Return flux budget of polychromatic laser guide stars

Transcription

1 Reun flux bude of polychomaic lase uide sas Huues Guille de Chaellus Jean-Paul Pique Ioana Cisina Moldovan To cie his vesion: Huues Guille de Chaellus Jean-Paul Pique Ioana Cisina Moldovan. Reun flux bude of polychomaic lase uide sas. Jounal of he Opical Sociey of Ameica Opical Sociey of Ameica pp <0.4/JOSAA >. <hal-08444> HAL Id: hal hps://hal.achives-ouvees.f/hal Submied on Ma 05 HAL is a muli-disciplinay open access achive fo he deposi and disseminaion of scienific eseach documens whehe hey ae published o no. The documens may come fom eachin and eseach insiuions in Fance o aboad o fom public o pivae eseach cenes. L achive ouvee pluidisciplinaie HAL es desinée au dépô e à la diffusion de documens scienifiques de niveau echeche publiés ou non émanan des éablissemens d enseinemen e de echeche fançais ou éanes des laboaoies publics ou pivés. Public Domain

2 Reun flux bude of polychomaic lase uide sas Huues Guille de Chaellus * Jean-Paul Pique and Ioana Cisina Moldovan Laboaoie de Specoméie Physique UMR 5588 CRS-Univesié Joseph Fouie 40 avenue de la Physique BP Sain Main d Hèes Fance Absac Polychomaic lase uide sa PLGS is one of he soluions poposed o exend he sky coveae by lae elescopes o 00 % by enablin a complee knowlede of all peubaion odes of he wavefon. The knowlede of he ip-il is deduced fom he monioin of he chomaic componens of he PLGS fom 0 nm o he visible o nea infaed. Hee we sudy he oiinal scheme o ceae he PLGS by esonan exciaion of he mesospheic sodium by wo pulsed lases ens of khz ep. ae ens of W aveae powe ens of ns pulse duaion a 589 and 59 nm especively. The efficiency of his pocess is invesiaed numeically by means of boh Bloch equaions and ae equaions models. The influence of numeous lase paamees is sudied. In he bes case havin opimized all lase paamees he eun flux a 0 nm should no exceed phoons /s /m fo *8 W lase aveae powe a he mesosphee. This maximum is obained fo a modeless lase whose spo diamee coesponds o 4 imes he diffacion limi. Fo a diffacion-limied spo he eun flux falls down o phoons /s /m. 007 Opical Sociey of Ameica OCIS codes: Adapive opics AO echniques equie he pesence of an inense lih souce in he viciniy of he objec obseved by he elescope []. This is a majo limiaion o complee sky coveae since bih sas ae ae especially a he alacic poles. Lase uide sas LGS enable o exend he sky coveae by povidin an inense and posiional lih souce []. Howeve hey canno compensae fo he so-called ip-il esulin in an unceainy on he posiion of he lase beacon elaive o he objec []. On he ohe hand if one is able o ceae a polychomaic lase beacon he absolue ip-il can be deduced fom he chomaic ip-il [4]. The lae could be measued by poinin he elaive displacemen of he chomaic componens of he PLGS on he camea. This echnique is vey aacive since i enables heoeically 00 % sky coveae povided he vibaions of he elescope esponsible fo an addiional lobal displacemen of he imae can be measued o compensaed. The key poin of he feasibiliy is iven by he pecision equesed on he measuemen of he diffeenial ip-il. The lae is diecly linked o he eun flux available i.e. o he inensiy of he polychomaic lase uide sa and o he size of he lase spo in he sodium laye. The Fench pojec ELPOA fo Polychomaic Lase Guide Sa fo Adapive Opics is oiinally based on he wo phoon exciaion of he mesospheic sodium by means of wo lases especively a 589 and 59 nm [5]. Recenly anohe exciaion scheme has been poposed: he diec exciaion of he mesospheic sodium a 0 nm. A deailed sudy of his lae pocess has been published a sho while ao []. In he pesen pape we focus on he wo-phoon exciaion scheme which has been paially invesiaed [7] bu fo which no exensive sudy has been published ye. We choose o esic ouselves o dye amplified lases wih pulse duaion of ens of ns and a few ens of khz epeiion ae. The pape is oanized as follows. Fis we ecall some of he popeies of his double exciaion and we pesen he wo

3 numeical models ha ae used o calculae he efficiency of he pocess. The fis one is he Beacon code based on he esoluion of Bloch equaions and developed by he CEA Commissaia à l Eneie Aomique [8]. We slihly modified he code by addin an addiional ouine o ake ino accoun a mulimode lase and a modeless lase we specifically developed fo he ELPOA pojec. The second one is a ae equaions model developed in ou eseach oup []. Boh models ae validaed by compain idenical siuaions. In he second pa we develop some saihfowad aspecs of he eun flux by calculain asympoic and bes case expecaions of he eun flux and we define he elaions beween he expeimenal paamees of he exciaion. In he hid pa we focus on he choice of he bes lase confiuaion sinle-mode phase modulaed modeless and mulimode. In he las pa we opimize he eun flux by invesiain he influence of diffeen lase paamees polaizaion specal widh pulse duaion delay and epaiion of he eney beween he wo lases. We finally ive a maximum value of he eun flux fo a iven easonable lase powe.. Inoducion..Two phoon esonan exciaion of he mesospheic sodium.. Genealiies The mesospheic sodium laye has been a subjec of inense sudies fo seveal yeas [9]. I is now eneally aeed ha i is locaed on aveae a 90 km above he sea level wih a heih of 5-0 km. The posiion of he laye vaies coninuously and pesens iniuin dynamic phenomena: he so-called sodium spoadics [0]. Is column concenaion vaies wih he seasons beween and aoms/cm. In he emainin of he pape we ake he value of aoms/cm. The oiinal idea of polychomaic lase uide sas is based on he exciaion of he 4D 5/ level of he sodium aom by a double esonan exciaion of he S / -P / ansiion a 589 nm D line followed by he P / -4D 5/ ansiion a 59 nm acually 58.8 nm [4]. Ohe wo-phoon exciaion schemes have been poposed such as he non-esonan absopion of wo phoons a 578 nm bu in eneal hey equie much hihe peak powes and lases in he picosecond eime wih low epeiion ae []. We chose no o discuss his case in his pape and we focus on he esonan case. The aom in he 4D 5/ level decays accodin o a adiaive cascade povidin lines a 0. nm 58.8 nm nm 589. nm 8 nm 40 nm 08 nm and 8 nm. To measue a sinifican vaiaion of he diffeenial chomaic il one need o conside he 0 nm line and a lowe fequency line 589. nm o 8 nm see fi.. Theefoe in he emainin of he pape he flux bude fo he polychomaic LGS will be chaaceized by he eun flux a 0 nm. We also pesen he esuls fo he eun flux a 589 nm since i mih be of inees fo he monochomaic LGS communiy: we conside ha he iadiaion by he second lase has a limied impac on he eun flux a 589 nm and ha he esuls pesened in his pape on he eun flux a 589 nm can also apply fo monochomaic LGS. We also make he hypohesis ha he ansmission of he amosphee on he eun ip a 0 nm and he oal efficiency of he deecion appaaus ae equal o 00 %. We nelec any non-adiaive pocess. Fo simpliciy easons we also conside he siuaion whee he lase powe is consan ove he shined aea i.e. op-ha spaial pofiles ae assumed fo boh lases. This hypohesis is discussed in 4. whee we compae he op-ha appoximaion wih aussian-shaped lase beams. All fluxes ae iven in numbe of phoons pe second and pe squae mee of ecepion elescope. They ae deduced fom he pobabiliy of emission by an individual sodium aom by: f. da. S p D whee : - in phoons/s/m is he eun flux a he wavelenh in nm. - f is he lase epeiion ae. - d a is he suface sodium densiy in he mesosphee d a 4.0 aoms/m. - S is he size of he lase spo in he mesosphee m. - D is he disance fom he ound level o he mesosphee D 90.0 m. - p is he pobabiliy of phoonic emission pe seadian pe aom and pe lase pulse. Amospheic ip-il 07 nm 4S/ 4P/ k59 k59 k0 k0 Telescope 8 nm 0 nm Lases 4D5/ 59 nm P/ 589 nm S/ Mesospheic sodium 59 nm 589 nm Fequency GHz Fequency GHz Fi. Oveview of he wo-phoon PLGS. The chomaic componens of he PLGS ae monioed by he elescope op. The scheme is based on he wophoon esonan exciaion of he mesospheic sodium boom. The Dopplehypefine widhs of he ansiions ae iven in GHz.

