherdeveloped,whichisreasonable.*Importantly,thespatialandtemporalvariationofgainpopulationweretakenintoaccountinthetheory.Thisisincontrast,forinstance,withtheHRSWandRNGHtravelingwavelaserinstabilities,wherethegainisspatiallyuniform.Suchvariationmakestheanalysismorecomplicatedbutwhatwaspresentedisconvincinglysuccessful.*Figure6isespeciallyinformative.Itilluminatesthemechanismbehindtheexcitationofsidebandswithfrequencyspacingsofmultiplefreespectralranges.Theinterplayofmultipletypesofgainisresponsiblewithdestructiveinterferencesuppressingthemostnearbymodes.Otherexplanations,suchas"zigzag"modes

[34],havebeenofferedbutlacksufficientreasoning(suchaswhytheywouldbepreferred).Symmetryofthesidebandsandtheirappearanceatlowerpumpingrates[34]fallsintolinewiththetheorypresentedinthismanuscript.*Athigherpumpingrates,frequencycombgeneration[35,37-39]hasbeenshowntooccurandfallsinlinewiththetheoryof"densestates"inthismanuscript.Withthemechanismrevealedandthestudyofhysteresispresentedhere,reliablegenerationofsuchcombsmaybepossiblewithlowerinputenergy.*Lastly,theinvestigationofFMandAMemissionisfundamentallyimportant.Theparameterregimesthatleadtoeachofthemareclearlydescribed.TheconsequencesofeachintermsofthelaserinstabilityaccountforthedifferentexperimentalresultsobservedbetweenthecoatedanduncoatedQCL(TL-4.6).Theonlycriticalremarkconcernsthetitle,which,whileelegant,issomewhatcrypticanddoesnotexpressthecontentorgeneralityofthepaper.Thekeyphenomenastudiedareinstabilitiesinstandingwavelasers,butnoneofthesewordsexplicitlyappearinthetitle(familiaritywithQCLsshouldnotbeassumed).Thus,thepapermaynotcatchtheattentionofsomeinthefield.

ReportoftheSecondRefereeThemanuscript"Thequantumcascadelaserasaself-pumpedparametricoscillator"dealswiththeinstabilitythresholdfromasinglemodelasertoamultimodebehavior.Itreportsatheoreticalmodel,whichiscomparedtoexperimentsbasedondevicesfromdifferentsuppliers.Experimentsandtheoryarewelldescribed,witharelevantselectionbetweenthetextwhichremainsinthemainpart,andthedetailswhichareputinthesupplementalmaterial.Theauthorstookthetimetowriteafullacademicpaper,whichwillbeveryusefulforamuchbroadercommunitythantheQCLgroups.Forthesereasons,IthinkthatthismanuscriptdeservespublicationasaregularpaperinPhysicalReviewA.However,Iwouldliketoaddafewcomments.1-Mymaincriticismisrelatedtothecomparisonbetweenexperimentsandtheory,whichisratherqualitative.Onlythreequantitiescanbequantitativelycomparedwithexperiments:p,delta-omega_sb,anddelta-omega_cr.Onlythelatteroneisdiscussedinthemaintextandtheorderofmagnitudeiscorrect.Nothingbettercanbeexpectedsincesomeparametersinvolvedintheformulaareknownwithonly1digit.panddelta-omega_sbarediscussedinthesupplementalmaterialVIII.AnestimatedvalueofT_1ismissingintheliterature.Theauthorsusedtheexperimentaldelta-omega_sbtoinferT_1.Thentheycalculatedtheexpectedp,withoutaconvincingagreement.Butstill,thereisnosensetogetT_1withmorethan1(ormaybe2)digit.Inmyopinion,atableinthemaintextcouldsummarizealltheseresults,onthemodelofTable1,whichisveryuseful.1bis-Anotherrelatedcommentisthereproducibilityoftheexperiments.Inthismanuscript,fourdifferentdevicesarecompared,whichisveryinteresting.However,eachofthemhasitsownspecificities.Itwouldhavebeenusefultocheckthedifferencesbetweentwodevicesfromthesamesupplierwiththesamespecifications.2-AttheendofsectionII,second-orderautocorrelationexperimentsarementionedtolookatthecoherencebetweenmodes.Thisisindeedacrucialissue.AsfarasIunderstandcorrectlythetheoreticalpartofthismanuscript,itisassumedthatthefirstmodeandthesidelobesarecoherentwitheachother.HoweverIthoughthatthisissuewasalreadyaddressed(andsettled),seeforinstanceNanophotonics,vol.5,272(2016).3-Itwasnotclearformeiftheissueofthedensemoderegimeisaddressedornot

inthismanuscript.Fromthetheoreticalpointofview,itseemsnot.4-InsectionII,justbelowTable1,itiswritten"theeffectiverefractiveindexisdeterminedfromtheFP-modespacingofthemeasuredspectra".Theinferredrefractiveindexthusincludesthedispersion.Thedifferencebetweentheactual"effectiverefractiveindex"andtheoneinferredfromtheFSRcouldbeabout10%.5-InSectionIIIAandafterwards,thetransitionfrequencyisreferredtoasomega_ab,whichmeansthatitisnarrow.However,aQLCisknowntopresentahomogeneousgain.Howwoulditbepossibletomergebothaspects?6-InSectionIIIA,theresonatorisassumedtobealinearFabry-Perotcavity,butthisassumptionisnotclearlywritten.7-InSectionIIIA,"thisassumption[oforthogonalmodes]turnsouttobequitegood".Couldyouprovideanexperimentalevidence?8-FirstsentenceofSectionIIIB.ThethresholdisdefinedthroughtheBeerlossrate.Thisnameisstrange,sinceitisusedasgain.9-Themodelconsiderstwodifferenteffects,namelythepopulationgratingandthepopulationpulsation(whichisanon-lineareffectconnecttoatwo-levelsystem).Ithinkthateacheffectcouldbeconsideredindependentlyinordertoevaluateditsinfluence.Inparticular,gamma_Dcouldprobablybechosenasone(whichmeans"nomobility")withoutanychangeintheconclusion.10-ThecurvesinFigure2aresmooth.Iwouldexpectslightchangesatthetransitions.

ReportoftheThirdRefereeIntheirmanuscriptT.S.Mansuripuretal.describeexperimentsandtheorythatexplainsthegenerationofnewfrequenciesinaQCL.Inparticular,twoquantitativedifferentregimesaredistinguished.OneinwhichresultsinanFMmodulationandonethatresultsinAMmodulationofthelaser.Inmyopinion,thedatasupportsthemainclaimofthemanuscriptwellandIthinkthemanuscriptisverysuitedforPRA.However,thepresentationcanbeimproved.Mainpoints:-TheauthorsmentionthegenerationofnewfrequenciesinmicroresonatorsviatheKerrnonlinearityandresultingfrequencycombs.However,Ithinkthisanalogyisover-emphasizedandtakestoomuchspaceandisdistracting.Itevenmakesupthefirstpartoftheintroductionalthoughtheauthorsadmitthatthesystemsarefairlydifferent.FortheKerrfrequencycombcase,thedispersionandthedetuningarethecrucialparameters.Neitheroftheseisusedinthemanuscript,tomyunderstanding.ThereforeIthinkthisintroductionisratherconfusing.Instead,Iwouldseethecomparisononlyasapartofthefinaldiscussionorpotentiallysomekindofoutlookasitisalreadypartiallyincluded.Thiswouldalsohelptofocusthepaperabitmore.-Afigurethatoutlinesthedifferencesbetweenthetworegimeswouldbeveryhelpful.Forexample,extendingFig.1byasketchofwhatisshowninFig.7wouldprobablymakeiteasiertounderstandthetwodifferentregimes.-Iliketheoverallwritingstyleofthepaper.However,sometimesitgoesabitoff.P2,middleleftforexample.Havingbasicallyafullparagraphinparenthesisdoesnothelpreadabilityandshiftsthefocusawayfromthemaintopic.Minorpoints:-Abstract:“Lorentziangain”isnotacommontermforme-Ref6ismaybenotthebestreferenceforsolitonsinmicroresonatorsasitisfocusedonafabricationtechnique-P3,topleft:“buttoourknowledgethediscrepancywasneverfullyresolved.”.Doesthismanuscriptnowfullyresolvethese?Orwhichpartisnowresolved?-P3,left:“becauselow-frequencysidebandscausethepopulationinversiontooscillateperfe…”Thisisnotevidenttome.-Itisemphasizedthatthecurrentofthelaserhastobeincreasedslowly.However,anactualrateassomethinglikeA/sisnotgivenanywhere,onlyastepsize.Whatistherate?Isthereanyinsightintowhatresultsinalongtimescalethatrequiressuchslowincreasestobeadiabatic?Isitjustthetemperaturecontrol?

-DoesthedispersionoftheQCLchangefordifferentpumpcurrentsasthegainchanges?Dotheystaynormal/anomalousasmeasuredotherwise?-P5,bottomleft:“isincreasedtoroll-over”isnotaprecisedescriptionforme-Fig.6,the\delta\omegaTatthebottomrightinsidethepanelsisverydifficulttoreadinprint-outsbecauseofthebackground

October(1st,(2016(Dear(Dr.(Cassemiro,((Thank(you(for(considering(our(manuscript(for(publication(in(Physical(Review(A.((We( thank( the( three( reviewers( for( their( thoughtful( reviews,( from(which( it( is( clear(that(they(took(the(time(to(understand(this(dense(manuscript,(and(are(grateful( that(all(three(have(found(the(manuscript(suitable(for(publication(in(Physical(Review(A.(((We( have( addressed( the( formatting( concerns( of( the( figures,( and( have( also(reformatted( the( supplemental( information( into( appendices.( Thank( you( for( this(suggestion,(since(it(is(much(simpler(to(have(all(the(information(in(one(place.((Some(figures(from(the(supplemental(material(were(also(moved(into(the(main(text(where(appropriate,(to(make(the(paper(easier(to(read.(We(now(respond(to(the(comments(of(the(three(reviewers(in(turn,(and(have(modified(the(manuscript(accordingly.((Response(to(the(First(Referee((“The( only( critical( remark( concerns( the( title,( which,( while( elegant,( is(somewhat( cryptic( and( does( not( express( the( content( or( generality( of( the(paper...”(Response:( We( have( taken( the( suggestion( to( change( the( title( to( attract( a( wider(community( than( the( QCL( one,( and( the( title( is( now( “SingleNmode( instability( in(standingNwave(lasers:(the(quantum(cascade(laser(as(a(parametric(oscillator.”(((Response(to(the(Second(Referee(((“1( N( My( main( criticism( is( related( to( the( comparison( between( experiments( and(theory,( which( is( rather( qualitative…( It( would( have( been( useful( to( check( the(differences( between( two( devices( from( the( same( supplier( with( the( same(specifications.”(Response:(We(have(taken(the(referee’s(suggestion(to(include(a(table(in(the(text(that(summarizes( these( differences.( We( have( also( mentioned( the( data( from( a( second(device,(nominally(identical(to(TLN4.6,(which(exhibits(sidebands(at(13(FSR(rather(than(the(26(FSR(of(TLN4.6.(We(point(out(that(for(TLN4.6:HR/AR,(we(had(already(included(data(from(two(nominally(identical(devices:(one(exhibits(sidebands(at(46(FSR(and(the(other(at(48(FSR,(very(similar(values.(We(recognize(that(many(of(our(comparisons(are(qualitative(at(this(point,(but(we(have(explicitly(stated(the(reasons(for(why(this(is(the(case,(and(have(clearly(outlined(how(future(work(can(make(these(comparisons(more(quantitative.( However,( the( most( important( comparison( between( experiment( and(data,( that( of( the( comparison( between( the( measured( sideband( spacing( and( the(theoretical(crossing(frequency(that(distinguishes(the(enhancement(and(suppression(regimes,(is(quantitative.((

“2N…( As( far( as( I( understand( correctly( the( theoretical( part(of( this( manuscript,( it( is( assumed( that( the( first( mode( and( the( side(lobes( are( coherent( with( each( other.( However( I( thought( that( this( issue(was( already( addressed( (and( settled),( see( for( instance( Nanophotonics,(vol.(5,(272((2016).”(Response:(Yes,(we(assume(that(the(primary(mode(and(sidebands(are(coherent(with(each( other,( and( explore( the( consequences( of( this( assumption.( (We( have( now(explicitly( written( this( in( the( text.)( The( best( evidence( we( have( that( justifies( this(assumption( is( the( mere( existence( of( the( harmonic( state:( if( the( modes( were( not(coherent(with(each(other,(there(would(be(no(reason(that(we(can(think(of(to(explain(the( modeNskipping( phenomenon.( Since( our( manuscript( is( the( first( to( discuss( the(harmonic( state,( there( is( no( prior( literature( that( explores( the( coherence( between(modes(in(the(harmonic(state,(although(there(is(literature(about(the(coherence(in(the(dense(state.(((“3( N( It( was( not( clear( for( me( if( the( issue( of( the( dense( mode( regime( is(addressed( or( not( in( this( manuscript.( From( the( theoretical( point( of(view,(it(seems(not.”(Response:( We( have( focused( this( paper( on( explaining( the( appearance( of( the(harmonic(state(in(a(theoretically(rigorous(way,(and(have(not(attempted(a(theory(of(the( transition( from( the( harmonic( state( to( the( dense( state.( However,( we( have(provided( data( on( this( second( transition,( and( we( have( moved( a( figure( from( the(supplemental(to(the(main(text(on(the(negative(differential(resistance(exhibited(at(the(transition(from(the(harmonic(to(the(dense(state.(This(observation(demonstrates(that(the( dense( state( emits( slightly(more( optical( power( than( the( harmonic( state( at( the(same(current,(and(provides(a(clue(for(future(work(in(this(direction.(((“4( N( …The( difference( between( the( actual( "effective( refractive( index"( and( the( one(inferred(from(the(FSR(could(be(about(10(%.”(Response:(The(only(reason(we(include(the(index(is(that(it(is(needed(to(calculate(the(diffusion(parameter(gamma_D.(We(understand(that(the(index(inferred(from(the(FSR(is( the(group( index,(while(what( is( really(needed( to(calculate(gamma_D( is( the(phase(index.(However,(there(are(larger(errorNsources(in(calculating(gamma_D,(such(as(the(unknown( amount( of( current( inhomogeneity,( than( the( precision( of( the( refractive(index,(so(we(don’t(want(to(focus(on(this(detail(and(distract(from(the(main(message(of(the(paper.(((“5( N(…( the( transition( frequency( is( referred( to(as(omega_ab,(which(means( that( it( is(narrow.(However,(a(QLC(is(known(to(present(a(homogeneous(gain.(How(would(it(be(possible(to(merge(both(aspects(?”(Response:( In( fact,(both(aspects(are(merged( in(our(work.(The( finite(dephasing(time(T_2( provides( the( homogeneously( broadened( gain,( centered( on( the( frequency(omega_ab.(((“6(N(In(Section(IIIA,(the(resonator(is(assumed(to(be(a(linear(FabryNPerot(cavity,(but(this(assumption(is(not(clearly(written.”(

Response:.(We(have(now(made(it(explicitly(clear,(and(have(also(included(a(reference(to(the(appendix(where(the(case(of(the(ring(cavity(is(treated.(((“7( N( In( Section( IIIA,( "this( assumption( [of( orthogonal(modes]( turns(out( to(be(quite(good".(Could(you(provide(an(experimental(evidence(?”(Response:( We( are( saying( that( the( intensity( profile( in( a( cavity( with( mirrors( of(reflectivity(R=0.25(closely(resembles(the(perfect(cosine(one(would(get(if(the(mirrors(had( R=1.( We( justify( this( based( on( the( following( calculation:( we( calculated( the(theoretical( LI( curve( for( a( laser( with( R=0.25( (taking( into( account( the( true( mode(shape)(and(another(with(R=1,(and(the(results(are(very(close.(We(also(calculate(the(mode(overlap(factors(in(the(two(cases,(and(they(are(also(very(close.(However,(for(an(HR/AR( laser( the( true(mode( shape( needs( to( be( taken( into( account,( and( cannot( be(replaced(simply(with(a(cosine(curve.(((“8( N( …The( threshold( is( defined( through( the( Beer( loss( rate.( This( name( is( strange,(since(it(is(used(as(gain.”(Response:( We( have( changed( the( name( to( the( “Beer( rate,”( and( explained( how( it(determines(the(maximum(amount(of(absorption(and(the(maximum(amount(of(gain,(depending(on(the(amount(of(population(inversion.(((“9( N( The( model( considers( two( different( effects,( namely( the( population(grating( and( the(population(pulsation…I( think( that( each(effect( could(be( considered(independently( in( order( to( evaluated( its( influence.( In( particular,( gamma_D( could(probably(be(chosen(as(one…without(any(change(in(the(conclusion.”(Response:( Indeed,(we( treat(each(effect( (the(population(grating(and( the(population(pulsations)(separately(in(two(different(theory(subsections(of(the(manuscript,(so(that(the( influence(of(each(effect(can(be(considered(separately.(However,(when(it(comes(time( to( evaluate( the( instability( threshold,( it( is( the( combined( effect( of( the( two(contributions( that(matters.( If(gamma_D=1(as( the(referee(suggests,( then( in( fact(one(can(show(that(the(effect(of(the(population(grating(dominates(to(such(a(large(degree(that( one( would( expect( the( sidebands( to( appear( at( adjacent( FP( modes.( Thus,( the(value(of(gamma_D(strongly(affects(the(conclusion.((“10( N( The( curves( in( Figure( 2( are( smooth.( I( would( expect( slight( changes(at(the(transitions.”(Response:(This(is(an(astute(observation.(There(are(indeed(small(kinks(in(the(output(power(at( the( transition( from(harmonic( to(dense( state( (on( the(order(of( tenths(of( a(percent)(but(they(are(so(small(that(they(are(difficult(to(measure(with(a(powermeter.(However,( these( kinks( can( be( seen( in( the( IV( curve( which( can( be( measured( more(sensitively( than( the(power.(We(have(moved(an( IV(curve( to( the(main( text( from(the(supplementary(to(highlight(this(feature.((((((

