Optimization of an autodyne laser interferometer for highspeed. confocal imaging

Transcription

1 Optimization of an autodyn lasr rfromtr for highspd onfoal imaging Eri Laot, * Wilfrid Glastr, Olivir Jaquin, Olivir Hugon, and Hugus Guillt d Chatllus Cntr National d la hrh Sifiqu/ Univrsité d Grnobl, Laboratoir Intrdisiplinair d Physiqu, UM 5588, Grnobl , ran *Corrsponding author: ri.laot@ujf-grnobl.fr In autodyn rfromtry, th bating btwn th rfrn bam and th signal bam taks pla insid th lasr avity and thrfor th lasr fulfills simultanously th rols of mittr and dttor of photons. In ths onditions, th lasr rlaxation osillations play a lading rol, both in th lasr quantum nois whih dtrmins th signal to nois ratio (SN) and also in th lasr dynamis whih dtrmins th rspons tim of th rfromtr. In th prsnt study, w hav thortially analyzd th SN and th rspons tim of a Lasr Optial dbak Imaging (LOI) stup basd on an autodyn rfromtr. Mor prisly, w hav ompard th imags quality of two lasrs having th sam put powr, th sam rlaxation frquny, but having two diffrnt valus of th LOI gain indud by two diffrnt valus of th lasr rspons tim. rom this study, w hav finally dtrmind th t lasr dynamial paramtrs and th t xprimntal onditions for high spd imaging at th shot nois limit. inally, w onlud that a lasr diod with a vry short rspons tim (in th nanosond rang) sms to b an rsting

2 andidat ompard to solid-stat mirohip lasr with a rspons tim of svral tns of mirosond. Analytial prditions ar onfirmd by numrial simulations. 0 Optial Soity of Amria OCIS ods: 0.375,

3 . INODUCION Whn a frquny shift is rodud btwn th two bams of an rfromtr, on ralizs th so-alld htrodyn rfromtry. sulting from this shift, th rfrn btwn th two wavs produs an nsity modulation at th bat frquny, whih an b masurd by a photo-dttor. In this papr, w rfr only to autodyn lasr rfromtry whr th htrodyn wav mixing taks pla insid th avity of th lasr sour and is finally indirtly dttd by a photodiod. Sin th dvlopmnt of th first lasr in 960, lasr htrodyn rfromtry has bom a usful thniqu on whih many high auray masurmnt systms for sifi and industrial appliations ar basd []. Sin th pionr work of K. Otsuka, on slf-mixing modulation ffts in lass-b lasr [] th snsitivity of lasr dynamis to frquny-shiftd optial fdbak has bn usd in autodyn rfromtry and mtrology [3], for xampl in slf-mixing lasr Dopplr vloimtry [4-7], vibromtry [8-0], nar fild mirosopy [,] and lasr optial fdbak imaging (LOI) xprimnts [3-6]. Compard to onvntional optial htrodyn dttion, frquny-shiftd optial fdbak shows an nsity modulation ontrast highr by svral ordrs of magnitud and th maximum of th modulation is obtaind whn th shift frquny is rsonant with th lasr rlaxation osillation frquny [7]. In this ondition, an optial fdbak lvl as low as -70 db (i tims wakr than th raavity powr) has bn dttd [5]. In prvious paprs [7-9], w hav dmonstratd that in autodyn rfromtry, th main advantag of th rsonant gain (dfind by th ratio btwn th avity damping rat and th population-invrsion damping rat of th lasr) is to rais th lasr quantum nois ovr th dttor nois in a rlativly larg frquny rang around to th lasr rlaxation frquny. 3