4 .. Physical paamees of sodium ansiions The pobabiliy of exciaion of he sodium aom is linked o he physical paamees of he ansiions involved. The Beacon code consides he dipole momen wheeas he ae equaions model uses he absopion coss secion. In his paaaph we ecall he elaion beween hese paamees fo he ansiions of he sodium aom and we ive coheen numeical values o compae boh models.... Dipole momen The pobabiliy fo an aom o undeo a sponaneous ansiion fom an excied sublevel mb o a sublevel ma is accodin o []: A m k b ma j b mb Dq ja m a 0 Usin he Wine-Ecka heoem he Einsein coefficien Aba of he ansiion b a is independen fom mb and is iven by: A k ba A mb ma ja maq jbm b 0 m m q a jb D ja jb k 0 When level b can decay o seveal ai lowe levels see fi. he pobabiliy of he ansiion b ai is elaed o he adiaive lifeime of level b b by: adiaive Aba i B ba B i ba i b 4 b whee Bba i is he banchin aio fom level b o level a i. The dipole momen can be expessed as: j D b j a i 0 jb Bba i b oe ha a diffeen convenion someimes makes use of he aveae dipole momen defined as: b collisions a i collisions Fi. Geneic mulilevel sysem. All saes can decay houh boh adiaive and non-adiaive pocesses. a b A ba i a i adiaive ai 5 ba i jb D jai jai This is he case fo insance in he Beacon poam whee he aveae dipole momen expessed in Debye is aken as a vaiable inpu.... Expession of he absopion coss secion Table. umeical values and wiin convenions used in he followin of he pape. To link he dipole momenum o he diffeenial absopion coss secion one needs o expess he homoeneous linewidh of he a i b ansiion: adiaive adiaive collisions a b a b 7 i i adiaive adiaive whee a / i a and i b / b a i bein he adiaive lifeime of he level a i []. Since he collision ae in he mesosphee is much smalle han he homoeneous widhs of he diffeen ansiions he collisions can be neleced. Fo a loenzian homoeneous line he fequencyineaed absopion coss secion d 0 is linked o he absopion coss secion a he cene of he line by: a i b a i b Tansiion a i b S / P / P /4D 5/ nm 58.8 nm ai 4 b 4 a ns i b ns Homoenous widh MHz Mean dipole momen Debye Absopion coss secion a b 0 i We also have [4]: b 0 A 4 ba a i i P / = ns P / 4D 5/ 8 9 = ns =0 MHz = MHz = 5. D =.9 D =54 ns 0= m 0=. 0-4 m

5 whee he coefficiens epesen he deeneacy of he levels; we obain he elaion: b a i aib B ba 0 a i i a i b umeical esuls ae pesened in able... Lase paamees The paicula se of wavelenhs involved in he wo-phoon polychomaic lase uide sa equies paicula lase ypes and foma. Up o now he exciaion of he mesospheic sodium a 589 nm o poduce a LGS has been demonsaed boh wih dye and solid sae lases. In he fis case lase physiciss use he fac ha he wavelenh of he ansiion coesponds o he maximum of he fluoescence cuve of hodamine G pumped by a een lase usually a 5 nm [5]. In he second case wo lase lines fom a YAG cysal especively a.0 µm and. µm ae mixed in a nonlinea cysal and povide 589 nm []. Howeve fo he second ansiion 59 nm no such coincidence has been evidenced ye excludin he use of simple solid sae soluions fo wophoon polychomaic lase uide sas. Dye lases appea as he naual choice fo he exciaion a 59 nm fo he same easons as a 589 nm. Concenin he opimizaion of he exciaion of he sodium aom o he 4D 5/ level he nonlineaiy of he exciaion equies hih peak powes limiin he use of CW lases. Fo he same mean inensiy he ain in ems of peak powe beween a CW lase and a pulsed lase 0 khz epeiion ae 50 ns pulse duaion is abou 000 leadin o a vey lae impovemen >0 in ems of pobabiliy of wo-phoon absopion. Mode-locked ps lases can also povide hih peak powes bu his possibiliy aises he difficuly of synchonisin he 589 nm and 59 nm ps pulses wih a pecision bee han a few ns due o he fac ha he adiaive lifeime of he inemediae level is ns. Moeove in ode o ovecome he sauaion limi a picosecond lase soluion would equie a vey hih epeiion ae. On he ohe hand pulsed dye lases appea as he simples and mos naual choice since hee exis commecial solid sae Q-swiched pulsed pump lases a 5 nm delivein moe han 80 W aveae powe. In he es of he aicle we esic ouselves o dye sysems delivein a few ens of nanosecond pulses a a epeiion ae in he ane 0-0 khz.. Pesenaion of he Bloch equaion model The Bloch equaion model we conside is he Beacon code published in 004 and available online [8]. I compues he populaion of all hypefine sublevels of he S / P / 4D 5/ 4P / and 4S / levels. The lase aom ineacion is eaed in he semi-classical appoximaion he aom is eaed by he quanum fomalism and he elecomaneic field is eaed classically in he plane wave appoximaion. The Dopple boadenin is also aken ino accoun. The code consides numeous paamees such as he lase powe densiy in W/cm he empoal shape of he pulse aussian/ hypeaussian/squae he polaizaion he specum sinle mode o phase modulaion he pulse duaion and he delay beween he 589 nm and 59 nm lase pulses. The paamees of he eleconic ansiions dipole momen lifeimes hypefine couplin coefficiens ae also aken as vaiables. The calculaion of he emission pobabiliy of sponaneous phoons akes ino accoun he ansiions beween hypefine levels accodin o enealized banchin aios. The anula disibuion of phoons is also consideed. The oupu file euns he ime evoluion of he populaions of he 48 hypefine sublevels involved he ime pobabiliy of emission of phoons a 589 nm 59 nm and 0 nm. I also ecalls he eal an imainay pas of he lase fields as well as hei specum. oe ha he Beacon code has been esed by is auhos and successfully compaed o anohe Bloch equaions model developed by Mois o accoun fo he exciaion of he monochomaic LGS a 589 nm [7]... Limiaions of he code... Hypoheses Beacon does no ake aomic collisions ino accoun. Because of he low pessue in he mesosphee collision pocesses occu on a ime scale of hundeds of micosecond much lone han any of he adiaive pocesses involved in he fluoescence of he sodium aom. The Zeeman spliin of he hypefine sucue due o he eah maneic field is also neleced. The level 4D / is locaed less han GHz apa fom he 4D 5/ level. Bu considein ha he oscillao senh of he P / 4D / ansiion is en imes less han fo he P / 4D 5/ ansiion his level has been neleced in he poam. Finally he ecoil expeienced by he sodium aom is no aken ino accoun. The eney shif associaed is abou 0 khz and is much smalle han he homoeneous widhs of he aomic ansiions consideed. Since only pulsed exciaions ae consideed each aom expeiences a bes a few cycles of absopion/fluoescence pe lase pulse which excludes any cumulaive effec.... Resicions In ode o limi he calculaion ime he wo-colou exciaion pocess is eaed disincly fom he adiaive cascade. amely he exciaion phase is based on he numeical esoluion of he Bloch equaion fo he 48 hypefine sublevels of he eney levels S / P / and 4D 5/. Then he cascade is descibed by a ae equaion model. Duin he fis phase he cascade i.e. fom he 4D 5/ level is eaed as a leakin. In he second sep his leakin is used o calculae he populaions of he diffeen levels involved in he cascade. This implies ha hee is no ecyclin of he populaions involved in he leakin duin he lase pulse. This bins a limiaion o he duaion of he lase pulses which should no exceed a few imes he adiaive lifeime of he 4D 5/ level i.e. 54 ns. An addiional esicion aises also fo hihly sauain lase pulses wih a empoal aussian shape. In ha case even if he duaion of 4