Response(to(the(Third(Referee((Referee:( “The( authors( mention( the( generation( of( new( frequencies( in(microresonators(via(the(Kerr(nonlinearity(and(resulting(frequency(combs.(However,(I(think(this(analogy(is(overNemphasized….”(Response:(We(understand( the( referee’s( concern(here.( In( fact,(we( even( considered(starting(the(introduction(with(a(discussion(of(the(singleNmode(laser(instability(rather(than( with( parametric( oscillation( in( microresonators.( However,( the( connection( is(very(deep,(and(we(had(a(specific(goal(in(mind(by(emphasizing(this(analogy.(We(are(confident(that(future(discoveries(in(QCL(physics(will(be(motivated(and(guided(by(the(current( state( of( understanding( of( microresonators.( Laser( physicists( in( the( 1960s(through(the(1980s(did(not(pay(much(attention(to(GVD,(or(to(the(detuning(between(the( lasing( mode( and( the( cold( cavity( mode( that( it( occupies.( However,( these(parameters( have( been( shown( by( the(microresonator( community( to( be( extremely(important( in( influencing( the(properties( of( the( sidebands.(Therefore,(we(hope( that(the( QCL( community( looks( to( the( recent( microresonator( literature( for( guidance,(rather(than(being(stuck(in(some(of(the(same(ruts(that(faced(the(laser(community(in(the(1980s.((“A( figure( that( outlines( the( differences( between( the( two( regimes( would( be(very( helpful.( For( example,( extending( Fig.1( by( a( sketch( of( what( is( shown( in( Fig.7(would(probably(make(it(easier(to(understand(the(two(different(regimes.”(Response:(In(our(opinion,(Fig.(7,(together(with(three(distinct(subsections(devoted(to(describing(the(incoherent,(FM,(and(AM(instabilities,(suffice(to(get(the(point(across.((((“I( like( the( overall( writing( style( of( the( paper.( However,( sometimes( it(goes( a( bit( off…Having( basically( a( full( paragraph( in( parenthesis( does( not( help(readability(and(shifts(the(focus(away(from(the(main(topic.”(Response:(As(with(the(referee’s(first(point,(we(understand(the(concern(here.(In(some(places(we(may(have( included(excess( information( that(makes(a( first( reading(of( the(paper( time( consuming.(However,(we(hope( that( such( excess( information(will,( on( a(second( reading,( serve( to( properly( place( all( of( the( presented( information( in( its(historical(context.(It(would(be(a(disservice(to(the(reader(to(not(mention(the(relation(between( population( pulsations,( the( ac( Stark( effect,( the( Mollow( triplet,( and( the(dressed(picture(of(atoms(in(fields.(((“Abstract:(“Lorentzian(gain”(is(not(a(common(term(for(me.”(Response:(It(is(not(a(common(term,(but(we(hope(the(meaning(is(clear.((“Ref( 6( is( maybe( not( the( best( reference( for( solitons( in( microresonators(as(it(is(focused(on(a(fabrication(technique.”(Response:(We(have(replaced(the(reference(with(one(to(Saha(et(al,(Opt.(Express(2013.((“P3,( top( left:( “but( to( our( knowledge( the( discrepancy( was( never( fully(

resolved.”.( Does( this( manuscript( now( fully( resolve( these?( Or( which( part(is(now(resolved?”(Response:(Our(manuscript(focuses(only(on(standingNwave(lasers(and(cannot(explain(the(discrepancy(in(travelingNwave(lasers(between(the(low(experimentally(measured(instability(threshold(and(the(large(theoretically(predicted(one.((“P3,( left:( “because( lowNfrequency( sidebands( cause( the( population(inversion(to(oscillate(…”(This(is(not(evident(to(me.”(Response:(This(statement(is(properly(explained(in(due(course,(so(we(have(added(the(phrase( “We( will( show( that…”( to( this( sentence( in( the( introduction( to( assure( the(reader(that(we(do(not(expect(them(to(understand(this(statement(until(later.((“…an( actual( rate( as( something( like( A/s( is( not( given(anywhere…(Is( there(any( insight( into(what(results( in(a( long(timescale(that(requires(such(slow(increases(to(be(adiabatic?(Is(it(just(the(temperature(control?”(Response:( We( have( added( the( rate( “roughly( 2( mA( per( second.”( When( the( laser(reaches(a(current(barely(above(threshold,(it(can(take(a(very(long(time(for(the(steady(state( to(be( reached,(because( the(exponential( growth(of( the( laser( field( is( still(quite(slow( barely( above( threshold.( We( have( also( seen( in( the( experiments( that( the(harmonic( regime( can( be( seen( even( if( adjusting( the( current( in( larger( increments.(However,( if( a( reader( tries( to( reproduce(our( experiments(we( recommend( ramping(the(current(up(slowly.((Does( the( dispersion( of( the( QCL( change( for( different( pump( currents( as(the(gain(changes?(Do(they(stay(normal/anomalous(as(measured(otherwise?(Response:( The( dispersion( does( change( to( a( small( degree( with( the( pumping,( as(shown(in(Villares,(Optica((2016).(We(have(added(a(new(appendix(to(show(the(GVD(measurements(that(we(have(done.(The(fact(remains(that(the(harmonic(state(exists(in(QCLs(that(exhibit(both(normal(and(anomalous(dispersion,(and(this( fact( is(the(main(point(that(we(wish(to(emphasize(in(this(manuscript(regarding(GVD.((“P5,( bottom( left:( “is( increased( to( rollNover”( is( not( a( precise(description(for(me.”(Response:(We(have(replaced(this(jargon(with(“persists(for(all(higher(currents.”((“Fig.( 6,( the( \delta( \omega( T( at( the( bottom( right( inside( the( panels( is(very(difficult(to(read(in(printNouts(because(of(the(background.”(Response:(We(have(made(those(letters(darker.((An( attached( manuscript( with( the( changes( highlighted( is( included( in( our(resubmission.((Sincerely,((Tobias(Mansuripur(Federico(Capasso((

The quantum cascade laser as a self-pumped parametric oscillator

Tobias S. Mansuripur,1 Camille Vernet,2, 3 Paul Chevalier,2 Guillaume Aoust,2, 4 Benedikt Schwarz,2, 5 FengXie,6 Catherine Caneau,7 Kevin Lascola,6 Chung-en Zah,6 David P. Ca↵ey,8 Timothy Day,8 Leo J.Missaggia,9 Michael K. Connors,9 Christine A. Wang,9 Alexey Belyanin,10 and Federico Capasso2, ⇤

1Department of Physics, Harvard University, Cambridge, MA 02138 USA2John A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, MA 02138 USA3Ecole Polytechnique, 91123 Palaiseau, France

4ONERA, The French Aerospace Lab, 91123 Palaiseau, France5Institute of Solid State Electronics, TU Wien, 1040 Vienna, Austria

6Thorlabs Quantum Electronics (TQE), Jessup, MD 20794 USA7Corning, Inc., Corning, NY 14831 USA

8Daylight Solutions, Inc., San Diego, CA 92128 USA9Massachusetts Institute of Technology, Lincoln Laboratory, Lexington, MA 02420 USA

10Department of Physics and Astronomy, Texas A & M University, College Station, TX 77843 USA

We report the observation of a clear single-mode instability threshold in continuous-wave Fabry-Perot quantum cascade lasers (QCLs). The instability is characterized by the appearance of side-bands separated by tens of free spectral ranges (FSR) from the first lasing mode, at a pump currentnot much higher than the lasing threshold. As the current is increased, higher-order sidebands ap-pear that preserve the initial spacing, and the spectra are suggestive of harmonically phase-lockedwaveforms. We present a theory of the instability that applies to all homogeneously-broadenedstanding-wave lasers. The low instability threshold and the large sideband spacing can be explainedby the combination of an unclamped, incoherent Lorentzian gain due to the population grating,and a coherent parametric gain caused by temporal population pulsations that changes the spectralgain line shape. The parametric term suppresses the gain of sidebands whose separation is muchsmaller than the reciprocal gain recovery time, while enhancing the gain of more distant sidebands.The large gain recovery frequency of the QCL compared to the FSR is essential to observe thisparametric e↵ect, which is responsible for the multiple-FSR sideband separation. We predict thatby tuning the strength of the incoherent gain contribution, for example by engineering the modaloverlap factors and the carrier di↵usion, both amplitude-modulated (AM) or frequency-modulatedemission can be achieved from QCLs. We provide initial evidence of an AM waveform emitted bya QCL with highly asymmetric facet reflectivities, thereby opening a promising route to ultrashortpulse generation in the mid-infrared. Together, the experiments and theory clarify a deep connectionbetween parametric oscillation in optically pumped microresonators and the single-mode instabilityof lasers, tying together literature from the last 60 years.

I. INTRODUCTION

In the last decade, significant e↵orts have spurredthe understanding of high-Q optically-pumped microres-onators. A monochromatic external pump beam is cou-pled to a mode of the microresonator, and at su�cientpump power the third-order �(3) Kerr nonlinearity, re-sponsible for the intensity-dependent refractive index,couples the pumped mode to fluctuations at other fre-quencies, which leads to interesting physics. Startingfrom an initial demonstration of third-order optical para-metric oscillation (OPO) [1, 2], in which the pump beamprovides su�cient parametric gain to allow a few pairs ofsidebands to oscillate, this technique has been extendedto generate wide-spanning frequency combs [3, 4], andmost recently temporal solitons [5, 6]. The many degreesof freedom one can manipulate in these systems, such asthe group velocity dispersion (GVD), the free spectral

⇤capasso@seas.harvard.edu

range of the resonator, the detuning of the pump fre-quency relative to the cold cavity mode that it pumps,and the pump power, among others, have provided a richnonlinear optical playground to observe diverse physicalphenomena.

A laser, much like an OPO, is an optical resonator inwhich circulating monochromatic light reaches high in-tensity, the di↵erence being that the light is internallygenerated rather than externally injected. Furthermore,the very gain medium that allows for lasing, simulta-neously provides a third-order nonlinearity, the popula-tion pulsation (PP) nonlinearity [7]. The PP nonlinear-ity is an intrinsic property of any two-level system thatinteracts with near-resonant amplitude-modulated (AM)light: the radiative transition rate between the states,and therefore the population of each state, is temporallymodulated by the AM light, resulting in so-called pop-ulation pulsations that act back on the light field in anonlinear way. The laser therefore contains the two in-gredients, high-intensity light and a non-linearity, nec-essary for parametric oscillation. Indeed, in the late1960s the importance of PPs in determining the above-

2

threshold spectral evolution of a homogeneously broad-ened, traveling-wave laser was realized. At the laserthreshold, one mode–which we call the primary mode–begins to lase and as the current is increased the popu-lation inversion remains clamped to its threshold value.It was first thought that this clamping should preventany other mode from reaching the oscillation threshold.This reasoning, however, neglects the fact that when aphoton of a di↵erent frequency is spontaneously emittedin the presence of the primary lasing field, a beat note–i.e., an intensity modulation at the di↵erence frequencyof the two fields–is created. The beat note creates a PPthat provides a parametric contribution to the gain ofthe spontaneously emitted photon. At a su�ciently highpumping level known as the instability threshold, thisparametric gain can–despite the fact that the popula-tion inversion is clamped to its threshold value–allow twosidebands to overcome the loss. The separation of thesesidebands from the primary mode is related to the Rabifrequency induced by the primary mode. This e↵ect isresponsible for both the Haken-Risken-Schmid-Weidlich(HRSW) instability [8, 9] and the Risken-Nummedal-Graham-Haken (RNGH) instability [10, 11]. Many yearslater, insightful work properly identified the fundamentalrole of PPs in the single-mode laser instabilities [12–18]and also chaos [19]. (We note that PPs are important notonly for inverted media. Historically, their e↵ects werefirst appreciated in microwave spectroscopy pump-probeexperiments by Autler and Townes [20] in 1955, and sooncame to be known as the ac Stark e↵ect. Through thelate 1960s and 1970s, significant work on sideband ampli-fication [21], resonance fluorescence [22, 23], and the Mol-low scattering triplet [24–28] culminated in the “dressed”description of atoms in strong fields [29]. In the 1980s,the PP nonlinearity was cast in the language of nonlinearoptics and applications such as four-wave-mixing (FWM)[30], phase conjugation [31], and optical bistability [14]were explored.)

Both the HRSW and RNGH single-mode instabilitiesapply to homogeneously-broadened traveling-wave lasers,and predict the appearance of sidebands on the primarylasing mode, as shown in Fig. 1(a). We remark that ingeneral, the temporal behavior of an electric field thatcontains three equally spaced frequencies can be moreamplitude-modulated or frequency-modulated (FM), de-pending on the spectral phase, as shown in Fig. 1(b). Onecan think of the intensity modulation (in other words,the beat note) of the AM and FM fields as resultingfrom the sum of two phasors rotating at frequency �!,each of which is created by the beat between a sidebandand the primary mode. As shown in Fig. 1(c), the twophasors either constructively interfere to create a largeintensity modulation (AM) or destructively interfere toeliminate the intensity modulation (FM). In both theHRSW and RNGH instabilities, the three-wave field isby necessity AM; a constant-intensity FM field wouldnot create the PP and the resulting parametric gain thatis required by the sidebands to reach the lasing thresh-

(a) Emission Spectrum

FM:

AM:

t

E(t)

t

E(t)

(b) E-field vs. Time (c) Beat Note Phasors

(d) Standing-Wave Cavity

FIG. 1. (a) The emission spectrum at the instability thresholdcomprises a primary mode and two weak sidebands. (b) Thetemporal behavior of the field E(t) depends on the relativephases of the three modes ��, �0, and �+, and shown are theAM and FM configurations. (c) The AM and FM fields can beunderstood in terms of the constructive and destructive addi-tion of two beat note phasors, where each phasor represents acontribution to the intensity modulation at the di↵erence fre-quency �! resulting from the superposition of each sidebandwith the primary mode. (d) In a standing-wave cavity, theintensity of each mode varies with position, and the spatialmodes corresponding to di↵erent frequencies do not perfectlyoverlap.

old. In the HRSW case, which applies to low-qualitycavities for which the photon lifetime is shorter than theatomic decay time, the sideband separation is smallerthan the mode spacing, or free spectral range (FSR), ofthe cavity. All three lasing frequencies fall within a sin-gle cold cavity resonance, which is made possible by aregion of anomalous dispersion created by the PP [12].In the RNGH instability, which applies to higher qual-ity cavities, the sidebands must coincide with cold cavitymodes in order to satisfy the roundtrip phase condition,resulting in a separation that is an integer multiple ofthe FSR. An important corollary of this requirement isthat to observe the e↵ect of the PPs, the FSR must besmaller than the gain recovery frequency (i.e., inverseof the gain recovery time T

1

). Why? The gain recov-ery time determines the fastest time scale at which thepopulation inversion can respond to an intensity mod-ulation, therefore the amplitude of the PP is only sig-nificant for sidebands detuned by an amount close to orsmaller than 1/T

1

. If the FSR is greater than 1/T1

, it isnot possible to simultaneously satisfy the roundtrip gainand phase conditions for a sideband to become unsta-ble. Ideally, the FSR should be significantly smaller than1/T

1

so that the FP modes densely populate the para-metric gain lobe, increasing the probability of satisfyingthe instability condition. Provided this condition is met,the RNGH instability predicts that a traveling-wave laserwith rapid dephasing must be pumped nine times above

3

threshold before the instability appears. Experimentalobservations of a rhodamine dye ring laser [16] showedsignatures of an RNGH-like instability, with two key dif-ferences: the instability threshold was only fractionallyhigher–not nine times higher–than the lasing threshold,and the sideband creation was accompanied by the dis-appearance of the primary mode. E↵orts to explain thediscrepancies between theory and experiment are well-summarized in [17, 32], but to our knowledge the dis-crepancy was never fully resolved.

In this work, we will investigate the single-mode in-stability in a standing-wave laser, shown schematicallyin Fig. 1(d). The distinguishing feature of the standing-wave laser is that the primary mode induces a populationgrating (PG) (as long as carrier di↵usion is limited), ane↵ect known as spatial hole burning (SHB). The gain ofother cavity modes is no longer clamped above thresh-old, but continues to increase with the pumping. There-fore, the instability threshold can be reached without theneed for PP parametric gain. We call this an incoherentinstability, and it occurs in media whose gain recoverytime is too slow for PPs to occur (FSR > 1/T

1

), suchas diode lasers. In gain media with a fast recovery time(FSR < 1/T

1

), both the incoherent gain and the para-metric PP contribution to the gain must be considered.The PP parametrically suppresses the gain of nearbysidebands, because low-frequency sidebands cause thepopulation inversion to oscillate perfectly out of phasewith the intensity modulation. On the other hand, thePP enhances the gain of larger-detuning sidebands, asoccurs in the RNGH instability. Depending on the rel-ative contributions of the incoherent and coherent gain,we show that the laser will either emit an FM or an AMwaveform at the instability threshold, to either minimizeor maximize the amplitude of the PPs. If the incoherentgain is large, nearby sidebands are favored and will yieldFM emission to minimize the amount of parametric sup-pression. If the incoherent gain is small, larger-detuningsidebands are favored and will yield AM emission to max-imize the amount of parametric enhancement. The possi-bility of both FM and AM emission from a standing-wavelaser is a novelty not shared by the traveling-wave laser,which, as we mentioned, can only produce an AM wave-form.