4 Morovr if th dttion bandwidth is narrowr than th lasr dynamial rlaxation width, th signal to nois ratio (SN) of a LOI stup is frquny indpndnt, and mor importantly, shot nois limitd. W hav also stablishd that to maximiz th dynamial rang of a LOI stup, th t valu of th shift frquny is not th rlaxation frquny, but th frquny at whih th amplifid lasr quantum nois is qual to th dttion nois lvl [8,9]. or high-spd imaging, w nd to dras th signal aquisition tim of our LOI stup and thrfor to work with a dttion bandwidth largr than th lasr dynamial rlaxation bandwidth. Undr this ondition th lasr trans dynami annot b ignord and th LOI SN boms frquny dpndnt. h main objtiv of this papr is to dtrmin th t lasr dynamial paramtrs and th t xprimntal onditions to obtain high quality imags (i.. shot-nois limitd) as fast as possibl. his papr is organizd as follows. irstly, aftr a basi dsription of our LOI st-up (i.. our autodyn rfromtr) for onfoal imaging, w brifly rall th xprssion of th LOI prmannt signal indud by th bating insid th lasr avity. Sondly, w dtrmin th stationary LOI SN for diffrnt valus of th xprimntal aquisition tim ompar to th lasr dynamial rspons tim. hirdly, for autodyn imaging, w alulat th lvl of th trans LOI signal apparing whn during th lasr sanning (i.. from on imag pixl to th nxt on), th targt undr invstigation prsnts disontinuous physial proprtis. inally, w dtrmin th t lasr paramtrs for high spd autodyn imaging with a shot-nois limitd dttion (i.. highst quality imag as fast as possibl). In ah stion, analytial prditions ar onfirmd by numrial simulations. 4

5 . AUODYNE SIGNAL A. LOI st up A shmati diagram of th LOI xprimntal stup (i.. th autodyn xprimntal rfromtr) is shown in ig.. ypially th lasr is a CW mirohip with an put powr P of svral milliwatts and a typial rlaxation osillation frquny in th mgahrtz rang and a damping rat of th rlaxation osillation ( ) in th kilohrtz rang [9-0]. h lasr is thrfor a lass-b lasr ( ). h lasr bam is snt on th targt, through a frquny shiftr. A part of th light diffratd and/or sattrd by th targt is thn rinjtd insid th lasr avity aftr a sond pass through th frquny shiftr. hrfor, th optial frquny of th rinjtd light is shiftd by. his frquny an b adjustd and is typially of th ordr of th lasr rlaxation frquny. or th gomtrial po of viw, th lasr bam waist and th lasr foal spot on th targt undr invstigation ar optially onjugatd. At this po, on an alrady noti that, ompard to a onvntional htrodyn stup, th autodyn stup shown hr dos not rquir omplx alignmnt. Indd, th LOI stup is vn always slf-alignd baus th lasr simultanously fulfills th funtion of th sour (i.. photons-mittr) and of th photo-dttor (i.. photons-rptor). h optial fdbak is haratrizd by th ltri fild omplx rfltivity ( r xp j ) of th targt, whr th phas dsri th optial round trip btwn th lasr and th targt, whil th fftiv powr rfltivity ( ) taks o aount th r targt albdo, th numrial aprtur of th olltion optis, th frquny shiftrs ffiinis and th transmission of all optial omponnts (xpt for th bam splittr whih is addrssd 5

6 sparatly) and th ovrlap of th rtro-diffusd fild with th Gaussian avity bam (onfoal fatur). ig.. Shmati diagram of th LOI rfromtr stup for sanning mirosopy. L, L and L 3 : Lnss, BS: Bam Splittr with a powr rfltivity, GS: Galvanomtri Sannr, S rquny Shiftr with a round trip frquny-shift, PD: Photodiod with a whit nois sptrum. h lok-in amplifir is haratrizd by its gration tim. h lasr is haratrizd by its put powr p (photons/s), its rlaxation frquny rfltivity. and its dynamial rspons tim. h targt is haratrizd by its fftiv h ohrnt ration (bating) btwn th lasing ltri fild and th frquny-shiftd rinjtd fild lads to a modulation of th lasr put powr. or th dttion purpos, a fration of th put bam of th mirohip lasr is snt to a photodiod by mans of a bam splittr haratrizd by a powr rfltivity. h photodiod is assumd to hav a quantum ffiiny of 00%. h voltag dlivrd by th photodiod is finally analyzd by a lok-in amplifir whih givs th LOI signal (i.. th magnitud and th phas of th rtro-diffusd ltri fild) at th dmodulation frquny [5,6]. h lok-in amplifir is haratrizd by its gration tim. Exprimntally, th LOI imags ar obtaind pixl by pixl (i.. 6