6 he lase pulse saisfies he pevious limiaion he effecive duaion of he lase pulse seen fom he sodium aom is inceased because of he wins of he empoal Gaussian funcion. Theefoe he esuls obained by he Beacon code fo son lase pulses wih a Gaussian shape have o be aken wih cauion. This does no apply o empoal squae pulses. The adiaive cascade 4P / D /5/ P // S / is neleced. Finally he hypefine sucue of he 4D 5/ is aken ino accoun bu all hypefine levels ae deeneae... Exapolaion fo lon imes and validaion The Beacon code is wien in he Foan lanuae. Since he vesion available online is no paallelized he aveae ime of calculaion 500 ns duaion wih 0.0 ns ime sep is ypically 4 hous fo a. GHz pocesso usin full CPU esouces. A lone ime ane would pohibi paameic sudies. Theefoe we chose o limi he ime ane o 500 ns. The ypical lase exciaions consideed hee ae shoe han 00 ns FWHM and we conside ha afe 00 ns he lase pulse has one away. Then he aom elaxes accodin o he lifeimes of he levels involved in he cascade. We deemine he asympoic value of he pobabiliy of emission pe aom afe 500 ns by fiin he empoal pobabiliy of emission by exponenial laws. Acually he diffeence beween he pobabiliy emission a 500 ns and he asympoic value neve exceeds 7 %. oe also ha his pocedue concens only he phoons a 0 nm since he lifeime of he 4P / level is 0 ns. Fo he flux a 589 nm and 59 nm he emission pobabiliies have eached by fa hei asympoic values a 500 ns since he lifeimes of he P / and 4D 5/ ae ns and 54 ns especively. Befoe sain inensive calculaions we pefomed seveal ess: we an he Beacon code in he same condiions as in [8] and we obained simila esuls. We also compaed he values we obained wih some of he esuls iven in [8] o accoun fo he PLGS expeimens PASS- in 000 in Pieelae and we find an aeemen bee han 8%... Modificaion of he Beacon code The Beacon code was developed iniially o simulae he behaviou of he sodium aom when submied o sinle mode o phase-modulaed lases. Howeve he developmen in ou oup of an inacaviy FSF Fequency Shifed Feedback - o modeless- lase lead us o include in he code an addiional ouine o simulae his paicula lih souce [9]. The peculiaiy of his lase is a coninuous specum a he oupu wih no appaen mode sucue. The modeless behaviou is obained by insein in he caviy an acouso-opics modulao opeain a an RF fequency of 40 MHz. The funcion of his modulao is o incease he opical fequency of he phoon each ime i makes a oundip in he lase caviy. This pohibis desucive and consucive inefeences: in ohe wods evey sponaneous phoon is suppoed and amplified by he caviy. To limi he specal widh ohe selecive elemens ae inseed in he lase caviy: a Lyo biefinen file and a Faby-Péo ealon. Fo he lase a 589 nm he specum obained has a GHz wide Gaussian-like pofile. The GHz widh is opimized o excie all he Dopple-hypefine sucue of he S / P / ansiion. We also plan o develop a simila modeless lase a 59 nm wih a GHz bandwidh hypefine widh of he P / and 4D 5/ levels + Dopple boadenin. The quesion is: how o include his paicula modeless elecomaneic field in he Beacon code? Despie many sudies published in he lieaue hee is no clea conclusion on he expession of he elecic field a he oupu of he FSF lase. Some expeimens lead o an incoheen specum [0] some ohes o a chipin comb []. We sudy wo possible models fo he elecic field of he modeless lase.... aow Fee Specal Rane Model FSR A way o descibe a modeless lase is o wie i as a mulimode lase wih a fee specal ane FSR naowe han he homoenous line widh of he aomic ansiion excied by he lase. In ha case we ake ino accoun he fac ha he modeless lase can excie all velociy classes of sodium aoms. The homoeneous linewidh of he S / P / esp. P / 4D 5/ ansiion is = 0 MHz esp. = MHz. We choose o model he modeless lase by a MHz fee specal ane mulimode lase. To avoid any aificial mode-lockin aefac we assume andom phases beween he modes. To check he consisency of ou appoach we vay he specal fee ane beween MHz and GHz. We obseve ha he eun flux deceases when he fee specal ane FSR exceeds a few ens of MHz. Fo a GHz FSR a sinle lase mode emains in he GHz wide D line and he same flux is obained han fo wo sinle mode lases.... Chipin Comb Model CCM We compae he pevious esuls wih an alenaive expession of he elecic field of a FSF lase consisin in a chiped comb specum. The expession we used is deived in []. The elecic field is wien as: i n / 0 s max i n E E s 0e e e wih he followin paamees: 0 ln and s / - max is he cenal anula fequency of he lase. - is he ound ip ime in he lase caviy.57 ns in ou case. - is he shif in fequency expeienced by a phoon pe ound ip 80 MHz. - is he lase specal widh of GHz fo he S / P / esp. GHz fo he P / 4D 5/ ansiion. 5