The quantum cascade laser (QCL) is precisely the kindof laser for which both the PG and PPs are important.An electron injected into the upper state has only ashort picosecond lifetime during which to di↵use beforeit scatters to the ground state–not enough time to tra-verse the half-wavelength mid-infrared (� ⇠ 3 -12µm)standing-wave from node to antinode. Therefore, thePG is not washed out. Also, the FSR (typically 8 to16GHz) is much less than the gain recovery frequency(1/T

1

⇡ 1THz), so the population inversion has no dif-ficulty following the beat notes in field intensity cre-ated when multiple modes lase simultaneously, yieldingPPs. We report the discovery that continuous-wave (cw)Fabry-Perot (FP) QCLs reach a well-defined instability

threshold, characterized by the appearance of sidebandswhose separation from the primary mode can be severalmultiples of the cavity FSR. This mode skipping is a clearsignature of the parametric PP interaction between theprimary mode and the sidebands, which strongly sup-presses sidebands at separations much smaller than thelarge gain recovery frequency of the QCL. The behav-ior is observed in QCLs that emit at wavelength 3.8µm,4.6µm, and 9.8µm, indicating that it is a universal fea-ture of mid-infrared QCLs, independent of the specificbandstructure of the active region. The strength of thePG can be tuned by coating the facets to adjust theirreflectivities. By comparing the measurements with thetheory, we argue that QCLs with uncoated facets emitan FM waveform. A QCL with one high-reflectivity facetand a su�ciently low reflectivity of the other facet shouldin principle emit an AM waveform, and we provide pre-liminary evidence that this is indeed the case, demon-strating a QCL whose sidebands are separated from theprimary mode by 46 FSR. While the PG and PP havebeen known to be important in QCLs, in previous worktheir e↵ects were treated separately [33]. Instead, we em-phasize that one should think of the PG–a spatial mod-ulation of the inversion–and the PP–a temporal modu-lation of the inversion–as working in tandem to create aphase-locked multimode state at low pump power.

As the current is increased past the instability thresh-old, higher-order sidebands that preserve the initial spac-ing appear. This suggests that the FP-QCL can emita harmonically phase-locked waveform without the needfor any external modulation or additional nonlinear ele-ments. Why have such spectra not been observed be-fore, except in a few cases [34, 35]? We have foundthe harmonic states to be extremely sensitive to opti-cal feedback. Simply placing a collimating lens betweenthe QCL and the spectrometer–even a poorly aligned,tilted lens with a focal length of a few cm–makes it dif-ficult to observe the harmonic state, and instead yieldsthe more familiar QCL spectrum in which all adjacentFP modes lase. It is also important to slowly increasethe current, which allows for a smooth transition fromthe single-mode to the harmonic regime. We argue thatthe harmonic state is an intrinsic regime of all QCLs. Thefact that it has only been observed 15 years after the in-vention of the cw QCL is a testament to the destabilizinginfluence of optical feedback [36].

In the last few years, comb generation in a QCL onadjacent FP modes has been demonstrated [37–39], andthe importance of parametric mode coupling is known[40, 41]. (These devices all had multi-stage inhomo-geneously broadened active regions, which distinguishesthem from the devices in our work.) Because these combshave so far always comprised adjacent cavity modes, con-sideration has only been given to the case where the fun-damental frequency of the PPs equals the FSR. This lowPP frequency strongly favors the emission of an FM wave-form. The remarkable degree of freedom to skip modes,never before considered, means that the temporal pe-

4

riodicity of the PPs is no longer pinned to the cavityroundtrip time (typically 60 to 120 ps), but is shortenedby a factor equal to the number of modes skipped, whichreaches 46 in one of our QCLs. This reduction of theperiod down to the order of the gain recovery time isthe crucial feature that allows for the possibility of AMemission.

Finally, we emphasize the deep connection betweenthe single-mode laser instability and mode proliferationin optically pumped microresonators. Both are cases ofparametric oscillation that are initiated by a nonlinear-ity, either PP or Kerr, transferring energy from a pumpbeam to two sidebands. For a passive microresonatorthe pump beam must be injected, while in the laser thepump beam is internally generated. This analogy, whichwe only begin to uncover here, can help guide future worktoward understanding the rich emission spectra of QCLs.We hope that the advancement of the QCL can parallelthe rapid progress seen in microresonators in the lastdecade, leading to a compact source of mid-infrared fre-quency combs for dual-comb spectroscopy of trace gasesand short pulse generation [42].

In Sec. II we present the experimental results, whichhelps to motivate the theory presented in Sec. III. In Sec.IV we compare the theory with the measurements, andfinally conclude in Sec. V.

II. EXPERIMENT

All four devices used in this study are cw, buried het-erostructure, FP-QCLs. Our device naming conventionidentifies the provider of the device (LL: MIT LincolnLaboratory, TL: Thorlabs, DS: Daylight Solutions) fol-lowed by the emission wavelength in microns. The activeregion of device LL-9.8 is a double phonon resonance de-sign using lattice-matched Ga

0.47

In0.53

As/Al0.48

In0.52

As,grown by metalorganic chemical vapor deposition, withthe well-known layer structure of [43], albeit with a nom-inal doping of n = 2.5 ⇥ 1018 cm�3, for which extensivebandstructure calculations have been done [44]. The de-vice length is 3mm and width is 8µm. Devices TL-4.6,TL-4.6:HR/AR, and DS-3.8 were grown using strainedGa

x

In1�x

As/Aly

In1�y

As and are described in [45], al-though the layer sequence is not given. The length is6mm and width is 5µm for these three devices. Bothfacets are left uncoated for LL-9.8, TL-4.6, and DS-3.8.The only coated device is TL-4.6:HR/AR, which has ahigh-reflectivity (HR) coating on the back facet (R ⇡ 1)and an antireflection (AR) coating on the front facet(R ⇡ 0.01), but is otherwise nominally identical to TL-4.6. Far-field measurements indicate that all devices ex-hibit single lateral-mode emission over the full range ofapplied current. It is worth mentioning that the short-wave QCLs, DS-3.8 and TL-4.6, have positive GVD andthe long-wave device LL-9.8 has negative GVD. We ex-pect this because their wavelengths lie on opposite sidesof the zero-GVD point of InP, but we have also confirmed

single-mode harmonic dense

LL-9.8

TL-4.6:HR/AR

TL-4

.6

DS-3

.8

DS-3.8

LL-9.8

TL-4.6:

HR/AR

TL-4

.6

slope=1

FIG. 2. Total power output of each QCL (from both facets)vs. current, color-coded to indicate the range over which thelaser operates in a single-mode, harmonic state, or dense state.Inset: the intracavity power normalized to the saturation in-tensity (calculated from the measured output power and thebest estimates for , T1, T2, and the facet reflectivities) isplotted vs. J/Jth.

Device ne↵ d [nm·e] Tup [ps] �D T2 [fs] �!FSR [GHz] �!sb [GHz] Jsb/Jth

LL-9.8 3.43 3 0.54 0.93 81 92 642 1.14

TL-4.6 3.23 1.63 1.7 0.49 74 48 1259 1.17

TL-4.6:HR/AR 3.25 1.63 1.7 0.49 74 48 2216 1.22

DS-3.8 3.25 1.5 1.74 0.40 43 49 977 1.12

TABLE I. Summary of relevant parameters of the devicesused in this study. ne↵ , T2, �!sb, and Jsb were measuredquantities. Tup and d were calculated from the bandstructurefor TL-4.6 and DS-3.8, and taken from [44] for LL-9.8, and�D was calculated assuming D = 77 cm2/s [47].

this by measurement [46]. Some relevant parameters foreach device are given in Table I: the e↵ective refractiveindex n

e↵

is determined from the FP-mode spacing of themeasured spectra; the dipole moment d and the upperstate lifetime T

up

are calculated from the bandstructure;the dephasing time T

2

is determined from a Lorentzianfit to either an electroluminescence or subthreshold mea-surement of each device. The output power of each laserwas measured with a calibrated thermopile (Ophir 3A-QUAD) placed close to the facet; the total output poweris plotted in Fig. 2, which is obtained from the front facetonly for TL-4.6:HR/AR and by doubling the single-facetpower of the uncoated lasers.

Our goal was to precisely examine the spectral evolu-tion of the QCL with increasing current, from the single-mode to the multimode regime. Specifically, we wantedto answer the question: at what pumping level does asecond mode start to lase, and what is the relationshipbetween the second frequency and the first? To answer

5

this question, we would begin each measurement with thelaser driven at a current beneath the laser threshold. Thecurrent was then slowly increased in steps of 1mA, andthe spectrum was monitored using a Fourier transforminfrared (FTIR) spectrometer (Bruker Vertex 80v), witheither a nitrogen-cooled InSb detector (for DS-3.8 andTL-4.6) or HgCdTe detector (for LL-9.8). The currentwas supplied by a low-noise driver (Wavelength Electron-ics QCL1500 or QCL2000), and the temperature of thecopper block beneath the QCL was stabilized to 15�C.The slow rate of increase of the current was necessaryto precisely identify the instability threshold, and alsoto prevent rapid temperature variations. To completelyeliminate the possibility of optical feedback due to reflec-tions from optical elements outside the laser cavity, theQCL was placed about 40 cm from the entrance windowto the FTIR and its output was not collimated with alens, but simply allowed to diverge. The high power ofthe devices and the sensitivity of the detectors was suf-ficient to measure spectra despite the small fraction ofcollected optical power.

Spectra measured in this manner are shown in Fig. 3for the three uncoated devices. Each spectrum is nor-malized to its own maximum and plotted on a logarith-mic scale covering 40 dB of intensity variation. All threelasers undergo a very similar spectral evolution. Abovethreshold, the laser remains single-mode for a substan-tial range of current until a clear instability threshold isreached, at which a 1mA increase in current results inthe appearance of new lasing modes. The new frequen-cies appear as symmetric sidebands on the primary lasingfrequency, with a separation that is many integer multi-ples of the FSR. The sideband spacing �!

sb

and pumpingJsb

at the sideband instability threshold are given in Ta-ble I for each device. Taking LL-9.8 as a first example,at J

sb

/Jth

= 1.14 a pair of equal-amplitude sidebandsseparated by 7 FSR from the primary mode suddenlyrise out of the noise floor to an intensity 20 dB weakerthan the primary mode. As the current increases further,higher-order sidebands appear that preserve the initialspacing, eventually yielding a spectrum at J/J

th

= 1.39of 11 modes, each separated by 7 FSR from its nearestneighbors. We refer to a spectrum of modes separated bymultiple FSR as a harmonic state. Above J/J

th

= 1.39,interleaving modes incommensurate with the harmonicspacing begin to appear. At J/J

th

= 1.47, there is an-other sudden transition at which all adjacent FP modesare populated; we refer to this as a “dense” state, andit persists as the current is increased to roll-over. Fordevice TL-4.6, sidebands with a separation of 26 FSRfrom the primary mode appear at J

sb

/Jth

= 1.17, andthe transition to the dense state occurs at J/J

th

= 1.30.For device DS-3.8, the sideband separation is 20 FSR atJsb

/Jth

= 1.12. As the current is increased, the sidebandspacing displays a sudden jump from 20 FSR to 25 FSR.At J/J

th

= 1.32 the laser jumps to a dense state forsomewhere between a few seconds and a minute beforereturning to a “noisy” harmonic state: one with promi-

983 mA

1115 mA

1140 mA

1500 mA

1180 mA

1280 mA

1360 mA

1380 mA

1414 mA

1440 mA

1116 mA

J/Jth = 1.00

1.14

1.16

1.53

1.20

1.31

1.39

1.41

1.44

1.47

1.14 7 FSR

435 mA

509 mA

520 mA

800 mA

545 mA

560 mA

565 mA

580 mA

585 mA

700 mA

510 mA

J/Jth = 1.00

1.17

1.20

1.83

1.25

1.29

1.30

1.33

1.34

1.61

1.17

380 mA

407 mA

416 mA

741 mA

424 mA

470 mA

480 mA (transient)

480 mA

500 mA

502 mA

408 mA

J/Jth = 1.04

1.12

1.14

2.04

1.16

1.29

1.32

1.32

1.37

1.38

1.12

26 FSR

20 FSR

25 FSR

29 FSR

(a) LL-9.8

(b) TL-4.6

(c) DS-3.8

FIG. 3. Spectra of the three uncoated QCLs (a) LL-9.8, (b)TL-4.6 and (c) DS-3.8 as the current is incremented, startingfrom below threshold.

6

990 mA

1000 mA

1100 mA

1500 mA

1140 mA

1180 mA

1240 mA

1280 mA

1360 mA

1440 mA

1040 mA

J/Jth = 1.01

1.02

1.12

1.53

1.16

1.20

1.27

1.31

1.39

1.47

1.06

LL-9.8: decreasing current

FIG. 4. Spectra of LL-9.8 as the current is decremented,starting from 1500 mA.

nent harmonic peaks but many incommensurate modespopulated as well. At J/J

th

= 1.38 the dense state ap-pears again, and this time persists for all higher currents.A measurement of the IV curve, shown in the supple-mental, exhibits negative di↵erential resistance when thelaser transitions from the noisy harmonic state to thedense state. This indicates that at a given voltage, thedense state can extract slightly more power from the gainmedium than the harmonic state.

After the laser enters the dense state, we decrease thecurrent slowly and study the spectral evolution. Thereis a remarkable hysteresis, as seen in the spectra for LL-9.8 in Fig. 4. The dense state persists all the way untilJ/J

th

= 1.01, when the single-mode finally reappears. Asimilar hysteresis occurs in TL-4.6 and DS-3.8 (see thesupplemental for spectra). For DS-3.8, a noisy harmonicstate can appear, and the laser can jump from a noisyharmonic state back to a dense state as the current isdecreased further. A general observation for all three de-vices is that the clean harmonic state cannot be recoveredonce the laser has entered the dense state.

Lastly, we present in Fig. 5 the spectral evolution ofTL4.6-HR/AR as the current is incremented. The behav-ior of this device is di↵erent from the uncoated devicesin two significant ways: 1) the sidebands appear witha separation of 46 FSR, much larger than any spacingseen previously, at J

sb

/Jth

= 1.22, and 2) the harmonicregime persists over a much larger range of output powerthan it does in the uncoated devices, as seen by the color-coding in Fig. 2. (A second device, nominally identicalto TL-4.6:HR/AR, developed sidebands with a spacingof 48 FSR at J

sb

/Jth

= 1.18.)When the spectral evolution measurement is repeated

many times for one device, starting from below thresholdand incrementing the current, we find that the instabilitythreshold J

sb

and sideband spacing �!sb

are always the

735 mA

875 mA

885 mA

1220 mA

900 mA

950 mA

1000 mA

1050 mA

1160 mA

1165 mA

880 mA

J/Jth = 1.01

1.20

1.22

1.68

1.24

1.31

1.38

1.44

1.60

1.60

1.21

46 FSR

TL-4.6:HR/AR

FIG. 5. Spectra of TL-4.6:HR/AR as the current is incre-mented, starting from below threshold.

same. As the current is increased past Jsb

, there can beslight variations from one experiment to another. Forexample, the jump from 20 to 25 FSR in TL-4.6 does notalways occur at the exact same current, but predictablywithin a range of about 20mA. The same is true of thetransition to the dense state.We would like to know the temporal behavior of the

emitted field in the harmonic state, specifically whetherit has more AM or FM character. Unfortunately this can-not be determined by measuring the strength of the beatnote of the laser output, because the smallest observedbeat frequency is greater than 100GHz, larger than theelectrical bandwidth of any mid-infrared photodetector.We therefore look to theory for insight on the matter,and plan on second-order autocorrelation experiments infuture work.

III. THEORY

The instability threshold is characterized by the ap-pearance of symmetric sidebands on the primary lasingmode. Our goal is to theoretically explain the frequencyseparation of the sidebands and the pump power at whichthey first appear. We begin with the general framework:the Maxwell-Bloch equations for a two-level system andthe spatial mode expansion of a laser cavity. Then, wefirst address the single-mode solution of the laser to de-termine how the primary mode and the population grat-ing evolve with increasing pumping. Secondly, we mustunderstand how the two-level system responds to a weakfield at a frequency di↵erent from that of the primarymode–the population pulsation. Finally, we will com-bine these two ingredients, the PG and the PP, to ex-plain the instability threshold. We find that the PGprovides an unclamped Lorentzian contribution to the

7

gain of the sidebands, which is responsible for the lowinstability threshold. The PP reshapes the gain, sup-pressing nearby sidebands and enhancing more distantones, and is responsible for the observed multiple-FSRsideband separation. Interestingly, we find that depend-ing on the relative contributions of the PG and the PPto the gain, the laser can emit either an FM or AM wave-form at the instability threshold.