7 po by po, lin aftr lin) by a full D galvanomtri sanning and th nssary tim nds to obtain an imag omposd of N pixls is roughly givn by: N. or high spd imaging (i.. high adn imaging), on nds to us a valu of as small as possibl. o dtrmin th SN of th obtaind LOI imags, nds to b ompard with th rspons tim of th lass-b lasr ( ). In this papr, whatvr th tmporal valus of (in th millisond or mirosond rang), w rfr to a fast rspons tim lasr whn: and to a slow rspons tim lasr whn:. B. LOI Modlling In th as of wak ( ) frquny shiftd optial fdbak, th dynamial bhavior of a rinjtd solid-stat lasr an b dsribd by th following st of quations [0, 7,8]: dn dt N N BN E t, (a) 0 N de dt BN E E os t t, (b) E whr, N is th population invrsion, E is th slowly varying amplitud of th lasr ltri fild, B is rlatd to th Einstin offi, is th day rat of th population invrsion, is th lasr avity day rat and N is th pumping rat. garding th nois, th lasr 0 quantum flutuations ar dsribd by th onvntional Langvin nois funtions N t and t E, whih hav a zro man valu and a whit nois typ orrlation funtion [-3]. 7

8 h lasr modl prsntd abov an b applid to thr lvls or four lvls lasrs with th ondition that th liftim of th uppr lvl of th pumping transition is vry short ompard to th liftim of th uppr lvl of th lasr transition. or xampl, this is ondition is satisfid in a thr lvls lasr suh as rbium lasrs as wll as for a four lvls lasr suh as nodymium lasr. In th st of Eqs. (), th fdbak tim dlay ( ), linkd to th optial round trip btwn th lasr and th targt is ompltly ngltd. It mans, that w only onsidr th as whr th round trip tim is shortr than th frquny shift ( ). C. LOI stationary signal In th st of Eqs. (), th priodi funtions xprss th bating (i.. th ohrnt ration) btwn th lasing and th fdbak ltri filds. h nt gain of th lasr is thn modulatd by th r-injtd light at th optial shift frquny. In th linar rgim, th photon put rat p (t) E t (numbr of photons pr sond) is thrfor priodially modulatd [7]: p t,, p G ( ) p os t, () whr p r is th man valu th photon put rat with B r N 0 th B normalizd pumping paramtr. In Eq. (), G ( ) dsri th amplifiation gain of th autodyn wavs mixing with: G ( ) ( ) (3) 8

9 whr for a lass-b lasr ( ), ( r ) is th frquny of th lasr rlaxation osillations and is th damping tim of th rlaxation osillations r [3]. At this po on an noti that for a lass-b lasr, whih is vry long ompard to th photon liftim in th avity ( ), is thrfor th lasr haratristi rspons tim [3]. In a LOI rfromtr, a partiularly rsting situation is th rsonan as ( ) whr th LOI signal gain (i.. th autodyn gain) boms: G ( ). (4) or a mirohip lasr, this ratio is typially of th ordr of [7,9] and th main advantag of th LOI dttion thniqu sms to om from this rsonant amplifiation of th optial wav mixing [3, 3, 7]. Using Eq. (), on an dfin th modulation ontrast (MC) of th autodyn wav mixing: MC p (, ), G ( ). (5) p In a LOI xprimnt, baus th lasr simultanously fulfills th funtions of th sour and of th dttor, w assum to simply dfin th saturation lvl as th fftiv rfltivity orrsponding to a maximum modulation of th lasr put powr (MC=): Sat (6) 4 G inally, using a lok-in amplifir, th LOI signal at th dmodulation frquny is givn by: 9

10 S LOI, p, G p. (7) 3. SAIONAY SN O AN AUODYNE INEEOMEE A. Stationary LOI SN With optial fdbak ( 0 ), th st of Eqs. () allows us to study th lasr quantum flutuations indud by th Langvin nois trms ( N t and t ). Using th Winr- Khhin thorm, on an obtain th powr dnsity sptrum of th lasr put powr quantum flutuations [7,, ]: E PD Lasr p t G. (8) p h LOI nois powr obtaind aftr th photodiod dttion (i.. aftr th bam splittr rfltion) and th lok-in amplifir filtring at th modulation frquny ( ) is thn givn by: N Lasr, PD, d (9a) Lasr whr for an gration tim :, (9b) 0