7 We compae he numeical esuls obained usin boh models fi.. We obain a ood aeemen beween hem. In he followin of he pape we choose o conside he FSR model o descibe he elecic field of he modeless lase Modeless lases P 589 = P 59 7 khz 50 ns Linea polaizaion spo size = 0.5 m Reun flux phoons/s/m Toal lase Powe W 589 nm FSR 0 nm FSR 589 nm CCM 0 nm CCM Fi.. Plo of he eun fluoescence flux iven by he Beacon code fo boh models of he modeless lase Gaussian ime pofile op-ha spaial pofile.. Expeimenal validaion and compaison wih he ae equaion model.. The ae equaion model REM We poposed ecenly a diffeen exciaion scheme o ceae he PLGS based on he diec exciaion of he 4P / level. We developed in [] a REM model o compae boh exciaion schemes. Hee we jus ecall he main definiions; addiional deails can be found in []. Fi. 4 shows he six levels simplificaion and noaion. Fi. 4. Eney diaam and elaxaion pahways of he wo-phoon exciaion in he REM. The six diffeenial ae equaions ha descibe he aomic sysem ae: d d L absopion 5 5 sponaneous emission d L simulaed emission 4 4 d d d d d d L L L L d d 7 d d d L L 5 8 Fo a non dissipaive aomic sysem we have a any ime: i i D 9 The homoeneous absopion has a loenzian pofile: 0 0 / / i i i i i 0 The lase pofile L is supposed o be sepaable. Consequenly i is expessed as he poduc of he lase empoal pulse shape phoons/s by he spaial disibuion of he phoons pe uni aea m - and he line pofile Hz -. L All numeical paamees of he REM ae hose used in [] excep and which have been ecalculaed in paaaph... able.. Flux bude : asympoic limis and eneal ends. Uppe- and asympoic limis In his paaaph we pesen saihfowad calculaions o esimae he ode of maniude of he eun flux a 589 nm and 0 nm boh in he weak and in he son sauaion limi. These consideaions ae based on simple hypoheses on he pobabiliy of absopion by he aoms. We conside ha each aom can be excied wih he same pobabiliy whaeve is Dopple shif mih be. This implies ha he sodium laye is S / 4 P / P / P / 4 S / 07 nm 40 nm D 5/ 58.8 nm Lase nm Lase 58.8 nm 8 nm 8 nm

8 excied by a lase whose specum coves all velociy classes i.e. by a modeless lase. To check he validiy of hese simple asympoic calculaions we compae hem o he esuls iven by he Beacon and REM codes wih a modeless lase. In he son sauaion limi we evaluae he numbe of phoons each aom can emi a bes. In he limi of he linea eime we evaluae he asympoic values of he eun fluxes. We show ha he bes-case limis ae coheen wih he numeical values and ha he lae end vey pecisely o he asympoic limis... Son sauaion limi The eun flux boh a 589 nm and 0 nm inceases wih he lase powe in a non-linea manne: fo hih lase powes sauaion sonly affecs he fluoescence yield. Howeve i is possible o ive a maximal value of he fluoescence a 589 nm and 0 nm when he lase powe ends o hih values. In he bes case an aom shined by a lase pulse of duaion lase and excied o a level wih a adiaive lifeime b can emi lase / b fluoescence phoons pe lase pulse. Thus akin a 50 ns duaion lase pulse a sonly sauaed aom can emi a bes 4 phoons a 589 nm pe lase pulse P = ns. / Then he maximum flux available a 589 nm is iven by: S. f. da D S is he suface of he lase spo f is he epeiion ae of he lase d a is he suface densiy of he sodium laye = 4 0 aoms/m. D is he aliude of he sodium laye. Fo he ansiion a 0 nm an aom can undeo a bes wo sponaneous decays fom he 4D 5/ sae pe lase pulse 4D 5 / = 54 ns. Fom his sae he aom can decay o he 4P / level wih a pobabiliy of / and emi a phoon a 0 nm wih a pobabiliy of /9. Bu in his lae case since he adiaive lifeime of he 4P / level is lae han he lase pulse duaion he aom canno be pomoed aain o he 4D 5/ level. Theefoe he maximum eun flux a 0 nm is simply iven by: S. f. da 0 9 4D oe ha he maximal flux a boh 589 nm and 0 nm evolves linealy wih he size of he lase spo and is independen fom he lase inensiy... Weak sauaion limi Hee we evaluae he asympoic behaviou of he 589 nm eun flux in he weak sauaion limi i.e. in he linea eime. Fo ha we adop a Bee-Lambe ype appoach. We conside a sinle sodium aom iniially in he S / level. The numbe of phoons seen by he aom pe second and coespondin o is velociy class is: I S whee is he homoeneous widh of he S / P / ansiion is he specal widh of he fis lase I he aveae lase powe a 589 nm and he anula fequency a 589 nm. Simila convenions ae aken fo he ansiion P / 4D 5/ a 59 nm wih he subscip. We conside hee ha he specal widh of he lase is lae han he homoeneous widh which is always he case fo ens of ns lase pulses. We also conside ha he lase widh is naowe D han he Dopple widh of he ansiion. The pobabiliy of exciaion numbe of exciaions pe second of his aom can be wien as: p The numbe of aoms shined by he lase is: na S. d a D Theefoe he eun flux a 589 nm is: na da I 589 p 0 7 D 4D 4D oe ha in he linea eime he one-phoon exciaion pocess is popoional o he lase inensiy and is independen fom he size of he lase spo. We esimae now he eun flux a 0 nm in he weak sauaion limi. Fo simpliciy easons we conside ha boh lases cove exacly he Dopple widhs of he ansiions. Duin a lase pulse he numbe of sodium aoms excied o he P / level is: I n P d / 0 f D a 8 Fo one of hese aoms he numbe of phoons a 59 nm pe lase pulse coespondin o hei velociy class is: I 59 f S D 9 The pobabiliy of exciaion o he 4D 5/ level pe aom in he P / level is heefoe: P / p 0 59 lase 0 The pefaco comes fom he fac ha he lase pulse is lone han he adiaive lifeime of he inemediae level P / which educes he effecive pobabiliy of an aom in he inemediae level o absob a phoon a 59 nm wihin is lifeime. Finally he eun flux a 0 nm is: 7