A. General Framework

We model the lasing transition as a two-level system,or a quantum dipole, subject to the electric field

E(t) = E(t)ei!0t + c.c. (1)

The response is characterized by the population inversionw (positive when inverted) and the o↵-diagonal elementof the density matrix �, which in turn obey the Blochequations (in the rotating wave approximation) [48],

� =

✓i�� 1

T2

◆� +

i

2wE (2)

w = i(E⇤� � E�⇤)� w � weq

T1

. (3)

where � = !ba

� !0

is the detuning between the fieldand the resonant frequency !

ba

of the two-level system,T1

is the gain recovery time, T2

is the dephasing time, ⌘ 2d/~ is the coupling constant where d is the dipolematrix element (assumed to be real) and ~ is Planck’sconstant, and w

eq

is the “equilibrium” population inver-sion that the system would reach in the absence of pho-tons, determined by the pumping. (Note that we havedefined T

1

to be the gain recovery time, which in QCLsis distinct from the upper state lifetime T

up

due to thenature of electron transport in the active region. See thesupplemental for a discussion of this subtlety.) We writethe macroscopic polarization P (dipole moment per vol-ume) as

P (t) = Pei!0t + c.c., (4)

where P = Nd�, and N is the volume density of dipoles.A characteristic of the two-level medium that will ap-

pear often is the “Beer loss rate”

↵ =Nd2T

2

!ba

cpµ/✏

~ , (5)

which is related to the more familiar Beer absorptioncoe�cient ↵ (with units of inverse length) that appearsin Beer’s law of absorption by ↵ = ↵c. We adopt theconvention of [49] and assume our dipoles to be embeddedin a host medium of permittivity ✏ and permeability µ.The speed of light c = 1/

p✏µ also denotes the value in

the background host medium.In the standing-wave cavity, the field envelopes vary in

space in addition to time. We follow a common approach

and decouple the spatial and temporal dependence, writ-ing the field as

E(z, t) =X

m=�,0,+

⌥m

(z)Em

(t)ei!mt + c.c., (6)

where !0

is the primary mode frequency and the sidebandfrequencies are !± = !

0

± �!. These three frequenciesare cold-cavity resonant frequencies, and are equidistantfrom one another because we have assumed zero GVD.We will henceforth assume that the primary mode !

0

lases at the resonant frequency of the two-level system,so � = 0. This is a reasonable approximation if the FSRis much smaller than the gain bandwidth. These twoassumptions, GVD = 0 and � = 0, simplify later mathe-matical formulas considerably and allow for easier under-standing of the essential physics. The full theory withoutthese assumptions is included in the supplemental infor-mation. The spatial modes ⌥

m

(z) are determined by thecavity geometry, and do not vary in time. We assumethe laser cavity to have mirrors with unity reflectivity, sothat the spatial modes are given by

⌥m

(z) =p2 cos(k

m

z) (7)

where km

is an integer multiple of ⇡/L and L is the lengthof the cavity. The mirror loss ln(1/

pR

1

R2

)/L is includedin the total optical losses of the cavity, ¯. The assump-tion of perfect reflectivity simplifies the problem in twoimportant ways: the spatial functions ⌥

m

(z) are orthog-onal, and they do not change shape as the pumping in-creases. This assumption turns out to be quite good evenfor semiconductor lasers with facet reflectivities around0.25. The approximation breaks down for our HR/ARcoated QCL, and here we will only give a qualitative de-scription of what happens and save the considerably morecomplicated theory for a future publication.

B. Population Grating

The threshold inversion is given by the ratio of the op-tical loss rate to the Beer loss rate, w

th

= ¯/↵. We definethe pumping parameter p ⌘ w

eq

/wth

. When p = 1, theprimary mode begins to lase at the frequency !

0

= !ba

.As the pumping p is increased, the field and inversion canbe solved for by the method of [33], which is detailed inthe supplemental information. We account for the pop-ulation grating, but not the coherence grating which hasbeen incorporated in recent work [50]. The primary modeE0

grows according to

|E0

|2 =p� 1

1 + �D

/2, (8)

where we have defined the dimensionless primary modeamplitude E

0

by normalizing by the saturation ampli-tude, E

0

⌘ pT1

T2

E0

. The di↵usion parameter �D

isgiven by �

D

= (1+4k20

DTup

)�1, whereD is the di↵usivity

8

of the excited-state electrons and Tup

is the upper-statelifetime. The parameter �

D

ranges from 0 (for infinitemobility) to 1 (for zero mobility). The population inver-sion in the presence of the primary mode, w

0

(z), varieswith p as

w0

(z) = wth

1 +

�D

2

p� 1

1 + �D

/2� �

D

p� 1

1 + �D

/2cos(2k

0

z)

�.

(9)Equations 8 and 9 are valid to first order in the primarymode intensity |E

0

|2, or equivalently, p � 1 ⌧ 1. Notethat for zero di↵usion (�

D

= 1), the slope e�ciency of thelaser is two thirds that of the infinite di↵usion (�

D

= 0)case. This is because for infinite di↵usion, the inversionis uniformly pinned to w

th

above threshold. For finitedi↵usion, as the pumping increases the population grat-ing grows in amplitude. At the same time, the averagevalue of the inversion increases, indicating that the inver-sion is not being converted into photons as e�ciently as itcould be if the electrons could di↵use from the field nodesto the antinodes. (In principle, one can extract �

D

frommeasurements of the slope of |E

0

|2 vs. p, which should bebetween zero and one. The inset of Fig. 2 shows |E

0

|2 vs.J/J

th

. All curves have a slope greater than one, whichsuggests that J/J

th

is an underestimate of p. See thesupplemental for how |E

0

|2 is determined from the mea-surements, and how the transparency current can causeJ/J

th

to underestimate p. Therefore, more characteriza-tion is needed to extract �

D

from the measurements.)

C. Population Pulsation

To understand the population pulsation, we can ignorethe spatial dependence of the intracavity field and con-sider only a single two-level system subject to an appliedfield

E(t) =X

m=�,0,+

Em

(t)ei!mt + c.c. (10)

Since we are interested in calculating the stability of thesidebands, the amplitudes E± should be thought of as in-finitesimal perturbations; as such, our entire treatmentretains only terms to first order in the sideband ampli-tudes E±. We write the total polarization as

P (t) =X

m=�,0,+

Pm

(t)ei!mt + c.c. (11)

The polarization at the sidebands can be calculated usingEqs. 2 and 3 [51], which gives

P+

=i✏

!ba

↵w0

E+

[1 + i�!T2

]+ ⇤E

0

E⇤0

E+

+ ⇤E0

E0

E⇤�

�

(12)

P� =i✏

!ba

↵w0

E�

[1� i�!T2

]+ ⇤⇤E

0

E⇤0

E� + ⇤⇤E0

E0

E⇤+

�,

(13)

where

⇤ =�(1 + i�!T

2

/2)h(1 + i�!T

1

)(1 + i�!T2

)2 + (1 + i�!T2

)|E0

|2i

(14)and w

0

is the saturated population inversion w0

=w

eq

/(1 + |E0

|2).The polarization at each sideband is neatly divided

into three contributions. Taking P+

as an example, thefirst term in Eq. 12 is the Lorentzian contribution thatthe sideband generates due to the linear susceptibilityof the dipole. The second and third terms are nonlin-ear contributions due to the PP at frequency �!: a self-mixing term of the sideband with the primary mode, anda cross-mixing term of the other sideband with the pri-mary mode. The frequency-dependent portion of thenonlinear susceptibility is ⇤, which is a dimensionlessfunction of the sideband detuning, the time constantsof the two-level system, and the primary mode intensity|E

0

|2. From the field and the induced polarization, wecan calculate the total power density generated, h�EP i.The quantity that most interests us is the gain g (with di-mension of frequency) seen by each sideband, defined asthe power generated at the sideband’s frequency, dividedby the energy density of the exciting sideband field.To develop a feel for the parametrically generated po-

larization and the resulting gain, we consider two instruc-tive cases. In both cases we take the sidebands to haveequal magnitudes, |E

+

| = |E�|, but choose the phases ofthe sidebands to give rise to an AM waveform in one caseand a constant-intensity FM waveform in the other case,as shown in Fig. 1(b). The gain of each sideband is foundto be

g = ↵w0

"1

1 + (�!T2

)2+Real(⇤)|E

0

|2 ·(

2 ; AM

0 ; FM

#.

(15)The first term is the Lorentzian contribution to the gain,and the second term is the parametric gain due to the PP.The factors of two and zero come from the constructive ordestructive addition, respectively, of the self-mixing andcross-mixing terms to the nonlinear polarization. Equiv-alently, one can say that the constant-intensity FM fielddoes not create a PP, and accordingly experiences noparametric gain. The parametric gain of the AM fieldis proportional to Real(⇤) and to the primary mode in-tensity |E

0

|2. (We note that one can quickly derive theoriginal RNGH instability for a traveling-wave laser fromEq. 15, which is done in the supplemental.) By expand-ing ⇤ in Eq. 14 in powers of |E

0

|2, it becomes clear thatthe PP interaction can be expressed in the perturbativeexpansion of traditional nonlinear optics as a third, fifth,seventh, etc. order nonlinearity. We will later calcu-late the instability threshold in the limit of small pri-mary mode intensity, and are therefore interested in thelowest-order nonlinearity. We obtain �(3), the dimen-sionless frequency-dependent portion of the third-order

9

PP nonlinear susceptibility, by evaluating ⇤ at |E0

|2 = 0,

�(3) =�(1 + i�!T

2

/2)

(1 + i�!T1

)(1 + i�!T2

)2. (16)

To better elucidate the nature of the PPs, the magni-tude, phase, and real part of �(3) are plotted in Fig. 6 asa function of the sideband detuning, for three di↵erentvalues of T

1

/T2

. At low frequencies �!, the populationinversion has no di�culty following the modulation ofthe field, which has two consequences: the amplitude ofthe PPs is large, and the PP is ⇡ out of phase with theintensity modulation of the exciting field. This can beunderstood simply in terms of rate equations: when thefield is stronger, the stimulated emission rate is larger,and the population inversion is therefore smaller. Thisscenario–higher inversion when the intensity is lower andlower inversion when the intensity is higher–is less e�-cient at extracting power from the two-level system rel-ative to the case of monochromatic or FM excitation;mathematically, this is described by a parametric gain(determined by the real part of �(3)) that is negative.We refer to this e↵ect as parametric suppression: a low-frequency PP reduces the gain of each sideband. As �!increases, the inversion can no longer as easily follow theintensity modulation, so the amplitude of the PPs de-creases and the phase of �(3) decreases from ⇡. For largeenough �!, the phase of �(3) decreases below ⇡/2, atwhich point Real[�(3)] becomes positive. We refer to thise↵ect as parametric enhancement: a high-frequency PPincreases the gain seen by each sideband. The crossingfrequency �!

cr

which separates the low-frequency sup-pression regime and high-frequency enhancement regimeis given by

�!cr

T2

⇡

s2/3

T1

/T2

, (17)

where we have made the approximation T1

/T2

� 1, validfor QCLs. The regions of parametric suppression andenhancement are highlighted in the plots of the phaseand real part of �(3) in Fig. 6. Finally, at very large�! the parametric gain approaches zero (from above),because the beat note becomes too short for the inversionto follow and the amplitude of the PP approaches zero.

It is worth pointing out that in the weak-field limit|E

0

|2 ⌧ 1 that we are interested in, �!cr

has no relationto the Rabi frequency ⌦

R

induced by the primary mode,

⌦R

T2

=|E

0

|pT1

/T2

. (18)

The Rabi frequency of course varies with the primarymode amplitude, while �!

cr

is independent of E0

in theweak-field limit. By comparing the factors

p2/3 and |E

0

|in the numerators of Eqs. 17 and 18, it’s clear that in thelimit |E

0

|2 ⌧ 1, �!cr

will always be greater than the Rabifrequency. Thus, the reason for the parametric enhance-ment when �! > �!

cr

should simply be ascribed to the

Parametric suppression

Parametric enhancement

Parametric enhancement

Parametric suppression

FIG. 6. The magnitude, phase, and real part of �(3) areplotted vs. �!T2 for three di↵erent ratios T1/T2 = 40, 10, 5.The parametric gain seen by the sidebands is determined byReal[�(3)]. Low-frequency PPs lead to a parametric suppres-sion of the gain. For the gain to be parametrically enhanced,�! must be large enough that the inversion can no longer fol-low the intensity in anti-phase; in other words, the phase of�(3) must be between �⇡/2 and ⇡/2.

10

fact that at high PP frequency, the phase lag betweenthe population inversion and the field intensity becomesappropriate for gain rather than absorption.

D. Instability Threshold

Now we can ask the question: what happens to thesingle-mode laser solution when it is perturbed by a weaksideband field? The source of the perturbation could bespontaneous emission, or even spontaneous parametricdownconversion of two primary mode photons into twosideband photons [52]. Our goal is to calculate the gainof the sideband modes averaged over the length of thecavity. The instability threshold is reached when thesideband gain equals the loss. Although our instabilityanalysis will not tell us about the steady-state reachedby the sidebands, one reasonable possibility is that thesidebands begin to lase, as seen in the experimental spec-tra.

To determine the sideband gain, we start with thepolarization in Eqs. 12-13 and account for the position-dependence by replacing E

m

(t) with Em

(t)⌥m

(z), and w0

with w0

(z) from Eq. 9. In keeping with our approxima-tion to order |E

0

|2, we replace ⇤ with �(3). The position-dependent polarization is then inserted as the sourceterm in Maxwell’s wave equation. From here, the cal-culation follows the same steps as the instability analysisdone for Kerr microresonators [53], and is detailed in thesupplemental information. After making the slowly vary-ing envelope approximation, and projecting the equationonto each of the orthonormal spatial modes, one finds afirst order di↵erential equation for each sideband ampli-tude. Unlike the earlier example where we hand-pickedthe phases of the sidebands to study the e↵ect of an AMand FM field, here the AM and FM sideband configu-rations emerge organically as the two “natural modes”of the system of two sideband equations. The naturalmodes [14] are the configurations of the three-wave fieldfor which the relative phases of the fields are preserved astime evolves; in other words, an AM field remains AM,and an FM field remains FM. (In the general case ofnonzero � and GVD, the natural modes can be a super-position of AM and FM.) The gain of the AM and FMnatural modes is given by

g¯ =

1 + �D

2

p�1

1+�D/2

1 + (�!T2

)2

+Real[�(3)]p� 1

1 + �D

/2·(

�self

+ �cross

= 3

2

; AM

�self

� �cross

= 1

2

; FM

(19)

where the �s are longitudinal spatial overlap factors

�self

=1

L

ZL

0

dz |⌥0

(z)|2|⌥±(z)|2 = 1 (20)

�cross

=1

L

ZL

0

dz ⌥0

(z)2⌥⇤�(z)⌥

⇤+

(z) = 1/2. (21)

By comparing the standing-wave sideband gain in Eq.19 to the sideband gain of a single two-level system inEq. 15, we see that the cavity introduces two modifica-tions. First, the Lorentzian gain contribution increaseswith p; this unclamped gain is a direct result of the PGthat develops in the presence of non-zero �

D

. Secondly,the partial overlap of the sideband spatial modes ⌥

+

and⌥� results in partial (rather than complete) interferenceof the self-mixing and cross-mixing contributions to thegain. To understand this, note from Fig. 1(a) that al-though the emitted waveform has equal-amplitude side-bands, within the cavity the plus and minus sidebandshave unequal amplitudes at most positions z, as shownby the red and blue modes in Fig. 1(d). Therefore, theself and cross-mixing contributions to the sideband po-larization at each position z cannot completely interfere,and the factors of 3/2 (AM) and 1/2 (FM) emerge afteraveraging over the full cavity length, as opposed to thefactors of 2 and 0 in Eq. 15. Thus, even when the laseremits an FM waveform, there is still a parametric con-tribution to the gain due to the incomplete destructiveinterference of the PP within the cavity.The instability occurs when p reaches a value such that

the sideband gain g in Eq. 19 equals the loss ¯for one par-ticular sideband detuning �!. (As discussed previously,we assume the FSR is small so that a FP mode alwaysexists very close to the unstable value of �!.) As p in-creases, the incoherent Lorentzian gain increases, but theparametric gain either increases or becomes more nega-tive depending on the sign of Real[�(3)] (which dependson �!). The three parameters T

1

, T2

, and �D

a↵ect therelative importance of the incoherent and coherent gainterms, and depending on the values of these parameters,one of three di↵erent classes of instability can occur: theincoherent instability, FM instability, and AM instabilty.In Fig. 7, each type of instability is illustrated by plottingthe sideband gain at the instability threshold, which wenow explain.

1. Incoherent Instability

The parametric gain can often be ignored. If T1

is largeenough, the interesting features of Real[�(3)] all occur forsideband detunings less than 1 FSR, and so the paramet-ric gain will be nearly zero for all values of �! greater than1 FSR. This is the case for diode lasers, where PPs aresignificant up to a few GHz (T

1

⇡ 1 ns), while the FSRis around 100GHz. Thus, only the incoherent gain termin Eq. 19 matters (although it is not a Lorentzian forbandgap lasers). As p increases beyond 1, the sideband

11

(a) Incoherent Instability

(b) FM Instability (strong grating)

(c) AM Instability (weak grating)

Incoherent Gain Threshold Gain

Total Gain (FM) Total Gain (AM)

Fabry-Perot Modes

- Very low instability threshold - Adjacent FP-mode sidebands

- Low instability threshold - Large sideband separation

- High instability threshold - Largest sideband separation

= 0.5 p = 1.06

= 0.04 p = 1.9

= 0.5 p = 1.001

FIG. 7. Overview of the three di↵erent types of instabilities.(a) The incoherent instability relies only on the unclampedgain due to the PG, and occurs when parametric e↵ects canbe neglected. (b) The FM instability occurs when the gaindue to a strong PG, despite the parametric suppression of thegain of small-detuned sidebands, raises the sidebands abovethe threshold. (c) The AM instability occurs when the gaindue to a weak PG, together with the parametric enhancementof the gain of large-detuned sidebands, raises the sidebandsabove the threshold. In (b) and (c), the value T1/T2 = 20was used. The FP modes (green) are not associated with theordinate, and simply provide a sense of the mode spacing.

gain increases but remains Lorentzian, so the sidebandsthat reach the instability threshold first will always bethe FP modes immediately adjacent to the primary las-ing mode [54]. In diode lasers, �

D

is small (⇠ 10�4), sop needs to be large before the second mode can appear.The value of �

D

= 0.5 in Fig. 7(a) is typical of short-wave QCLs. We see that if coherent e↵ects were negligi-ble in QCLs, we would expect the sidebands to appear atp = 1.001, barely above threshold. The much higher in-stability threshold measured in the experiments, togetherwith the observation that the sidebands do not appear atthe nearest-neighbor FP modes of the primary mode, in-dicates that coherent e↵ects play an essential role in theQCL instability.