11 is assumd to b a first ordr powr filtr. By ombining Eqs. (8) and (9), on finally obtains for a lass-b lasr ( analytial xprssion of LOI nois indud by lasr quantum nois: ), th following N Lasr, p t, (0) and by ombining Eqs. (7) and (0), on finally obtains th stationary LOI SN: SN S, LOI,,. () N, Lasr ig. shows th volution of th stationary LOI SN ( S LOI N Lasr ) vrsus th normalizd shift frquny ( ) for diffrnt valus of th lok-in gration tim ( ) ompard to lasr rspons tim ( ).

12 00 a) S LOI / N Lasr 0 b) ) Normalizd frquny ( / ) ig.. Stationary LOI SN ( S N ) vrsus th normalizd shift-frquny ( ) for diffrnt LOI Lasr valus of th lok-in gration tim: a) onditions ar 7 0, b) ) 0. h xprimntal 0 and /. h lasr is a lass-b lasr with: p 3. 0 photons / s ( P 60 mw at 064 nm ) 356 khz and ( r. 0, 5 0 s, 5 0 s ). or ah gration tim th ontinuous lin shows th xat valu of th LOI SN [Eq. ()], whil th dash lin shows th orrsponding LOI shot-nois limit [Eq. (3)]. B. Aquisition with a fast rspons tim lasr ( ) If th lok-in gration tim is long ompar to th lasr rspons tim ( ), th nois [Eq. (0)] is thn simply givn by: N Lasr, p t (a) ( )

13 or idntially, by roduing th xprssion of th gain givn by Eq. (3): N Lasr, p t G. (b) At this po, on an noti that th rsonant amplifiation gain G ( ) prsnt in th LOI signal [s Eq. (7)], is also prsnt in th LOI nois and, as a rsult, th SN of th LOI stup is frquny indpndnt: SN,, p, (3) As mntiond abov, th rlaxation frquny sms to b of no partiular importan [s ig. (a) for omparison]. At this po, on an noti that th ondition:, SN, physially, mans that during th gration tim ( ), only,, min bak-rfltd photons ar dttd: p. (4), min h LOI st-up is thrfor shot nois limitd and th bam splittr rfltivity ( ) apparing in Eqs. (7) and (0) and finally in th right hand trm of Eq. (4), an b rprtd as th quantum ffiiny of th LOI dttion whr th lasr modulation produd insid th lasr avity signal is finally indirtly dttd by th photodiod aftr th bam splittr. Consquntly, and through th rst of th manusript, Eq. (3) is what w will all th LOI shot nois limit. 3

14 C. Aquisition with a slow rspons tim lasr ( ) If now th lok-in gration tim is short ompard to th lasr rspons tim ( ), th lasr quantum nois [Eq. (0)] is thn approximatly givn by: N Lasr, p t (5) 4 Now, to analyz th LOI SN, thr diffrnt ass nd to b studid dpnding on th rval btwn th shift frquny and th rlaxation frquny. irstly, if w work vry far away from rsonan ( ), thn th LOI nois [Eq. (5)] and LOI signal [Eq. (7)] an rsptivly and approximatly b givn by: N Lasr, p t (6) 4 S LOI p, (7) and finally on obtains (far away from th rsonan frquny), th following approximat xprssion of th LOI SN: SN,, p (8) 4