9 Reun flux phoons/s/m d a P/ I I 0 9 D lase f 0 0 D D 4 S The pobabiliy o populae he 4D 5/ is popoional o boh he populaion of he P / level and he inensiy of he lase a 59 nm. When boh lase inensiies ae aken equal his pobabiliy evolves as he squae of he lase powe. oe also ha he eun flux a 0 nm in he eime of weak sauaion is popoional o he invese of he spo size and o he invese of he epeiion ae of he lase... umeical esimaions... Reun flux vs. lase inensiy Hee we conside he case of wo modeless lases opeain a 7 khz epeiion ae. The pulse duaion is fixed a 50 ns and he ime shape is a squae funcion. The specal widh of he lase a 589 nm esp. 59 nm is se o GHz esp. GHz. Boh lases have he same aveae inensiy I. The size of he lase spo is 0.5 m. We compae he numeical daa fom boh Beacon and he REM wih he asympoic laws obained in he pevious paaaph. In he limi of weak lase powe we have: I phoons/s/m and I iven in W I phoons/s/m and I iven in W. In he son lase powe limi we have: phoons/s/m phoons/s/m. oe on Fi. 5 he ood aeemen beween he asympoic and bes case limis and he esuls of he Beacon and REM codes boh in he weak and in he son sauaion eime. I is also woh noin ha he aio beween he eun flux a 589 nm and 0 nm is a leas equal o ~0 in he son sauaion limi and inceases owads he linea eime Beacon: 0 nm 589 nm Rem: 0 nm 589 nm modeless lases spo size = 0.5 m 7 khz 50 ns linea polaizaion empoal squae pofile spaial op-ha Aveae powe pe lase W Fi. 5. Reun flux of he PLGS wih espec o he aveae lase powe. Scae: Beacon and REM daa; doed lines: bes-case limis dash-do lines: asympoes in he linea eime.... Reun flux vs. spo size In his paaaph we sudy he asympoic behavious in he eun flux vs. lase spo size epesenaion. We evidence ha he eun flux fo he one-phoon exciaion monochomaic LGS exhibis a vey diffeen behaviou fom he eun flux of he PLGS. We sill conside he case of wo modeless lases opeain a 7 khz. The squae pulse duaion is se a 50 ns. The aveae powe fo each lase is 5 W. In he small spo size limi i.e. in he son sauaion limi he bes-case expessions fo he flux a 589 nm and 0 nm ae deduced fom he pevious elaions. We obain: S phoons/s/m and S iven in m S phoons/s/m and S iven in m. In he lae spo size limi i.e. in he weak sauaion limi he asympoic expessions fo he flux a 589 nm and 0 nm ive especively: phoons/s/m S phoons/s/m and S iven in m. As can be seen on fi. he aeemen beween boh models is sill ood fo boh wavelenhs in he small and lae spo limi. oe ha he eun flux a 589 nm ends o is asympoic value fo lae spo sizes. This is no he case fo he eun flux of he PLGS which exhibis a maximum. This comes fom he fac ha he eun flux a 0 nm is limied by wo phenomena: he sauaion of he aom on he side of small spo sizes and he ininsic non-lineaiy of he wo-phoon exciaion pocess fo lae spo sizes.. Fiue of mei of he polychomaic lase uide sa The epesenaion of he eun flux vs. spo size is also peinen o deemine he fiue of mei of he PLGS. The spo size is deemined by seveal paamees: he qualiy of he lase beam he diamee and qualiy of he lase pojeco he Fied paamee 0 amon ohes. The solid veical line on fi. epesens he diffacion limied spo houh a 50 cm pojeco. The pecision on he measuemen of he ip-il is linked o wo paamees: he inensiy of he available eun flux and he size of he imae on he ip-il senso a CCD camea o a 4-quadan phoodiode fo example. Concenin he fis paamee fo a iven imae size in a X-Y epesenaion he pecision in he X diecion X is ideally deemined by he phoon noise i.e. i is popoional o / bein he numbe of phooelecons deeced by he AO sysem pe acquisiion fame. is simply popoional o he eun flux 0. The dependence of he pecision of he ip-il measuemen wih he size of he imae of he fluoescence is no so saihfowad. The poin is ha Beacon uses he lase suface densiy as an inpu assumin implicily 8

10 Reun flux phoons/s/m op-ha spaial pofiles. Howeve in ealiy he fluoescence spo does usually no mach exacly he lase spo in he mesosphee he eason bein ha on one side sauaion ends o incease he appaen widh of he spo and ha on he ohe side he nonlineaiy of he wo-phoon pocess in he case of he PLGS ends o educe he size of he fluoescence spo compaed o he size of he lase spo. Theefoe in he eneal case when is consan he pecision in he X diecion X is popoional o he diamee of he spo i.e. whee is he aea of he fluoescence spo which o usually diffes fom S he aea of he lase spo. The pecision on he ip-il evolves as: X Y 0 Minimizin his aio leads o he opimal PLGS confiuaion in ems of lase chaaceisics powe polaizaion specum empoal shape and spo size a he mesosphee. Theefoe we can define he fiue of mei fo he PLGS as 0. The bes soluion in ems of lases should maximize his paamee. The difficuly aises fom he knowlede of he fluoescence spo: he popaaion of he lase beam o he mesosphee leads o wavefon disoion and speckle effecs ha ae usually difficul o conol. Howeve fo illusaion puposes i is ineesin o conside he ideal case whee he lase beam has a op-ha pofile in he mesosphee. In ha case since he powe densiy of boh lases is consan ove he aea shined by he lases he fluoescence spo maches exacly he lase spo and hee is a simple popoionaliy beween and S. Theefoe he fiue of mei of he PLGS evolves as: 0 S. The bes siuaion coesponds o a eun flux as hih as possible and o a lase spo as small as possible. The dashed cuve in fi. epesens he fiue of mei fo he case discussed in he pevious paaaph. The veical scale is loaihmic wih abiay unis. Hee he bes case whee he unceainy on he posiion of he chomaic componen of he PLGS is he smalles coesponds o he limi of hih sauaion ha is o say owads small lase spos. The opimizaion of he fiue of mei in he eal case of aussian lase modes disoed by amospheic popaaion would consiue an ineesin discussion exceedin he fame of his pape Beacon: 589 nm 0 nm Rem: 589 nm 0 nm Modeless lases x5 W 7 khz 50 ns linea polaizaion. spaial op-ha empoal squae pofile E Spo size m Fi.. Reun flux of he PLGS wih espec o he spo size. Scae: Beacon and REM daa; doed lines: bes-case limis dash-do lines: asympoes in he linea eime. The dash cuve epesens he fiue of mei of he PLGS pocess in ab. unis see.. The solid veical line a 0.05 m coesponds o a diffacion limied spo houh a 50 cm pojeco.. Influence of lase paamees: saihfowad aspecs In his paaaph we pesen endencies in he behaviou of he eun flux when specific expeimenal paamees ae vaied. This is of paicula impoance when one wans o spae calculaion ime: we show ha fom he knowlede of a sinle cuve eun flux vs. lase inensiy o vs. lase spo size fo a iven se of lase paamees i is possible o deduce wihou any fuhe calculaion he eun flux cuve when one o some of he lase paamees is ae vaied. This is quie useful when sain fom a equesed eun flux and lase spo size one wans o define he coespondin lase soluion in ems of mean powe ep. ae ec : his possibiliy is illusaed in he final conclusion of his pape. The eun flux depends on many paamees: he lase peak powe he epeiion ae he size of he lase spo ec. Howeve hese paamees ae somehow inedependen and his can be used o avoid addiional simulaions and o save compuin ime. We deail hee he elaions beween he lase paamees and we show how his can be used o pedic diffeen cases... Influence of he epeiion ae Fo a iven peak powe inensiy he eun fluxes 589 and 0 ae popoional o he numbe of aoms excied i.e. o he suface of he spo S and o he lase ep. ae f. Howeve hee is no popoionaliy beween he lase peak inensiy I peak and he eun flux. The lae can be wien as: S. f. F I peak whee F is an unknown nonlinea funcion. We also have I peak popoional o I. Theefoe S. f 9