2. FM Instability

When the Lorentzian gain increases quickly with p dueto a strong PG, sidebands that fall within the paramet-ric suppression regime, �! < �!

cr

, can reach the insta-bility threshold. This is counterintuitive: why should asideband lase when the parametric interaction providesnegative gain? The answer is that the Lorentzian gainfavors sidebands with as small a separation as possible,and if it is large enough it can pull sidebands abovethreshold in spite of the negative contribution from theparametric gain. In this scenario, FM sidebands have alower instability threshold than AM sidebands becausethe parametric contribution to the gain is less negative,since 1/2 < 3/2 in Eq. 19. Such a case is illustrated inFig. 7(b) for �

D

= 0.5 and T1

/T2

= 20. At p=1.06, FMsidebands reach the instability threshold, while AM side-bands are too strongly suppressed to reach the instability.A key feature of the instability is that the unstable side-band will be several FSR away from the primary mode(provided that the FSR is small), while still satisfying�! < �!

cr

.

3. AM Instability

When the Lorentzian gain increases little with p due toa weak PG, only sidebands that fall within the paramet-ric enhancement regime, �! > �!

cr

, will be able to reachthe instability. In this case, AM will have a lower insta-bility threshold than FM because AM receives a largerparametric enhancement (since 3/2 > 1/2 in Eq. 19).Such a case is illustrated in Fig. 7(c) for �

D

= 0.04 andT1

/T2

= 20. At p = 1.9, AM sidebands reach the in-stability threshold while the FM sidebands are not su�-ciently enhanced to reach the instability. Strictly speak-ing, p=1.9 falls outside the region of validity of our per-turbative treatment (p � 1 ⌧ 1), so the specific valuesin this plot are not exactly accurate, but the qualita-tive features are correct. The unstable sidebands satisfy�! > �!

cr

, and so their separation will be even greaterthan for the FM instability. The original RNGH insta-

12

bility is precisely this AM instability, in a traveling-wavelaser. For traveling waves, the Lorentzian gain is clampedat threshold regardless of the di↵usion parameter, so theinstability can only be reached by the parametric en-hancement of AM sidebands.

To access both the FM and AM instability regimes ex-perimentally, we need to tune the strength of the PG.The electron di↵usivity can be reduced by lowering thetemperature, and indeed temperature has a strong e↵ecton the emission spectra of QCLs [33], although the ef-fect is not yet well-understood. In this work, we chooseto manipulate the PG by adjusting the facet reflectiv-ities. Increasing the disparity of the reflectivities ofthe two mirrors reduces the contrast of the standing-wave, because the wave traveling from the higher to thelower-reflectivity facet becomes larger than the counter-propagating wave [55]. For a su�ciently large disparity,the incoherent gain contribution is small enough that thelaser can only undergo the AM instability. In a practicalsense, engineering the facet coatings allows one to trans-form a standing-wave cavity into more of a traveling-wavecavity. It is for this reason that we chose to study anHR/AR coated laser, where the AR coating has as low areflectivity as current technology allows, to maximize thecavity asymmetry. The full mathematical treatment ofmirrors with non-unity reflectivity is complicated by thefact that the spatial modes ⌥

m

(z) are no longer orthog-onal, and also that that the ⌥

m

(z) and the longitudinaloverlap factors � vary with the pumping. This theorywill be presented in future work.

IV. DISCUSSION

In order for a mode to oscillate, it must satisfy twoconditions: 1) the roundtrip gain must equal the loss,and 2) the roundtrip phase must equal a multiple of 2⇡.Our theory in Sec. III has treated only the gain condi-tion. The same approach was taken in the descriptionof the original RNGH instability [10–14]; the underlyingassumption is that the cavity modes are densely spaced,so that a pair of sidebands that satisfies the instabilitycondition for the gain will always be “close enough” totwo cavity modes that satisfy the phase condition. How-ever, the experimental and theoretical developments ofthe last decade concerning optical parametric oscillationin externally pumped microresonators have shown thatthe phase condition has a large e↵ect on the oscillationthreshold and sideband spacing [53]. In microresonatorexperiments, the detuning between the external pumpfrequency and the center frequency of the cold cavitymode is a degree of of freedom that must be precisely con-trolled to achieve the lowest possible instability thresh-old. In a laser this detuning is not a degree of freedom,but it can vary with the pumping and should be prop-erly accounted for. A parameter that has no analogyin microresonators is �, the detuning between the las-ing mode !

0

and the two-level resonance !ba

, which also

varies with the pumping and is di�cult to control in ex-periments. To precisely predict the instability thresholdwould require knowledge of both of these detunings, aswell as the GVD.

At this stage, the simplest and most important appli-cation of the theory is to help determine whether theobserved sidebands are parametrically enhanced or sup-pressed. Because the theory assumes end mirrors withunity reflectivity, we can only expect Eq. 19 to applyreasonably well to the uncoated QCLs. For each device,�D

is calculated using the theoretical value of Tup

(calcu-lated from the bandstructure) and the di↵usion constantD = 77 cm2/s [47], giving �

D

= 0.4 (DS-3.8), 0.49 (TL-4.6), and 0.93 (LL-9.8). For these large values of �

D

thePG is strong, and we find from numerically solving Eq.19 that the FM instability has a lower threshold thanthe AM instability, regardless of the value of T

1

. In thesupplemental information, we show that the theory pre-dicts sideband spacings �!

sb

that are consistent with theexperimental observations, but underestimates the insta-bility threshold p

sb

. We attribute this discrepancy to theaforementioned detunings and GVD that our theory ne-glects.

A more direct method to discriminate between theparametric suppression and enhancement regimes is tocompare the observed sideband spacing �!

sb

to the cross-ing frequency �!

cr

. If �!sb

< �!cr

, the sidebands areparametrically suppressed and therefore the FM insta-bility has the lower threshold. Thus, we reason that thesidebands must be FM because the AM state would bean unstable equilibrium; an AM waveform, if slightly per-turbed, would evolve to an FM waveform. Similarly, if�!

sb

> �!cr

, the sidebands are parametrically enhanced,so FM sidebands would be an unstable equilibrium andwe conclude that they must be AM. Notably, this rea-soning depends only on the behavior of �(3) as a func-tion of �!; it is therefore independent of GVD and canbe applied to both the uncoated and HR/AR lasers. Tocalculate �!

cr

from Eq. 17, we use our measured valuesof T

2

but still need an estimate for the gain recoverytime T

1

. Pump-probe experiments [56, 57] and theory[58] have shown that T

1

around 2 ps. From Eq. 17, wesee that �!

cr

decreases with increasing T1

, so if we takeT1

= 3ps as a generous upper bound on the gain recov-ery time, we establish a lower bound of �!

cr

at 2270GHz(DS-3.8), 1730GHz (TL-4.6), and 1660GHz (LL-9.8).The measured values of �!

sb

for each uncoated laser–977GHz (DS-3.8), 1259GHz (TL-4.6), and 642GHz (LL-9.8)–are all substantially smaller than the lower boundon �!

cr

. This is consistent with the prediction that theuncoated lasers have a lower FM instability thresholdthan AM threshold, and with these two results we arereasonably confident that the uncoated lasers emit para-metrically suppressed FM sidebands. In stark contrast,TL-4.6:HR/AR exhibits a large sideband separation of�!

sb

= 2216GHz. If we use the accepted value of T1

equal to 2 ps, we find �!cr

= 2120GHz. The observedsideband spacing is slightly larger than �!

cr

, suggesting

13

the enhancement regime. While a smaller gain recov-ery time or non-perturbative calculation would raise �!

cr

slightly, this is our first hint that TL-4.6:HR/AR emitsparametrically enhanced AM sidebands.

The di↵erence in the range of intracavity power overwhich the harmonic state persists in the uncoated vs.coated lasers, as shown in Fig. 2, is additional evidencethat the uncoated devices operate in the suppressionregime and the HR/AR device operates in the enhance-ment regime. Here we propose a qualitative explanationof this feature. Consider a laser operating in the suppres-sion regime. While the FM state is more stable than theAM state in this regime because it minimizes the amountof gain suppression, it would be even more favorable forthe laser to emit a temporally incoherent state, in a sense“washing out” the PPs (in much the same way that a spa-tially incoherent state washes out the PG), thus avoidingany suppression of the gain altogether. We surmise thatthe observed dense state is precisely such a temporallyincoherent state; by lasing on an incoherent superposi-tion of adjacent cavity modes, the laser maximizes theamount of incoherent gain that it extracts while avoidingthe large parametric suppression that would a✏ict a co-herent state with such a small one-FSR spacing. (Fromthe measurement of negative di↵erential resistance in DS-3.8 shown in the supplemental, we know that the densestate indeed extracts more gain than the harmonic state.)This could explain why the uncoated lasers only exhibitthe harmonic state over a small range of current: thelaser soon finds a way to transition from the parametri-cally suppressed FM state to the favored incoherent state.The fact that TL-4.6:HR/AR exhibits the harmonic stateover a large current range suggests that the harmonicstate is more stable than the incoherent state, which canonly be true in the parametric enhancement regime. Ata su�ciently high current, when the spectral span of theharmonic state approaches the gain bandwidth, the inco-herent dense state finally becomes favored for its abilityto lase on adjacent modes, despite no longer benefittingfrom the parametric enhancement.

We have argued that the dense state is a temporallyincoherent state that manages to avoid parametric gainsuppression. If this is the case, why do the uncoatedQCLs choose to emit a harmonic state at all, and not sim-ply jump from the single-mode state to the dense state asthe current is increased? In fact, the spectral hysteresisshown in Fig. 4 for LL-9.8 (and in the supplemental forthe other uncoated devices) proves that the dense state isthe favored lasing state down to barely above threshold.However, this state can only be reached by decreasingthe current after the laser has already entered the densestate at high current. When the laser starts in a single-mode state and the current is increased, there is clearlya barrier that prevents the transition to the dense state.In general, introducing noise allows a system to overcomeenergy barriers and explore a larger volume of its statespace. It is likely that delayed optical feedback servesas such a noise source, and explains why it is di�cult to

observe the harmonic state when optical feedback is noteliminated.

V. CONCLUSION

We have experimentally identified the single-mode in-stability of QCLs, which is characterized by the appear-ance of sidebands at FP modes not adjacent to the pri-mary lasing mode. We have seen the behavior in QCLsat three di↵erent wavelengths, each based on a di↵erentactive region design, and with both positive and negativeGVD. Therefore, the phenomenon is a general feature ofthe electron-light dynamics of QCLs. The instability isreached due to the combined contributions of an inco-herent gain due to the spatial population grating, and acoherent parametric gain due to the temporal populationpulsations. Our theory predicts both an FM instabilityin situations where the incoherent gain contribution islarge, and an AM instability when the incoherent gaincontribution is small. To explore the second possibility,we coated the QCL facets with an HR and an AR coatingto reduce the incoherent gain contribution; indeed, thismodification substantially increases the sideband spac-ing, and it is likely that the waveform is AM. Followingthe first appearance of sidebands at the instability thresh-old, our measurements show that increasing the pumpinggenerates more sidebands which preserve the initial spac-ing. This suggests that a cw QCL can self-start into aphase-locked frequency comb, and must be investigatedfurther. We have also placed our observations and the-ory within historical context, explaining the relation tooptically pumped microresonators and the single-modeinstability in traveling-wave lasers.The future direction of this work is clear. At first,

we can take guidance from the well-established under-standing of microresonators and exploit their similaritywith QCLs to further our understanding. The calcula-tion of the instability threshold will be extended to ac-count for GVD, so that the cold-cavity modes are notnecessarily equidistant. We must also better understandthe nature of the single-mode solution; specifically, howdoes its detuning from the resonant frequency !

ba

, andalso its detuning from the cold-cavity mode that it occu-pies, a↵ect the nature of the instability threshold? Wemust account quantitatively for the non-unity facet re-flections and the precise shape of the mode profile withinthe cavity. Experimentally, second-order autocorrelationexperiments are needed to establish the temporal natureof these short-period waveforms.

ACKNOWLEDGEMENTS

We acknowledge support from the National Sci-ence Foundation under awards ECCS-1230477, ECCS-1614631, and ECCS-1614531. We gratefully acknowledgethe O�ce of Naval Research (ONR) for assistance in de-

14

veloping the 3.85 µm QCL used in this study. This workwas performed in part at the Center for Nanoscale Sys-tems (CNS), a member of the National NanotechnologyCoordinated Infrastructure (NNCI), which is supportedby the National Science Foundation under NSF award

no. 1541959. CNS is part of Harvard University. BS wassupported by the Austrian Science Fund (FWF) withinthe doctoral school Solids4Fun (W1243) and the projectNanoPlas (P28914-N27). TSM is grateful to Jacob Khur-gin for the remark that initiated this research, and toRobert Amaral for the co↵ee that kept it going.

[1] T. J. Kippenberg, S. M. Spillane, and K. J. Vahala,Phys. Rev. Lett. 93, 083904 (2004).

[2] A. B. Matsko, A. A. Savchenkov, D. Strekalov, V. S.Ilchenko, and L. Maleki, Phys. Rev. A 71, 033804 (2005).

[3] P. Del’Haye, T. Herr, E. Gavartin, M. L. Gorodetsky,R. Holzwarth, and T. J. Kippenberg, Phys. Rev. Lett.107, 063901 (2011).

[4] T. Herr, K. Hartinger, J. Riemensberger, C. Y. Wang,E. Gavartin, R. Holzwarth, M. L. Gorodetsky, and T. J.Kippenberg, Nat. Photonics 6, 480 (2012).

[5] T. Herr, V. Brasch, J. D. Jost, C. Y. Wang, N. M. Kon-dratiev, M. L. Gorodetsky, and T. J. Kippenberg, Nat.Photonics 8, 145 (2013).

[6] M. H. P. Pfei↵er, A. Kordts, V. Brasch, M. Zervas,M. Geiselmann, J. D. Jost, and T. J. Kippenberg, Optica3, 20 (2016).

[7] W. E. Lamb, Phys. Rev. 134, A1429 (1964).[8] H. Haken, Zeitschrift fur Phys. 190, 327 (1966).[9] H. Risken, C. Schmid, and W. Weidlich, Zeitschrift fur

Phys. 194, 337 (1966).[10] H. Risken and K. Nummedal, J. Appl. Phys. 39, 4662

(1968).[11] R. Graham and H. Haken, Zeitschrift fur Phys. 213, 420

(1968).[12] S. Hendow and M. Sargent, Opt. Commun. 40, 385

(1982).[13] S. T. Hendow and M. I. Sargent, Opt. Commun. 43, 59

(1982).[14] L. W. Hillman, R. W. Boyd, and C. R. Stroud, Opt.

Lett. 7, 426 (1982).[15] L. A. Lugiato, L. M. Narducci, D. K. Bandy, and N. B.

Abraham, Opt. Commun. 46, 115 (1983).[16] L. W. Hillman, J. Krasinski, R. W. Boyd, and C. R.

Stroud, Phys. Rev. Lett. 52, 1605 (1984).[17] L. A. Lugiato, L. M. Narducci, E. V. Eschenazi, D. K.

Bandy, and N. B. Abraham, Phys. Rev. A 32, 1563(1985).

[18] L. W. Hillman, J. Krasinski, K. Koch, and C. R. Stroud,J. Opt. Soc. Am. B 2, 211 (1985).

[19] R. S. Gioggia and N. B. Abraham, Phys. Rev. Lett. 51,650 (1983).

[20] S. Autler and C. Townes, Phys. Rev. 100, 703 (1955).[21] B. Senitzky, G. Gould, and S. Cutler, Phys. Rev. 130,

1460 (1963).[22] B. Senitzky, IEEE Trans. Microw. Theory Tech. MTT-

16, 728 (1968).[23] M. C. Newstein, Phys. Rev. 167, 89 (1968).[24] B. Mollow, Phys. Rev. 188, 1969 (1969).[25] C. R. Stroud, Phys. Rev. A 3, 1044 (1971).[26] F. Schuda, C. R. Stroud, and M. Hercher, J. Phys. B

At. Mol. Phys. 7, L198 (1974).[27] F. Y. Wu, R. E. Grove, and S. Ezekiel, Phys. Rev. Lett.

35, 1426 (1975).

[28] H. Carmichael and D. Walls, J. Phys. B At. Mol. Phys.9, 1199 (1976).

[29] C. Cohen-Tannoudji and S. Reynaud, J. Phys. B At. Mol.Phys. 10, 345 (1977).

[30] R. W. Boyd, M. G. Raymer, P. Narum, and D. J. Harter,Phys. Rev. A 24, 411 (1981).

[31] D. J. Harter and R. W. Boyd, IEEE J. Quantum Elec-tron. 16, 1126 (1980).

[32] L. A. Lugiato, F. Prati, D. K. Bandy, L. M. Narducci,P. Ru, and J. Tredicce, Opt. Commun. 64, 167 (1987).

[33] A. Gordon, C. Y. Wang, L. Diehl, F. X. Kartner,A. Belyanin, D. Bour, S. Corzine, G. Hofler, H. C. Liu,H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Ca-passo, Phys. Rev. A 77, 053804 (2008).

[34] R. L. Tober, J. D. Bruno, S. Suchalkin, and G. Belenky,J. Opt. Soc. Am. B 31, 2399 (2014).

[35] Q. Y. Lu, M. Razeghi, S. Slivken, N. Bandyopadhyay,Y. Bai, W. J. Zhou, M. Chen, D. Heydari, A. Haddadi,R. McClintock, M. Amanti, and C. Sirtori, Appl. Phys.Lett. 106, 051105 (2015).

[36] M. C. Soriano, J. Garcıa-Ojalvo, C. R. Mirasso, andI. Fischer, Rev. Mod. Phys. 85, 421 (2013).

[37] A. Hugi, G. Villares, S. Blaser, H. C. Liu, and J. Faist,Nature 492, 229 (2012).

[38] G. Villares, A. Hugi, S. Blaser, and J. Faist, Nat. Com-mun. 5, 5192 (2014).

[39] G. Villares, S. Riedi, J. Wolf, D. Kazakov, M. J. Suess,M. Beck, and J. Faist, Optica 3, 252 (2016).

[40] J. B. Khurgin, Y. Dikmelik, a. Hugi, and J. Faist, Appl.Phys. Lett. 104, 081118 (2014).

[41] G. Villares and J. Faist, Opt. Express 23, 1651 (2015).[42] A. Schliesser, N. Picque, and T. W. Hansch, Nat. Pho-

tonics 6, 440 (2012).[43] D. Hofstetter, M. Beck, T. Aellen, J. Faist, U. Oesterle,

M. Ilegems, E. Gini, and H. Melchior, Appl. Phys. Lett.78, 1964 (2001).

[44] J. Faist, D. Hofstetter, M. Beck, T. Aellen, M. Rochat,and S. Blaser, IEEE J. Quantum Electron. 38, 533(2002).

[45] F. Xie, C. Caneau, H. P. Leblanc, N. J. Visovsky, S. C.Chaparala, O. D. Deichmann, L. C. Hughes, C. Zah,D. P. Ca↵ey, and T. Day, IEEE J. Sel. Top. QuantumElectron. 17, 1445 (2011).

[46] D. Hofstetter and J. Faist, IEEE Photonics Technol. Lett.11, 1372 (1999).

[47] J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford Uni-versity Press, Oxford, 2013).

[48] L. Allen and J. Eberly, Optical Resonance and Two-LevelAtoms (Dover Publications, New York, 1987).

[49] A. Siegman, Lasers (University Science Books, Sausalito,1986) pp. 473–89.

[50] N. Vukovic, J. Radovanovic, V. Milanovic, and D. L.Boiko, (2016), arXiv:1601.03212.

15

[51] R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press,San Diego, 2003).

[52] D. A. Kleinman, Phys. Rev. 174, 1027 (1968).[53] Y. K. Chembo, D. V. Strekalov, and N. Yu, Phys. Rev.

Lett. 104, 103902 (2010).[54] N. J. Holonyak, R. M. Kolbas, R. D. Dupuis, and P. D.

Dapkus, IEEE J. Quantum Electron. QE-16, 170 (1980).

[55] T. S. Mansuripur, G. M. De Naurois, A. Belyanin, andF. Capasso, Optica 2, 48 (2015).

[56] H. Choi, L. Diehl, Z. K. Wu, M. Giovannini, J. Faist,F. Capasso, and T. B. Norris, Phys. Rev. Lett. 100,167401 (2008).

[57] H. Choi, L. Diehl, Z. K. Wu, M. Giovannini, J. Faist,F. Capasso, and T. B. Norris, IEEE J. Quantum Elec-tron. 45, 307 (2009).

[58] M. A. Talukder, J. Appl. Phys. 109, 033104 (2011).

The quantum cascade laser as a self-pumped parametric oscillator

(Supplemental Information)

Tobias S. Mansuripur,1 Camille Vernet,2, 3 Paul Chevalier,2 Guillaume

Aoust,2, 4 Benedikt Schwarz,2, 5 Feng Xie,6 Catherine Caneau,7 Kevin Lascola,6

Chung-en Zah,6 David P. Ca↵ey,8 Timothy Day,8 Leo J. Missaggia,9 Michael

K. Connors,9 Christine A. Wang,9 Alexey Belyanin,10 and Federico Capasso2, ⇤

1Department of Physics, Harvard University, Cambridge, MA 02138 USA

2John A. Paulson School of Engineering and Applied Sciences,

Harvard University, Cambridge, MA 02138 USA

3Ecole Polytechnique, 91123 Palaiseau, France

4ONERA, The French Aerospace Lab, 91123 Palaiseau, France

5Institute of Solid State Electronics, TU Wien, 1040 Vienna, Austria

6Thorlabs Quantum Electronics (TQE), Jessup, MD 20794 USA

7Corning, Inc., Corning, NY 14831 USA

8Daylight Solutions, Inc., San Diego, CA 92128 USA

9Massachusetts Institute of Technology,

Lincoln Laboratory, Lexington, MA 02420 USA

10Department of Physics and Astronomy,

Texas A & M University, College Station, TX 77843 USA

⇤ capasso@seas.harvard.edu

1

I. COMMENTS ON POWER VS. CURRENT CURVES

In the single-mode regime, the intracavity intensity of the single-mode determines the

strength of the parametric interaction with the sidebands. Therefore, we would like to

calculate the intracavity intensity from the measured output power. We remain true to the

distributed loss approximation, for which the output power is given by

Pout =↵mhE2iLwhp

µ/✏(1)

where ↵m = ln[1/(R1

R2

)]/(2L), the length, width, and height of the cavity are L, w, and

h, and the time-average intensity of the single-mode is hE2i = 2|E0

|2. We are assuming a

uniform field intensity in the transverse dimensions, and therefore not worrying about the

transverse overlap factor. We can rearrange this equation for the intracavity intensity

|E0

|2 ⌘ 2T1

T2

|E0

|2 =2d2T

1

T2

pµ0

/✏0

~2ne↵

↵mLwhPout. (2)

With this equation, we can convert the measured total output power of each laser into the

intracavity intensity, using our measured values of the refractive index ne↵

and the dephasing

time T2

, our best estimates for d and T1

, and in the case of the HR/AR laser we have used

R1

= 1, R2

= 0.01. The result is plotted in Fig. S1 as a function of J/Jth

, which is the same

as the inset in Fig. 1 of the main text.

The theoretical formula for the intracavity intensity is

|E0

|2 = p� 1

1 + �D/2, (3)

where p ⌘ weq/wth is the pump parameter. We emphasize that p is not the same as J/Jth

.

The slope of |E0

|2 vs. p is always between 2/3 and 1, depending on the di↵usion parameter

�D. The reference line in Fig. S1 is drawn with a slope of one to indicate that each of the

|E0

|2 vs. J/Jth

curves has a slope greater than one. Therefore, we conclude that J/Jth

must

underestimate p. One factor that contributes to this underestimation is the transparency

current Jtrans

: a fixed amount of current that must be delivered to the active region simply

to raise the inversion from a negative number to zero. To understand this simply, suppose

that the equilibrium inversion scales like weq / J �Jtrans

, and that Jtrans

remains a constant

number at threshold and above. Then the pump parameter p ⌘ weq/wth is expressed in

terms of J as

p =J � J

trans

Jth

� Jtrans

. (4)

2

DS-3.8

LL-9.8

TL-4.6:

HR/AR

TL-4

.6

slope=1

single-mode harmonic dense

FIG. S1. The measured output power of each QCL (from both facets) is converted to the intracavity

intensity |E0

|2, and plotted against J/Jth

. Each curve is color-coded to indicate the range over which

the laser operates in a single-mode, harmonic state, or dense state. Note that the quantity |E0

|2

is only meaningful in the single-mode regime, because we are interested in the intensity of the

single-mode before the harmonic regime sets in. A line with a slope of one is drawn as a reference.

For example, suppose that for a laser with Jth

= 500 mA the harmonic state kicks in at

550 mA, or J/Jth

= 1.1. If the transparency current was Jtrans

= 250 mA, (in other words,

half of the threshold current, which is reasonable for QCLs), then the pump parameter

at the harmonic state onset would be p = (550 � 250)/(500 � 250) = 1.2. Thus, J/Jth

underestimates p.

A more rigorous study is required to determine Jtrans

for each laser, which can be done by

measuring many lasers of the same active region but di↵erent lengths. Once Jtrans

is known,

the slope of |E0

|2 vs. p should fall between 2/3 and 1 and in principle a value for �D can be

extracted, allowing one to quantify the amount of di↵usion present.

II. HYSTERESIS OF IV CURVE

The IV curve of device DS-3.8 shown in Fig. S2 demonstrates a hysteresis. When starting

below threshold and increasing the current (red), the voltage of the laser decreases (negative

di↵erential resistance) when the noisy harmonic state transitions to the dense state at 523

mA. (In the spectra shown in Fig. 3(c) of the manuscript, this transition occurs at 502 mA.

3

Current (A)

Volta

ge (V

) increasing current

decreasing current

noisy harmonic state becomes dense state

dense state becomes noisy harmonic state

4.8 mV

FIG. S2. The IV curve of DS-3.8 exhibits a hysteresis as the current is increased (red) and decreased

(blue). The hysteresis is correlated with the transition from the noisy harmonic state to the dense

state.

The exact current at which the transition occurs is not identical from one experiment to

another, but occurs predictably within a range of about 20 mA.) Once the laser reaches the

dense state and the current is subsequently decreased (blue), the laser persist in the dense

state at currents below 523 mA. This is responsible for the hysteresis loop. For the same

current, the voltage is 4.8 mV smaller when in the dense state than in the noisy harmonic

state. At 494 mA, the dense state transitions to the noisy harmonic state, and the two

voltage curves overlap again.

The lower voltage of the dense state indicates a larger radiative photocurrent, which

implies that the output power of the laser is slightly larger in the dense state than in the

harmonic state, for the same current pumping. (While we could have measured the output

power to demonstrate this, the IV measurement is more sensitive.) Thus, the dense state is

more e�cient at extracting photons from upper state electrons and is, in one sense, more

stable. Therefore, once the laser enters the dense state, it likes to remain there even as the

current is decreased.

4

990 mA

1000 mA

1100 mA

1500 mA

1140 mA

1180 mA

1240 mA

1280 mA

1360 mA

1440 mA

1040 mA

J/Jth = 1.01

1.02

1.12

1.53

1.16

1.20

1.27

1.31

1.39

1.47

1.06

(d) LL-9.8: decreasing current

439 mA

440 mA

485 mA

800 mA

530 mA

535 mA

580 mA

620 mA

680 mA

740 mA

480 mA

J/Jth = 1.01

1.01

1.11

1.84

1.22

1.23

1.33

1.43

1.56

1.70

1.10

J/Jth = 1.05

1.06

1.10

2.04

1.12

1.14

1.19

1.28

1.38

1.89

1.07

381 mA

385 mA

399 mA

741 mA

406 mA

416 mA

434 mA

466 mA

501 mA

686 mA

391 mA

983 mA

1115 mA

1140 mA

1500 mA

1180 mA

1280 mA

1360 mA

1380 mA

1414 mA

1440 mA

1116 mA

J/Jth = 1.00

1.14

1.16

1.53

1.20

1.31

1.39

1.41

1.44

1.47

1.14 7 FSR

435 mA

509 mA

520 mA

800 mA

545 mA

560 mA

565 mA

580 mA

585 mA

700 mA

510 mA

J/Jth = 1.00

1.17

1.20

1.83

1.25

1.29

1.30

1.33

1.34

1.61

1.17

380 mA

407 mA

416 mA

741 mA

424 mA

470 mA

480 mA (transient)

480 mA

500 mA

502 mA

408 mA

J/Jth = 1.04

1.12

1.14

2.04

1.16

1.29

1.32

1.32

1.37

1.38

1.12

26 FSR

20 FSR

25 FSR

29 FSR

(a) LL-9.8: increasing current

(b) TL-4.6: increasing current

(c) DS-3.8: increasing current

(e) TL-4.6: decreasing current

(f) DS-3.8: decreasing current

FIG. S3. The spectra of the three uncoated devices as the current is ramped up and as it is ramped

down. Of the six plots, only (e) and (f) were not included in the main text.

5

735 mA

875mA

885 mA

1220 mA

890 mA

935 mA

960 mA

1040 mA

1070 mA

1100 mA

880 mA

J/Jth = 1.01

1.20

1.22

1.68

1.22

1.29

1.32

1.43

1.47

1.51

1.21 46 FSR

(b) TL-4.6:HR/AR: decreasing current

735 mA

875 mA

885 mA

1220 mA

900 mA

950 mA

1000 mA

1050 mA

1160 mA

1165 mA

880 mA

J/Jth = 1.01

1.20

1.22

1.68

1.24

1.31

1.38

1.44

1.60

1.60

1.21

46 FSR

(a) TL-4.6:HR/AR: increasing current

FIG. S4. The spectra of TL-4.6:HR/AR as the current is (a) ramped up and (b) ramped down.

Only plot (b) was not presented in the main text.

III. ADDITIONAL DATA ON SPECTRAL HYSTERESIS

In Fig. S3, the spectra of the three uncoated devices are plotted as the current is increased

starting from below threshold, and also as the current is decreased. For devices LL-9.8 and

TL-4.6, once the dense state is reached it persists as the current is decreased until the single-

mode state reappears. Device DS-3.8 also exhibits noisy harmonic states as the current is

decreased. For all three devices, it is true that the clean harmonic state never reappears

once the laser has reached the dense state.

In Fig. S4, the spectra of TL-4.6:HR/AR are plotted as the current is increased and then

decreased. As the current is decreased, the dense state persists until a noisy harmonic state

appears at 1.29Jth

. Interestingly, in this device–unlike the uncoated ones–the clean harmonic

state with one pair of sidebands reappears at 885 mA, which is quite close to the instability

threshold of 880 mA found when the current is ramped upwards. The reappearance of the

harmonic state may also suggest that the sidebands are in the parametric enhancement

regime, in addition to the two pieces of evidence discussed in the main text; namely, the

large sideband spacing (46 FSR) and the large range of intracavity power over which the

harmonic state persists.

6

IV. GAIN RECOVERY TIME VS. UPPER STATE LIFETIME

In a QCL, the upper state lifetime Tup

tells us how long an electron sits in the upper

state before making a nonradiative transition to the lower state. From here, it takes some

additional time to travel through the injector region and tunnel into the upper level of the

next stage. This additional amount of time is the bottleneck that determines the gain re-

covery time. The Maxwell-Bloch equations, by making the two-level approximation, cannot

account for the full complexity of the QCL, and only provide us with one carrier relaxation

time, which we have called T1

. This begs the question: does T1

represent the upper state

lifetime or the gain recovery time? The answer is that it depends on what you want to calcu-

late. In the steady-state single-mode regime, we find that the output power and population

inversion are functions of T1

due to di↵usion; here, we argue that T1

should represent the

upper state lifetime Tup

, because Tup

tells us how much time an electron in the upper state

has to di↵use before transitioning to the lower state. It is for this reason that Tup

appears

in the definition of �D, �D = (1 + 4k2DTup

)�1, rather than T1

. In dynamical situations, on

the other hand, the intensity of the field varies with time and we are interested in the how

the population inversion responds. We argue that this response is determined by the gain

recovery time, not the upper state lifetime, because the response definitely depends on how

long it takes an electron to get from one active stage to the next. To summarize it concisely,

the upper state lifetime is used for the calculation of the population grating (PG), but the

gain recovery time is used for the calculation of the population pulsations (PPs). Since the

bulk of our manuscript deals with PPs, we chose simply to call T1

the gain recovery time,

rather than name a new time scale such as Tgr

, for instance. We hope that this does not

confuse the reader.

V. THEORY: SINGLE-MODE SOLUTION

This section gives a more detailed derivation of the single-mode solution, including the

intracavity power as a function of pumping, and the population inversion as a function of

position and pumping.

For a two level system with upper state |ai and lower state |bi, the material equations in

7

the non-rotating frame and the field equation are

d⇢abdt

= �i!ba⇢ab �id

~ E(t)w � ⇢abT2

(5)

dw

dt=

�2id

~ E(t)(⇢ab � ⇢⇤ab) +weq � w

T1

+D@2w

@z2(6)

@2E

@z2� 1

c2@2E

@t2= Ndµ

@2

@t2(⇢ab + ⇢⇤ab). (7)

We emphasize that these equations are in the non-rotating frame, whereas the equations we

have used in the main text [1] were already in the rotating frame and the RWA had already

been applied. However, since we are here dealing with two counter-propagating waves, we

chose to more closely follow the approach in [2]. We make the following ansatzes:

E(z, t) =1p2

⇥ER(z, t)e�i(!t�kz) + EL(z, t)e�i(!t+kz) + c.c.