11 Reun flux phoons/s/m Reun flux phoons/s/m Reun flux phoons/s/m S. f. F I. 4 S. f We calculae he eun flux when he epeiion ae is inceased by a faco k fo iven spo size and aveae powe. We have: S k. f I S. k. f F I S. k. f k. S f I 5 E Spo size m 589 nm 00 khz 589 nm 0 khz 589 nm 0 khz 0 nm 00 khz 0 nm 0 khz 0 nm 0 khz Fi. 7. Reun flux of he PLGS iven by Beacon wih espec o he spo size fo diffeen ep. aes. The lases aveae powe is consan; Lase powe = *8 W cicula polaizaion modeless lase 80 ns aussian ime shape opha spaial pofile. This means ha in a aph ivin he eun flux vs. spo suface an incease of he ep. ae by a faco of k esuls in a homoheic ansfomaion on he hoizonal scale by a faco of /k fi. 7. This is of paicula impoance if one consides he fiue of mei of he PLGS: in he sauaion eime and fo a consan lase powe i is desiable o incease he epeiion ae o shif he flux cuves o small spo sizes esulin in inceasin he fiue of mei. We also have: S k. f I k. S f I / k which means ha in a aph ploin he eun flux vs. lase powe an incease of he ep. ae by a faco of k esuls in a homoheic ansfomaion on he hoizonal scale by a faco of k and on a veical scale by a faco of k fi Lase inensiy W 589 nm 00 khz 589 nm 0 khz 589 nm 0 khz 0 nm 00 khz 0 nm 0 khz 0 nm 0 khz Fi. 8. Reun flux of he PLGS iven by Beacon wih espec o he lase inensiy pe lase fo diffeen epeiion aes spo size = 0.5 m cicula polaizaion modeless lase 80 ns aussian ime shape op-ha spaial pofile... Influence of he oal lase powe We conside now he case whee he aveae lase powe is inceased by a faco k all ohe paamees bein unchaned. Then he eun flux is: S f k. I S. f. F k. I S. f k. S / k. f. F I S / k. f 7 k. S / k f I This means ha in a eun flux vs. spo size plo an incease of he lase powe by a faco of k esuls in a homoheic shif by a faco of k alon boh he hoizonal and he veical axis see fi. 9. Ineesinly he aph linkin he eun flux o he spo size cuve is unchaned on he lef side of he plo. This simply means ha in he son sauaion limi inceasin he lase powe does no incease he fiue of mei of he PLGS all ohe paamees bein consan nm 0 khz x54 W 589 nm 0 khz x8 W 589 nm 0 khz x W 0 nm 0 khz x54 W 0 nm 0 khz x8 W 0 nm 0 khz x W slope = E Spo size m 0

12 Reun flux phoons/s/m Fi. 9. Reun flux of he PLGS iven by Beacon wih espec o he spo size fo diffeen lase powes ep. ae = 0 khz cicula polaizaion modeless lase 80 ns aussian ime shape op-ha spaial pofile... Influence of he lase spo size Finally we conside he case whee he spo aea is inceased by a faco of k all paamees bein consan. In ha case he eun flux is iven by: I k. S f I k. S. f. F k. S f I / k k. S. f 8 This indicaes ha in a eun flux vs. lase inensiy plo an incease of he lase spo size by a faco of k esuls in a homoheic shif by a faco of k alon boh he hoizonal and he veical axis fi Lase inensiy W 589 nm.5 m 589 nm 0.5 m 589 nm 0. m 0 nm.5 m 0 nm 0.5 m 0 nm 0. m Fi. 0. Reun flux of he PLGS iven by Beacon wih espec o he lase inensiy pe lase fo diffeen lase spo sizes ep. ae = 0 khz cicula polaizaion modeless lase 80 ns aussian ime shape op-ha spaial pofile.. Choice of he lase sysem As we menioned i befoe we esic his sudy o he case of ens of nanoseconds lase pulses wih a epeiion ae of few ens of khz. In his paaaph we compae fou ypes of lases: a sinle mode lase a phase-modulaed lase a modeless and a mulimode lase.. Sinle-mode vs. phase modulaed lases The eun flux of he PLGS is conolled by wo limiaions: on one side he wo-phoon absopion is a non-linea phenomenon and equies hih lase inensiies. On he ohe side he sauaion limis he fluoescence yield fo hih peak inensiies. A nice way o educe he sauaion is o dispach he lase inensiy ove as many sodium velociy classes as possible i.e. o boaden he lase specum. This enables o decease he sauaion fo small spo sizes and o impove he fiue of mei see.. Phase modulaion PM of he lase beam is an efficien way o add sidebands o he lase specum and o impove he ovelap wih he aomic specal lines. The phase modulaion elies on he eleco-opic EO effec. A sinusoidal volae is applied o he cysal esulin in a peiodic modulaion of he linea phase of he lase field a he oupu of he cysal. This ime-dependen phase leads o addiional lines in he powe specum of he lase field. oe ha o incease he specal widh addiional modulaos can be inseed in seies on he lase beam pah. The fequency shif of he addiional specal lines is a linea combinaion of he fequencies applied o he EO modulaos wheeas hei inensiy is dicaed by non-ivial behavious excep when a sinle phase-modulaion is applied: in his case he specum es simple and is descibed by Bessel funcions. The insananeous phase shif expeienced by he lih acoss a sinle EO modulao is simply: M sin whee M is called he modulaion inensiy in MHz and is he fequency applied o he modulao in MHz. -x0 9 -x0 9 -x0 9 0 x0 9 x0 9 x0 9 4x0 9 5x0 9 -x0 9 -x0 9 -x0 9 0 x0 9 x0 9 x0 9 4x0 9 5x0 9 -x0 9 -x0 9 -x0 9 0 x0 9 x0 9 x0 9 4x0 9 5x0 9 -x0 9 -x0 9 -x0 9 0 x0 9 x0 9 x0 9 4x0 9 5x0 9-4x0 9 -x0 9 -x0 9 -x0 9 0 x0 9 x0 9 x0 9 4x0 9-4x0 9 -x0 9 -x0 9 -x0 9 0 x0 9 x0 9 x0 9 4x0 9 Deunin Hz 80 MHz 0 MHz 00 MHz 00 MHz 80 MHz 50 MHz 00 MHz 000 MHz 80 MHz 000 MHz 00 MHz 500 MHz Deunin Hz 80 MHz 450 MHz 00 MHz 00 MHz 80 MHz 000 MHz 00 MHz 500 MHz 5 MHz 5 MHz Fi.. Speca coespondin o diffeen phase modulaions wih espec o he hypefine-dopple lines of he ansiions a 589 and 59 nm. The boom ih cuve is he specum fo a lase a 59 nm modulaed a 5 MHz wih an inensiy equal o 5 MHz. All ohe aphs concen he double phase modulaion of he fis lase. All PM fequencies ae 80 and 00 MHz. The associaed inensiies ae iven in paenheses. oe ha all speca ae cened on he F= line of he D line excep he boom lef one cened on he D line. The Beacon code enables o inoduce up o hee phase modulaions on each of he lases. Fis we esic ouselves o a sinle phase modulaion on he lase a 59 nm and a double modulaion on he lase a 589 nm as his was he case duin he PASS- campain [8]. We conside he case of 50 ns pulses *5 W aveae powe linea polaizaion and 7 khz epeiion ae. The ime pofiles ae aussian and he spaial one is a op-ha. Seveal confiuaions have been esed hei speca ae ploed on fi. and he numeical esuls ae pesened on fi..