⇤(8)

⇢ab(z, t) = ⌘⇤R(z, t)e�i(!t�kz) + ⌘⇤L(z, t)e

�i(!t+kz) (9)

w(z, t) = wDC

(z, t) + w2

(z, t)ei2kz + w⇤2

(z, t)e�i2kz. (10)

(We use the subscript “DC” rather than “0” for the spatial average of the population inver-

sion, wDC

, because the subscript 0 is used throughout the text to refer to the primary mode.

No such ambiguity occurs for the subscript “2.”) Plugging the ansatzes into the di↵eren-

tial equations, and making the RWA as well as the slowly-varying envelope approximation

(SVEA) yields the following equations:

d⌘⇤Rdt

=�i

2p2(ERwDC

+ ELw2

)�✓

1

T2

+ i�

◆⌘⇤R (11)

d⌘⇤Ldt

=�i

2p2(ELwDC

+ ERw⇤2

)�✓

1

T2

+ i�

◆⌘⇤L (12)

dwDC

dt=

ip2(ER⌘R + EL⌘L � c.c.) +

weq � wDC

T1

(13)

dw2

dt=

ip2(ER⌘L � E⇤

L⌘⇤R)�

w2

T1

� 4k2Dw2

(14)

1

c

@ER@t

= �@ER@z

+ip2↵

T2

⌘⇤R � `0

2ER (15)

1

c

@EL@t

= +@EL@z

+ip2↵

T2

⌘⇤L � `0

2EL (16)

where = 2d/~, ↵ = N!T2

d2p

µ/✏/~ is the Beer absorption coe�cient of the material, and

� = !ba�! is the detuning of the field from the atomic resonance frequency. The loss term

8

`0

has been added to the field equation heuristically, and in this context it represents only

the waveguide loss.

We solve for the single-mode solution by setting the time-derivatives to zero and the

slowly-varying envelope functions to be constants. In doing so, we are now making the

distributed loss approximation because we are not allowing the fields to grow in space.

Thus, `0

must now be taken to be the total loss, waveguide plus mirror loss. We take � = 0

for simplicity, because the single-mode will lase very close to the peak of the gain spectrum.

We denote the steady-state field amplitudes by ER = EL = E0

and find the LI curve

|E0

|2 = p� 1

1 + �D/2(17)

where �D = (1 + 4k2DT1

)�1 is the di↵usion parameter. Based on the discussion in Sec.

IV, however, we know T1

represents the upper state lifetime Tup

, so we define �D = (1 +

4k2DTup

)�1. The steady-state population w0

(z) is given by

w0

(z) = wth

1 +

�D2

p� 1

1 + �D/2� �D

p� 1

1 + �D/2cos(2k

0

z)

�. (18)

VI. THEORY: POPULATION PULSATIONS

This section gives a more detailed derivation of the population pulsations, and demon-

strates how to include nonzero detuning � and GVD into the formalism.

We begin by imagining a small volume of dipoles subject to a spatially uniform E-field

to develop an understanding of the non-linear e↵ects caused by the Bloch dynamics. The

electric field is given by

E(t) = E(t)ei!t + c.c. (19)

The Bloch equations in the rotating wave approximation are

� =

✓i�� 1

T2

◆� +

i

2wE (20)

w = i(E⇤� � E�⇤)� w � weq

T1

(21)

where � is the o↵-diagonal element of the density matrix in the rotating frame, w is the

population inversion (positive when inverted), � = !ba � ! is the detuning between the

applied field and the resonant frequency of the two-level system, T1

is the (longitudinal)

population relaxation time, T2

is the (transverse) dephasing time, ⌘ 2d/~ is the coupling

9

constant where d is the dipole matrix element (assumed to be real) and ~ is Planck’s constant,

and weq is the equilibrium population inversion in the absence of any electric field which is

determined by the pumping. (Note that these equations are identical to Eqs. 3.19(a)-(c)

in [1], except that we have allowed E to be complex and left the o↵-diagonal component of

the density matrix in complex notation rather than writing � = (u + iv)/2.) With these

conventions, the macroscopic polarization P (dipole moment per volume) in a region with

a volume density of N dipoles is given by

P (t) = Nd�ei!t + c.c. (22)

First, we consider the e↵ect of a monochromatic field at frequency !, obtained from Eqs.

20-21 by setting E(t) = E0

and all time derivatives to zero. The result is a steady-state

polarization �0

and population inversion w0

given by

�0

=iT

2

2(1� i�T2

)w

0

E0

(23)

w0

=weq

1 + 2T1T2|E0|21+(�T2)

2

(24)

Note that the population inversion w0

is saturated as the field strength E0

increases: this is

responsible for saturable loss (when weq < 0) and saturable gain (when weq > 0).

A. Two-frequency operation

Next, we consider the E-field

E = E0

+ E+

ei�!t (25)

which consists of the strong field E0

at frequency ! superposed with the much weaker field

E+

detuned from ! by �!. A polarization will of course be induced at ! + �!. However, a

polarization at !��! also results due to the beat note at �! which modulates the intensity:

the resulting modulation of the population inversion with time (i.e., a population pulsation)

leads to nonlinear frequency mixing. We express the full polarization as

P (t) =X

m=�,0,+

Pmei!mt + c.c. (26)

10

where !+

⌘ ! + �! and !� ⌘ ! � �!. We can solve for the polarization as done in [3],

keeping only terms to first order in the weak field E+

, which gives

P0

=i✏

!ba

↵w0

E0

1� i�T2

(27)

P+

=i✏

!ba

↵w0

E+

1� i(�� �!)T2

+ ⇤+

+

E0

E⇤0

E+

�(28)

P� =i✏

!ba

↵w0

⇤+

�E0E0E⇤+

(29)

where

⇤+

+

=�[1 + i(�+ �!)T

2

](1 + i�!T2

/2)

[1� i(�� �!)T2

](1 + i�T2

)h(1 + i�!T

1

)[1 + i(�+ �!)T2

][1� i(�� �!)T2

] + (1 + i�!T2

)|E0

|2i

(30)

⇤+

� =�(1� i�!T

2

/2)

(1� i�T2

)h(1� i�!T

1

)[1 + i(�� �!)T2

][1� i(�+ �!)T2

] + (1� i�!T2

)|E0

|2i

(31)

are the self-mixing and cross-mixing coupling coe�cients, respectively. We consider the

dipoles to be embedded in a host medium of permittivity ✏ and permeability µ. (We adopt

the convention of [4]: ✏, µ and the speed of light c = 1/p✏µ always take their values in

the background host medium.) Many of the material properties of the two-level system are

lumped into the “Beer loss rate”

↵ =Nd2T

2

!bacpµ/✏

~ , (32)

which is related to the more familiar Beer absorption coe�cient ↵ (with units of inverse

length) that appears in Beers law of absorption by ↵ = ↵c. (Note, however, that in our

expressions for the polarization due to the two-level system, all factors of ✏ and µ drop

out; that is, these expressions do not contain the polarization contributions due to the

background medium.) The central mode amplitude E0

has been normalized such that E0

⌘

pT1

T2

E0

. Note that P0

is una↵ected to first order in E+

. The polarization P+

comes from

two contributions. First, there is the linear contribution from the Lorentz oscillator which

E+

would induce even in the absence of the strong field E0

. Second, there is a contribution

due to the PP which is described by the term ⇤+

+

. The term P� is due solely to the PP and

is governed by ⇤+

�. Note that the full polarization is directly proportional to the steady-

state population inversion w0

; this will be important when we generalize our results to

standing-wave cavities, where w0

varies with position.

11

Now that we have the polarization, we can calculate the gain seen by the sideband field.

We define the gain g (with dimension of frequency) of the sideband as the power density

generated at ! + �! by the interaction of the field with the dipoles–considering only field

and polarization terms oscillating at ! + �!–divided by the energy density of the exciting

sideband field, or

g+

⌘ �hEP i+

2✏|E+

|2 (33)

=i!

+

(E+

P⇤+

� E⇤+

P+

)

2✏|E+

|2 (34)

1. � = 0

Here we consider the case of zero detuning, � = 0, which simplifies the mathematical

expressions considerably and is a prerequisite to understanding the case of non-zero detuning.

Under this simplified scenario, we denote the self-mixing coe�cient ⇤+

+

by ⇤, where

⇤ =�(1 + i�!T

2

/2)h(1 + i�!T

1

)(1 + i�!T2

)2 + (1 + i�!T2

)|E0

|2i , (35)

and it is simple to show that the cross-coupling coe�cient ⇤+

� is simply ⇤⇤. The gain of the

sideband field is found to be

g+

= ↵w0

1

1 + (�!T2

)2+ Real(⇤)|E

0

|2�. (36)

(We have used (!+�!)/!ba ⇡ 1.) Thus, the gain can be nicely divided up into a contribution

from the Lorentz oscillator and a contribution from the PP. All of this is proportional to

↵w0

: ↵ gives you the gain of a weak field tuned to line-center in a perfectly inverted medium

(or alternatively, the loss seen by a weak field tuned to line-center in a material in its ground

state), and w0

gives you the expectation value of finding an electron in the excited state

(equal to 1 when excited, -1 when in the ground state, and 0 at transparency). Note that

Real(⇤) can be positive or negative, which we will discuss shortly.

B. Three-frequency operation

Of course, the polarization created at ! � �! will create a field at that frequency, which

is precisely why in the experiments we always observe the two sidebands appearing simul-

taneously. One sideband cannot exist in isolation when the mixing terms naturally couple

12

them together. Therefore, we need to consider the field

E = E0

+ E+

ei�!t + E�e�i�!t. (37)

The polarization at each sideband frequency now contains a Lorentzian term, a self-mixing

term, and a cross-mixing term:

P+

=i✏

!ba

↵w0

E+

[1� i(�� �!)T2

]+ ⇤+

+

E0

E⇤0

E+

+ ⇤�+

E0

E0

E⇤�

�(38)

P� =i✏

!ba

↵w0

E�

[1� i(�+ �!)T2

]+ ⇤�

�E0E⇤0

E� + ⇤+

�E0E0E⇤+

�(39)

where ⇤�� and ⇤�

+

are obtained by making the substitution �! ! ��! in the expressions for

⇤+

+

and ⇤+

�, respectively, given in Eqs. 30-31 .

1. � = 0

Let us again focus on the case � = 0, for which the polarization at each sideband

simplifies to

P+

=i✏

!ba

↵w0

E+

[1 + i�!T2

]+ ⇤E

0

E⇤0

E+

+ ⇤E0

E0

E⇤�

�(40)

P� =i✏

!ba

↵w0

E�

[1� i�!T2

]+ ⇤⇤E

0

E⇤0

E� + ⇤⇤E0

E0

E⇤+

�, (41)

where ⇤ is simply ⇤+

+

evaluated for � = 0. We see the nice property that when � = 0,

⇤+

+

= ⇤�+

(⌘ ⇤), and ⇤�� = ⇤+

� (⌘ ⇤⇤); in other words, the self- and cross-mixing coupling

coe�cients are equal.

The gain g+

of the positive sideband is

g+

= ↵w0

(1

1 + (�!T2

)2+ Real

"⇤|E

0

|2 1 +

E2

0

E⇤�

|E0

|2E+

!#), (42)

and a similar expression holds for the minus sideband. This equation tells us that the PP

contribution to the gain depends on the phase and amplitude relationships of E0

, E�, and

E+

, which is not too surprising because the amplitude of the PP itself is sensitive to these

parameters. Without loss of generality, we can take E0

to be real. If E+

= E⇤�, then the

two sidebands’ contributions to the beat note at �! add constructively, resulting in a field

whose amplitude modulation (AM) is twice the strength of a field with only one sideband.

13

If E+

= �E⇤�, then the two sidebands’ contributions to the beat note at �! destructively

cancel and there is no longer any amplitude modulation at frequency �!. We refer to such

a field as frequency-modulated (FM). We see from Eq. 42 that the AM sidebands therefore

experience a PP contribution to the gain that is twice as large as the single sideband case,

while the FM sidebands experiences only the background Lorentzian gain, consistent with

the fact that there is no PP in this case. We summarize this with the formula for the gain

g of each sideband for the case of equal-amplitude sidebands (|E+

| = |E�|),

g = ↵w0

2

4 1

1 + (�!T2

)2+ Real(⇤)|E

0

|28<

:2 ; AM

0 ; FM

3

5 . (43)

Note that for a superposition of AM and FM, the gain due to the PP will fall between 0

and 2 times the factor Real(⇤)|E0

|2.

VII. THEORY: INSTABILITY THRESHOLD

This section gives a more detailed derivation of the instability threshold, combining the

populating grating and the population pulsations into.

When a continuous-wave (cw) laser is pumped at its lasing threshold, only a single

frequency of light–the one nearest the gain peak that also satisfies the roundtrip phase

condition–has su�cient gain to overcome the roundtrip loss and begins to lase. As the

pumping is increased, the single-mode solution yields to multimode operation; this is known

as the single-mode instability. Our goal is to determine 1) how hard to pump the laser to

reach the single-mode instability and 2) which new frequencies start lasing.

Consider a laser pumped above threshold that is lasing on a single-mode, which we

refer to as the primary or central mode. If another mode is to lase, it must be seeded by

a spontaneously generated photon at a di↵erent frequency. This photon will necessarily

create a beat note through its coexistence with the primary mode, resulting in a population

pulsation. The gain seen by the new frequency must therefore account for this parametric

gain in addition to the background Lorentzian gain. Furthermore, the PP couples the

sideband to the symmetrically detuned sideband frequency on the other side of the primary

mode, so we should in general assume the presence of both sidebands. Because the instability

threshold depends on the cavity geometry, we will consider a traveling-wave laser as well

as a standing-wave laser. In both cases, the strategy is the same. First, we solve for the

14

single-mode intensity E0

and the population inversion w0

(z) as a function of the pumping,

entirely neglecting the sidebands. Knowing this, we can then calculate the sideband gain in

the presence of the primary mode.

We start with the wave equation

@2E

@z2� 1

c2@2E

@t2= µ

@2P

@t2. (44)

Following the approach used to calculate the optical parametric oscillation threshold in

optically pumped microresonators [5], we expand the field in terms of the cold cavity modes,

E(z, t) =X

m=�,0,+

Em(t)⌥m(z)ei!mt + c.c. (45)

The spatial modes obey the normalization condition

1

L

Z L

0

dz |⌥m(z)|2 = 1. (46)

When group velocity dispersion (GVD) is non-zero, the two modes !+

and !� will not be

equidistant from !0

. We have also assumed that the spatial and temporal dependence of the

modes can be separated. This is a good approximation in the case of a laser, because we

know the intracavity field will be sharply resonant at the modes. The spatial variation of the

polarization can be described by making the substitution Em ! Em⌥m(z) and w0

! w0

(z)

into the polarization Eqs. 38-39, which results in the polarization

P (z, t) =X

m=�,0,+

Pm(z, t)ei!mt + c.c. (47)

where

P+

(z, t) =i✏

!ba

↵w0

(z)

E+

⌥+

(z)

[1� i(�� �!)T2

]+ ⇤+

+

(z)|⌥0

(z)|2⌥+

(z)|E0

|2E+

+ ⇤�+

(z)⌥0

(z)2⌥⇤�(z)e

i!tE2

0

E⇤�

�

(48)

P�(z, t) =i✏

!ba

↵w0

(z)

E�⌥�(z)

[1� i(�+ �!)T2

]+ ⇤�

�(z)|⌥0

(z)|2⌥�(z)|E0|2E� + ⇤+

�(z)⌥0

(z)2⌥⇤+

(z)ei!tE2

0

E⇤+

�.

(49)

We have introduced ! ⌘ 2!0

� !+

� !�, the deviation of the cold cavity modes from equal

spacing. Note that the ⇤s now depend on z due to the term in their denominators dependent

on the primary mode amplitude. Because we no longer demand that the two sidebands have

the same detuning �!, ⇤+

+

and ⇤+

� should, strictly speaking, be calculated using the detuning

15

�!+

= !+

� !0

, while ⇤�� and ⇤�

+

should depend on �!� = !0

� !�. In practice, we can

ignore this di↵erence in the ⇤s; the term ei!t captures the most important e↵ect of GVD.

Plugging everything into the wave equation gives

X

m

✓d2⌥m

dz2+

!2

m

c2⌥m

◆Emei!mt � 2i

c2

X

m

!mdEmdt

⌥mei!mt = µ

X

m

�!2

m(Pm � Pm,loss)ei!mt

(50)

where the slowly-varying-envelope approximation allowed us to ignore second time deriva-

tives of Em on the left-hand side, and first and second derivatives of Em on the right-hand

side. The spatial modes ⌥m(z) are chosen so that the first term on the LHS equals zero. The

loss of each mode has been added to the equation in the form of a polarization contribution;

we assume each mode has the same linear loss, which can be expressed

Pm,loss(z, t) =i✏

!ba

¯⌥m(z)Em(t). (51)

Equation 50 couples all of the modes Em. We can project this equation onto each mode

by multiplying by ⌥n(z) and integrating over the length of the laser cavity, thus taking

advantage of the orthonormality of the spatial modes ⌥m(z), and then equating terms which

oscillate at the same frequency (since terms with di↵erent frequencies will not a↵ect the

time-averaged gain seen by a mode). The result is one equation for the central mode

E0

=

�¯

2+

↵

2(1� i�T2

)

Zdz

Lw

0

(z)|⌥0

(z)|2�E0

, (52)

one for the positive sideband

E+

= �¯

2E+

+↵

2

E+

1� i(�� �!)T2

Zdz

Lw

0

(z)|⌥+

(z)|2

+ |E0

|2E+

Zdz

Lw

0

(z)⇤+

+

(z)|⌥0

(z)|2|⌥+

(z)|2

+E2

0

E⇤�e

i!t

Zdz

Lw

0

(z)⇤�+

(z)⌥0

(z)2⌥⇤�(z)⌥

⇤+

(z)

�, (53)

and one for the negative sideband

E� = �¯

2E� +

↵

2

E+

1� i(�+ �!)T2

Zdz

Lw

0

(z)|⌥�(z)|2

+ |E0

|2E�Z

dz

Lw

0

(z)⇤��(z)|⌥0

(z)|2|⌥�(z)|2

+E2

0

E⇤+

ei!tZ

dz

Lw

0

(z)⇤+

�(z)⌥0

(z)2⌥⇤+

(z)⌥⇤�(z)

�. (54)

16

These three equations will be used to understand the instability threshold. In general, one

must first apply the steady-state condition E0

= 0 to Eq. 52 which, together with the Bloch

equation relating the field to the inversion, will yield the amplitude of the primary mode E0

along with the resulting population inversion w0

(z), both as a function of the pumping weq.