13 Reun flux phoons/s/m 0 nm phoons/s/m 8x0 4 sinle mode lases F= P.M.: boh lases 7x0 4 x0 4 5x0 4 4x0 4 x0 4 x0 4 x nm: F= D F= F= F= 59 nm: Spo diamee m Fi.. Reun flux a 0 nm iven by Beacon wih espec o he lase spo diamee fo diffeen phase modulaion funcions. Lase paamees: *5W 50 ns linea polaizaion aussian ime pofile spaial op-ha. All PM fequencies ae 80 and 00 MHz. The associaed inensiies ae iven in paenheses. All values ae iven in MHz. The menion D o F= indicaes he posiion of he cenal fequency of he phase modulaion. Geneally speakin phase modulaion shows an eviden ain ove sinle mode lases which end o each hei maximal eun flux fo vey lae spo sizes. Concenin he choice of he phase modulaion he bes esuls in ems of eun flux coespond o a lase cened on F= wih a elaively small specal widh. When he specal widh of he fis lase is inceased o excie all velociy classes of he D line he eun flux a 0 nm is lowe. The difficuly associaed wih he phase modulaion is ha in pinciple any fequency line on he fis lase should o have is counepa on he second one: a velociy class excied by he fis lase needs also o be eached by he second one fo an efficien populaion of he 4D 5/ level. If one wans o cove he whole D line his would equie exemely complicaed phase modulaion funcions. Moeove fom an expeimenal poin of view any jie on he lase speca has an impoan impac on he efficiency of he wophoon ansiion. To check he possible benefis of ovelappin he wo lase specal lines we pesen he esuls obained by applyin he same PM on boh lases. We choose o esic he exciaion of he fis ansiion o he F= line of he D ansiion and we use he followin paamees: he modulaion fequencies ae 80 MHz and 00 MHz and he especive inensiies ae 0 MHz and 00 MHz. The esuls ae pesened also on fi.. Bu supisinly we obseve ha in he case of idenical phase modulaions on boh lases he plo eun flux a 0 nm vs. spo diamee is no inceased: conay o wha could be expeced he maximum eun flux is smalle. We also can noe ha he maximum of he fluoescence cuve is shifed o he small spos bu a he expense of lowein he eun flux.. Modeless lase The benefis bouh by he phase modulaion ove he sinle mode opeaion can be inceased by usin a modeless lase covein all velociy classes and fo which he poblem of machin he fequency lines of boh lases is auomaically solved by he coninuous naue of he specum. As a poof of demonsaion we focus on he lase siuaions poposed in [5]. We an he Beacon code fo he diffeen siuaions poposed in he publicaion. In he case called ELPOA he lases ae boh phase-modulaed hei mean powe a he mesosphee is 8 W he epeiion ae is 5 khz he pulse duaion is 50 ns and he lases ae linealy polaized. In boh cases ELPOA and he lase paamees ae: 8 W aveae powe 0 khz epeiion ae aussian 80 ns pulse duaion and cicula polaizaions. The spaial mode of he lases is a op-ha; he diffeence bein ha in ELPOA he lases ae phase-modulaed wheeas hey ae modeless in he ELPOA case. Since [5] does no explici he phase modulaion funcion we ook he same phase modulaion fo ELPOA and ELPOA : 80 MHz 0 MHz and 00 MHz 00 MHz fo he fis lase and 5 MHz 50 MHz fo he second one [8]. The esuls iven by he Beacon code ae ploed in fi.. Concenin he diffeence beween ELPOA and we obseve ha he flux cuves of ELPOA ae shifed o he ih compaed o ELPOA : his coesponds o he fac ha he ELPOA case is moe sauaed han he ELPOA case smalle epeiion ae and shoe lase pulse. In ELPOA he value of he maximal flux a 0 nm is inceased by a faco of ~. Theefoe he fiue of mei as defined in. is impoved in he case of he modeless lase which appeas naually as he bes soluion fo he PLGS [4]. 0 7 flux 0 nm ELPOA [5] flux 0 nm ELPOA [5] flux 0 nm ELPOA [5] 0 E Spo size m lase ype: 589 nm ELPOA 589 nm ELPOA 589 nm ELPOA 0 nm ELPOA 0 nm ELPOA 0 nm ELPOA Fi.. Reun fluxes a 589 nm and 0 nm fo he cases eaed in [5]. The ed scae daa ae he numeical values of he flux iven in he pape. In his wok we ook he lase chaaceisics of ELPOA : cicle ELPOA : squae and ELPOA : ianle. See ex and conclusion fo deails.. Mulimode lase To illusae he benefis of he modeless lase vs. sinle-mode o phase-modulaed lases we can epesen he modeless lase as he limi of mulimode lases when he fee specal ane becomes naowe han he homoeneous linewidh of he ansiions. In his paaaph we sudy he influence of he oal numbe of lase modes in he absopion line by chanin he fee specal ane of he lases. Boh he modified vesion of