This information is then used in Eqs. 53-54 to determine the minimum level of pumping weq

at which a pair of sidebands with detuning �! experiences more gain than loss. This is the

instability threshold.

So far, we have kept Eqs. 52-54 as general as possible to account for arbitrary spatial

profiles, GVD, and detuning � between the lasing mode and the peak of the gain spectrum.

From here on we will simplify the problem by taking ! = 0 (zero GVD) and � = 0, and

apply these conditions to the simplest possible traveling-wave and standing-wave cavities.

A. Traveling-wave cavity

For the traveling-wave laser, the spatial modes are

⌥m(z) = e�ikmz (55)

so every point in the cavity sees the same intensity. At and above threshold, the population

inversion is everywhere saturated to the threshold inversion, so w0

is independent of z. For

� = 0, the inversion is

w0

= wth ⌘¯

↵(56)

and the intensity of the primary mode is given by

|E0

|2 = p� 1 (57)

where we have made use of the normalized primary mode amplitude E0

⌘ pT1

T2

E0

, and p

is the pumping parameter defined as p ⌘ weq/wth. Because |E0

|2 is independent of z, all of

the ⇤s are independent of z. Furthermore, since both w0

and the ⇤s are independent of z,

they can be pulled out of the spatial integrals in Eqs. 53-54. These integrals are then equal

to one, where we have used the zero GVD condition ! = 0 in order for the cross-overlap

17

integral (the last integral in each equation) to equal one. The sideband equations become

E+

= �¯

2E+

+↵wth

2

E+

1 + i�!T2

+ ⇤|E0

|2E+

+ ⇤E2

0

E⇤�

�(58)

E� = �¯

2E� +

↵wth

2

E�

1� i�!T2

+ ⇤⇤|E0

|2E� + ⇤⇤E2

0

E⇤+

�(59)

which can be written in matrix form0

@ E+

E⇤�

1

A =

0

@M+

R+

R⇤� M⇤

�

1

A

0

@ E+

E⇤�

1

A (60)

where

M+

= M⇤� = �

¯

2+

↵wth

2

✓1

1 + i�!T2

+ ⇤|E0

|2◆

(61)

R+

= R⇤� =

↵wth

2⇤|E

0

|2. (62)

(In the last step, we have finally taken the freedom to choose E0

to be real, which we can do

at this point without loss of generality.)

Now, if we assume a solution of the form E± ⇠ e�t, we find the two solutions for �

� =1

2[M

+

+M⇤� ±

q(M

+

�M⇤�)2 + 4R

+

R⇤�]. (63)

The net gain seen by each sideband is given by Real(2�) (the factor of two is for intensity

gain rather than amplitude gain), which includes the gain minus the loss. Subtracting o↵

the loss, the gain g seen by each sideband is

g = ↵wth

2

4 1

1 + (�!T2

)2+

8<

:2 Real(⇤)|E

0

|2 ; AM

0 ; FM

3

5 (64)

where the two solutions correspond to AM and FM sideband configurations. Finally, we

recognize that the gain is pinned at threshold, so ↵wth = ¯, and we write down the sideband

gain normalized to the loss

g¯ =

1

1 + (�!T2

)2+ Real(⇤)|E

0

|2 ·

8<

:2 ; AM

0 ; FM. (65)

When the gain g exceeds the loss ¯, the weak sideband amplitudes experience exponential

growth, therefore the single-mode solution becomes unstable. Note that the Lorentzian term

18

T1 / T2 = 1 T1 / T2 = 10 T1 / T2 = 100

p=1 p=5 p=10 p=14.9

p=1 p=3 p=6 p=9.59

p=1 p=3 p=6 p=9.06

increasing p

FIG. S5. The sideband gain gAM/¯ of a traveling-wave laser, given in Eq. 66, is plotted at various

pump strengths, for three di↵erent values of Z: 1, 10, and 100. The largest value of p in each plot

is equal to the instability threshold given in Eq. 69.

is always less than 1. This is a direct result of uniform gain clamping in the traveling-

wave laser, which clamps the net gain of the mode at the peak of the Lorentzian to zero,

and therefore any mode detuned from the peak will see slightly more loss than gain. FM

sidebands therefore never become unstable because they only see the Lorentzian gain. On

the other hand, AM sidebands induce a PP and with it a coherent gain term, which can

provide enough extra gain on top of the Lorentzian background to allow the sidebands to

lase,

gAM

¯ =1

1 + (�!T2

)2+ 2 Real(⇤)|E

0

|2. (66)

To get a feel for the sideband gain, we have plotted gAM/¯ in Fig. S5 at various pump

strengths p for Z = 1, 10, and 100, where Z ⌘ T1

/T2

. Graphically, we see that at large

enough p sidebands will become unstable. Analytically, it is a simple matter to calculate

how hard to pump the laser p before the sidebands appear, starting from Eq. 66. We start

by replacing |E0

|2 with p � 1, and note that this substitution must also be made in ⇤,

which implicitly varies with |E0

|2. Then, setting gAM/¯ equal to one, we can solve a simple

quadratic formula for �!2,

(�!T2

)2 =�1 + 3Z(p� 1)±

p[1� 3Z(p� 1)]2 � 8Z2p(p� 1)

2Z2

. (67)

Finally, we must apply some physical reasoning: as p is increased past 1, the sideband gain

increases. Right at the moment when the instability threshold is reached, �!2 must take

19

on a single value. Thus, we set the radical in Eq. 67 to zero and solve for p. After solving

another simple quadratic equation, we find that

p = 5 +3

Z± 4

r1 +

3

2Z+

1

2Z2

. (68)

How do we choose between the plus and minus sign? By plugging this expression for p back

into Eq. 67, it is simple to check that only the plus sign yields real-valued solutions for �!.

Thus, we have found the instability threshold, which we denote pRNGH ,

pRNGH = 5 +3

Z+ 4

r1 +

3

2Z+

1

2Z2

(69)

because it is the well-known instability threshold found by Risken and Nummedal (see Eq.

3.10 in [6]) and Graham and Haken (see Eq. 7.35 in [7]). Plugging this value of p into Eq.

67 yields the value of �! of the sidebands when the instability sets in

(�!RNGHT2

)2 =4

Z2

+6

Z

1 +

r1 +

3

2Z+

1

2Z2

!. (70)

One thing to notice is that in the limit Z � 1 (transverse relaxation must faster than

longitudinal relaxation), the instability threshold pRNGH ! 9 from above and �!RNGHT2

!p

12/Z.

B. Standing-Wave Cavity

As before, we restrict ourselves to the case � = 0 and ! = 0. We will see that calculations

for the standing-wave cavity are significantly more complicated than for the traveling-wave

cavity. The spatial variation of the primary mode causes the inversion w0

and the coupling

⇤ to both depend on z, which makes the integrals more di�cult to compute. For this reason,

we treat the problem to first order in the primary mode intensity |E0

|2, which allows us to

compute the integrals analytically. However, the theory can be extended to higher order at

will, or the integrals can always be computed numerically.

For the standing-wave laser with perfectly reflecting end mirrors, the spatial profile of

each mode is given by

⌥m(z) =p2 cos(kmz). (71)

The spatial modulation of the intensity is responsible for the spatial modulation of the

population inversion w0

(z), though mitigated somewhat by carrier di↵usion. We calculated

20

w0

(z) in Sec. V of the supplementary information. The result is

w0

(z) = wth

1 +

�D2

p� 1

1 + �D/2� �D

p� 1

1 + �D/2cos(2k

0

z)

�. (72)

to first order in |E0

|2, where wth = ¯/↵, and �D = (1+4k2DT1

)�1 is the di↵usion parameter.

The spatial variation of the inversion has important consequences. For one, it reduces the

power of the laser, which is given by

|E0

|2 = p� 1

1 + �D/2. (73)

Secondly, the gain is no longer uniformly clamped by the primary lasing mode, which will

allow new modes to lase even in the absence of PPs.

The spatial variation of the primary lasing mode also causes ⇤ to vary with position. In

keeping with our approximations, we can expand ⇤ to zeroth order in |E0

|2 because in our

equations ⇤ always multiplies |E0

|2, so the final result is first order in |E0

|2. We define the

zeroth order expansion of ⇤ to be

�(3) =�(1 + i�!T

2

/2)

(1 + i�!T1

)(1 + i�!T2

)2, (74)

where the symbol �(3) was chosen to emphasize that this term now plays the role of a

third-order nonlinear coe�cient.

We start with the sideband Eqs. 53-54, replace w0

(z) with Eq. 18, ⇤(z) with �(3), and

keep only terms to first order in |E0

|2. The resulting equation for the growth of the positive

sideband is

E+

= �¯

2E+

+↵wth

2

"1 + �D

2

|E0

|2

1 + i�!T2

E+

+ �(3)|E0

|2E+

Zdz

L|⌥

0

(z)|2|⌥+

(z)|2

+�(3)E2

0

E⇤�

Zdz

L⌥

0

(z)2⌥⇤�(z)⌥

⇤+

(z)

�, (75)

and a similar equation can be written down for E�. We define the longitudinal overlap

integrals

�self

=

Z L

0

dz

L|⌥

0

(z)|2|⌥+

(z)|2 = 1 (76)

�cross

=

Z L

0

dz

L⌥

0

(z)2⌥⇤�(z)⌥

⇤+

(z) = 1/2. (77)

21

The implication is that the self-mixing interaction of a sideband with itself, mediated by

the primary mode intensity, is twice as large as the cross-mixing interaction of one sideband

generating gain for the other sideband, again mediated by the primary mode intensity. This

is true only for the cosine-shaped modes that we have assumed, and the overlap integrals

will change when the longitudinal spatial profile changes, as when the non-unity reflectivity

of the facets is taken into account. The sideband Eqs. 53-54 become

E+

= �¯

2E+

+↵wth

2

"1 + �D

2

|E0

|2

1 + i�!T2

E+

+ �self

�(3)|E0

|2E+

+ �cross

�(3)E2

0

E⇤�

#(78)

E� = �¯

2E� +

↵wth

2

"1 + �D

2

|E0

|2

1� i�!T2

E� + �self

�(3)

⇤|E0

|2E� + �cross

�(3)

⇤E2

0

E⇤+

#, (79)

which we express as 0

@ E+

E⇤�

1

A =

0

@M+

R+

R⇤� M⇤

�

1

A

0

@ E+

E⇤�

1

A (80)

where

M+

= M⇤� = �

¯

2+

↵wth

2

1 + �D

2

|E0

|2

1 + i�!T2

+ �self

�(3)|E0

|2!

(81)

R+

= R⇤� =

↵wth

2(�

cross

�(3)|E0

|2). (82)

As we did for the traveling-wave laser, the sideband gain is easily calculated from these two

coupled first-order di↵erential equations. Normalizing the gain to the total loss, we find

g¯ =

1 + �D2

|E0

|2

1 + (�!T2

)2+ Real[�(3)]|E

0

|2 ·

8<

:�self

+ �cross

= 3

2

; AM

�self

� �cross

= 1

2

; FM. (83)

There are two things to notice here. As the laser pumping is increased, the term �D|E0|2/2

grows, and consequently the gain is not clamped at the threshold value. This is due to spatial

hole burning, or more precisely, the imperfect overlap of the standing-wave modes together

with a finite amount of carrier di↵usion. We view this background gain as a Lorentzian-

shape whose amplitude increases with the pumping, and is therefore fully capable of pulling

the sidebands above threshold, without any additional PP contribution to the gain.

Secondly, the PP contribution to the gain never vanishes. Even when the sidebands are

phased such that an FM waveform is emitted from the laser, there is still a PP within the

laser cavity. The reason for this is the imperfect overlap of the two sidebands’ spatial modes,

22

which means that at any given position within the cavity, the plus and minus sideband are

likely to have di↵erent amplitudes. Therefore, even if the two sidebands are phased such

that their contributions to the beat note at �! destructively interfere with each other, the

destruction is not perfect. The amplitude of the PP varies with position in the cavity, and

in locations where the two sideband amplitudes are equal the PP will not exist, but the

spatially averaged e↵ect of the FM PP yields the factor of 1/2 in Eq. 83. By the same

token, sidebands phased for AM will not fully constructively interfere, yielding a factor of

3/2 for the PP contribution to the gain rather than the factor 2, as it would be for the

traveling-wave laser.

VIII. NUMERICALLY CALCULATING THE INSTABILITY THRESHOLD

In this section, we demonstrate the predictions of the theory for the three uncoated lasers,

and compare the results with the measurements.

Because the theory assumes end mirrors with unity reflectivity, we can only expect Eq.

83 to apply reasonably well to the uncoated QCLs. For each device, �D is calculated using

the theoretical value of Tup

(calculated from the bandstructure) and the di↵usion constant

D = 77 cm2/s [8], giving �D = 0.4 (DS-3.8), 0.49 (TL-4.6), and 0.93 (LL-9.8). For these

large values of �D, the incoherent gain increases rapidly with the pumping, and we find from

Eq. 83 that the FM instability will have a lower threshold than the AM instability, regardless

of the value of T1

. The gain recovery time T1

of each QCL is not as easily calculable as Tup

because it depends on a few other time constants of the active region, such as the escape time

of the electron from one injector region to the next active region. Therefore, we treat T1

as a

variable and calculate the instability threshold psb

and sideband spacing �!sb

as a function of

T1

. The resulting curves are shown in Fig. S6. By comparing the curves with the measured

values of �!sb

, we can deduce the values T1

= 1.83 ps (DS-3.8), 1.15 ps (TL-4.6), and 0.91

ps (LL-9.8). For these values of T1

, the theory predicts an instability threshold of psb

= 1.02

(DS-3.8), 1.09 (TL-4.6), and 1.04 (LL-9.8). It is encouraging that these fitted values of T1

are close to the accepted value of the QCL gain recovery time, which has been shown by

pump-probe experiments [9, 10] and theory [11] to be around 2 ps. However, the predicted

psb

is significantly lower than the measured values Jsb

/Jth

= 1.12 (DS-3.8), 1.17 (TL-4.6),

and 1.14 (LL-9.8), and the discrepancy is made worse by the fact that J/Jth

is likely an

23

0.93 (LL-9.8) 0.49 (TL-4.6) 0.40 (DS-3.8)

experimental

(a)

(b)

FIG. S6. Numerical solutions of the instability threshold obtained by setting the gain g in Eq. 83

equal to the loss ¯, yielding both (a) the sideband separation �!sbT2

and (b) the pumping psb. The

experimentally measured values of �!sb

T2

are compared to the theory to infer T1

/T2

, which also

gives the theoretical prediction for the instability threshold psb.

underestimate of p (see the discussion in Sec. I of the supplement). The fact that the theory

underestimates the instability threshold is perhaps not surprising, as we have only made

sure that one of the two necessary conditions for sideband oscillation is satisfied (gain, not

phase). We hope that future work which accounts for the detuning �, the detuning between

the lasing mode and the cold cavity mode it occupies, and GVD can accurately predict the

instability threshold, which would be a milestone in the understanding of lasers, and also

yield a novel laser characterization method of lifetimes and di↵usion rates by comparing

24

measured values of psb

and �!sb

to an established theory.

[1] L. Allen and J. Eberly, Optical Resonance and Two-Level Atoms (Dover Publications, New

York, 1987).

[2] A. Gordon, C. Y. Wang, L. Diehl, F. X. Kartner, A. Belyanin, D. Bour, S. Corzine, G. Hofler,

H. C. Liu, H. Schneider, T. Maier, M. Troccoli, J. Faist, and F. Capasso, Phys. Rev. A 77,

053804 (2008).

[3] R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic Press, San Diego, 2003).

[4] A. Siegman, Lasers (University Science Books, Sausalito, 1986) pp. 473–89.

[5] Y. K. Chembo and N. Yu, Phys. Rev. A 82, 033801 (2010).

[6] H. Risken and K. Nummedal, J. Appl. Phys. 39, 4662 (1968).

[7] R. Graham and H. Haken, Zeitschrift fur Phys. 213, 420 (1968).

[8] J. Faist, Quantum Cascade Lasers, 1st ed. (Oxford University Press, Oxford, 2013).

[9] H. Choi, L. Diehl, Z. K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, Phys.

Rev. Lett. 100, 167401 (2008).

[10] H. Choi, L. Diehl, Z. K. Wu, M. Giovannini, J. Faist, F. Capasso, and T. B. Norris, IEEE J.

Quantum Electron. 45, 307 (2009).

[11] M. A. Talukder, J. Appl. Phys. 109, 033104 (2011).

25