14 Reun 0 nm phoons/s/m Reun flux phoons/s/m he Beacon code and he REM can accoun fo mulimode speca bu fo simpliciy easons we only pesen daa obained fom he REM; fo calibaion a few poins have been also un by Beacon and ae consisen wih he REM esuls. In his case we conside *5 W mulimode lases linea polaizaion 50 ns pulses 7 khz ep. ae and we vay he fee specal ane beween MHz and GHz. Recall ha he fis value descibes he modeless lase and he second one a sinle mode lase. Boh lases a 589 nm and 59 nm have he same FSR. The esuls ae pesened on fi. 4. appoximaed by hypeaussian pofiles. In fi. 5 we compae he eun fluxes beween he aussian and he squae pofiles all ohe paamees bein consan aussian squae Beacon: 589 nm 589 nm 0 nm 0 nm Rem: 589 nm 589 nm 0 nm 0 nm FSR = GHz FSR = 00 MHz FSR = 00 MHz FSR = 50 MHz FSR = 0 MHz FSR = MHz 4x0 4 x0 4 x0 4 x Modeless lases x5 W 7 khz 50 ns linea polaizaion E Spo size m Spo size m Fi. 4. Reun flux a 0 nm vs. spo size fo diffeen values of he fee specal ane REM code. I is woh noin ha deceasin he FSR of he lases simply shifs he cuve eun flux vs. spo size o he small spos his is also he behaviou we obain fo he monochomaic LGS a 589 nm. The maximum eun flux is weakly affeced by he value of he FSR bu in ems of maximizin he fiue of mei of he PLGS i is desiable o decease he spo size wihou deceasin he eun flux. This is achieved by deceasin he sauaion i.e. by inceasin he densiy of lase modes he asympoic limi bein he modeless lase which appeas as he opimal soluion fo he PLGS. 4. Opimizaion of he lase sysem In he las pa we es he influence of seveal paamees on he eun flux of he PLGS he idea bein o impove he fiue of mei of he PLGS in ems of boh inceasin he eun flux and minimizin he spo size. We discuss hee successively he influence of he empoal shape of he lase pulse aussian vs. squae he spaial mode aussian vs. opha he polaizaion cicula vs. linea he specal shape of he fis lase he pulse duaion 50 ns vs. 80 ns he ime delay beween he lase pulses and he epaiion of he oal inensiy beween he wo lases. 4. Influence of he empoal lase pofile Beacon and REM codes accep boh opions: a squae o a aussian ime pofile. Fo he Beacon code he opion is se in he daa inpu file: he squae funcion is asympoically Fi. 5. Compaison of he eun flux fo squae and aussian ime pofiles Beacon and REM codes. Boh empoal ime shapes show a ood qualiaive aeemen he diffeence does no exceed 0 % in he limi of son sauaion. oe also ha he slope of he eun flux vs. suface fo small spos is sicly fo he squae pofile. The eun flux fo aussian pulses is a lile bi inceased compaed o he squae pulses: his comes fom he fac ha due o he wins of he aussian funcion he effecive duaion of he lase pulse seen by he aom is lone han in he case of squae pulses. Theefoe he maximum numbe of phoons an aom can emi duin a lase pulse is inceased and he eun flux is inceased consequenly. 4. Influence of he spaial lase mode The Beacon code akes he peak powe densiy of he lase in W/cm pe pulse as an inpu vaiable. Theefoe i implicily assumes a op-ha disibuion of he lase inensiy. In ealiy he siuaion is diffeen since he lase beams ae usually aussian a he ound and can be disoed afe popaaion houh he amosphee dependin on he exension of he lase beam wih espec o he Fied paamee 0. Fo simpliciy easons we assume ha boh lase beams exhibi a aussian mode a he mesosphee. We also assume ha hese modes have he same spaial exension. In he paaaph we sudy he diffeence beween he op-ha appoximaion and he aussian one. To ake ino accoun he aussian pofile we sample he aussian funcion by a sep-like funcion. The Beacon code is an fo each discee sep and he se of eun flux values is used o evaluae he eun flux fo he aussian case. To compae boh cases we define he elaive widhs of he op ha and he aussian mode accodinly o Sieman: he adius of he op-ha pofile is equal o w / w bein he wais of he aussian funcion [5]. As ploed in fi. he aeemen beween he aussian shape and he op-ha is ood in he linea eime lae spo sizes.

15 Reun flux phoons/s/m 0 nm phoons/s/m Howeve in he sauaed eime he aussian shape leads o a lae incease in he eun flux up o a faco of 4 fo vey sauaed cases. oe also ha he maximum of he cuve 0 S is shifed o he small spo sizes by a faco of and he maximum flux is slihly hihe in he aussian han in he op-ha case. This behaviou is due o he fac ha he aussian case is less sauaed since he spaial exension of he beam is no limied conay o he op-ha pofile. oe ha in he case of he aussian spaial mode he effecive size of he fluoescence spo is no moe equal o he lase spo size defined by he hoizonal axis. oe also ha i is possible expeimenally o eneae op-ha spaial pofiles bu a he expense of loosin a subsanial lase powe. 8x0 4 7x0 4 x0 4 5x0 4 4x0 4 x0 4 x0 4 x0 4 7 khz 50 ns x5 W Spo diamee m Phase modulaion linea Phase modulaion cic. Modeless lases linea Modeless lases cic op-ha Beacon: Rem: aussian Beacon: Rem: Modeless lases *5 W 7 khz 50 ns linea polaizaion E Spo size m 589 nm 0 nm Fi.. Compaison of he eun flux fo squae and aussian spaial pofiles Beacon and REM codes. Boh ime pofiles ae aussian. 4. Influence of he polaizaion Anohe possibiliy of he BEACO code is o se he polaizaion of he lase beams. This is of paicula inees since he hypefine sucue is aken ino accoun and he exciaion pobabiliy of he sodium aom can be opimized accodin due o he values of he Clebsch-Godan CG coefficiens. Fo insance fo he exciaion of he S / P / ansiion he hypefine ansiion m F = m F = has he hihes CG coefficien. This aumen has been sudied in deail in [7] and a subsanial ain is obained when cicula polaizaion is used insead of linea. oe ha o ake full advanae of his one needs o use ahe lon lase pulses o accumulae he sodium aoms ino lae m F saes. Fo he wo phoon exciaion we invesiaed he benefi of cicula vs. linea polaizaion by compain boh cases fo boh phase modulaed and modeless lases. The esuls iven by he Beacon code ae ploed in fi. 7. Fi. 7. Evoluion of he eun flux a 0 nm wih espec o he spo diamee. Boh modeless and PM lases ae pesened wih linea and cicula polaizaions. The paamees of he phase modulaion ae: 80 MHz 0 MHz and 00 MHz 00 MHz a 589 nm; and 5 MHz 5 MHz a 59 nm Beacon code. In boh cases he cicula polaizaion inceases he eun flux a 0 nm up o 40 % fo phase modulaed lases. I is also woh noin ha he value of he spo diamee ivin he maximum eun flux is unchaned afe polaizaion chane. Aain he bes esul in ems of flux and spo size is obained fo he modeless lase. 4.4 Influence of he specal widh In his paaaph we discuss he benefis of adjusin he specum of he modeless lase a 589 nm elaively o he D line. Up o now we consideed a Gaussian lase specal line wih a GHz FWHM o excie all velociy classes associaed o he S / P / ansiion. Bu a quick look shows ha he ovelap of a Gaussian funcion wih he double-gaussian shape of he D line is no opimal. Theefoe we invesiaed he possibiliy of exciin only he F= line of he D ansiion. This confiuaion happens o be he mos efficien fo he phase modulaion; see.. This line is a eula Doppleboadened Gaussian funcion wih a GHz FWHM. We compae he eun fluxes vs. spo size fo he GHz and fo he GHz specal widh of he fis lase. oe ha he second lase emains unchaned modeless GHz specal widh. The esuls ae pesened in fi. 8. 4