15 A brif omparison of Eq. (8) and of Eq. (3) (whih givs th LOI shot nois limit), shows that for:, th LOI SN is lowr than th LOI shot nois limit and boms frquny indpndnt whn working vry far away from th rsonan frquny. In agrmnt with th thortial prditions, ig. () shows that th LOI SN (ontinuous lin) saturats and thrfor boms frquny indpndnt at vry low and vry high frqunis. On an also orv that in ths onditions, th LOI SN is smallr than th orrsponding LOI shot nois limit (dash lin) by a multipliativ fator givn by:. Sondly: if w now work at rsonan ( ), thn th LOI nois and th LOI signal ar givn by: N Lasr, p t (9) S LOI, p (0) and finally on asily obtains th rsonant valu of th LOI SN: SN,, p p () h middl part of Eq. () shows that th LOI SN is now indpndnt from th gration tim ( ). his surprising fft ariss baus th lasr nois powr sptrum [Eq. (8)] hav a Lorntzian typ profil lading to a limit nois valu [S Eq. (0)] whn th dttion 5

16 bandwidth is larg ompard to th rsonan width ( ). A brif omparison of Eq. () and Eq. (3) shows that, for a lasr with a slow rspons tim ( ), th LOI SN at rsonan is highr than th standard LOI shot nois limit. In agrmnt with Eq. (), ig. () shows that th LOI SN (ontinuous lin) at rsonan is highr than th orrsponding LOI shot nois limit (dash lin) by a multipliativ fator givn by:. hirdly, in th rmdiat situation, whr:, on an dtrmin th frquny shift ( ) for whih th LOI SN is qual to th LOI shot nois limit. By qualizing Eqs. (a) and (5) on obtains: () inally, w onlud this stion, by rminding that th stationary LOI SN ( S LOI N Lasr ) is frquny indpndnt [s ig. (a)] and abov all shot nois limitd for a lasr with fast rspons tim ( ). On th othr hand, for a lasr with slow rspons tim ( ), th stationary LOI SN is frquny dpndnt [ig. ()], largr than th LOI shot-nois limit nar th rlaxation frquny (i.. whn: ) and smallr than th LOI shot-nois limit far way from th rlaxation frquny (i.. whn: ). Mor prisly th LOI SN is largr (by a fator givn by ) than th LOI shot nois limit 6

17 whn working at th rsonan frquny and smallr (by a fator givn by ) whn working vry far away from th rsonan frquny. D. Numrial simulation of th stationary LOI SN By using a ung-kutta mthod, w hav numrially solvd th st of diffrntial quations () to dtrmin th stationary LOI SN for diffrnt xprimntal onditions (i.. for diffrnt valus of ompard to ). Mor prisly, w hav ompard th dynamial bhavior of two lasrs having th sam put powr ( p ) and th sam rlaxation frquny ( ), but having two diffrnt valus of th LOI gain ( G ) indud by two diffrnt valus of th lasr rspons tim ( high quality imaging (i.. larg LOI SN). ). Hr, our aim is to dtrmin th t lasr for ig. 3 shows a omparison of th stationary LOI SN obtaind with th two lasrs, for diffrnt valus of th shift frquny ( ) and for diffrnt valus of th lok-in gration tim ( ). or ah xprimntal ondition (, ), th stationary LOI signal (obtaind with 0 0 ) and th LOI nois (obtaind with 0 ) hav bn dtrmind from an avrag of 00 masurmnts ah (no sanning ours) to rdu th SN unrtay. or th lasr having th shortst tim rspons tim ( µs 4 ) and thrfor th lowst valu of th LOI gain ( G 4 0 ), th numrial rsults shown on ig. 3(a) ar in good agrmnt with th analytial prditions givn by Eq. (3). Indd, using a lasr with a fast rspons tim ( ), th simulatd LOI SN is almost frquny indpndnt and th numrial 7

18 simulations show valus that ar vry los to th analytial prditions. Indd, by using Eq. (3), with a lasr put powr of P 60 mw, a targt fftiv rfltivity of and a bam splittr rfltivity 0. 5 on obtains for th LOI SN th valus givn in abl, whih orrspond to th LOI shot nois limit. A losr look at th rsults of ig. 3(a) shows a small frquny dpndn whih sms to b mor important for th shortst gration tim. his small frquny dpndn is also in agrmnt with th thortial prdition [s ig. (b) for omparison]. or th lasr having th longst tim rspons ( 333 µs ) and thrfor th highst valu of th LOI gain ( G ), th rsults of th numrial simulations shown on ig. 3(b) ar now frquny dpndnt and muh mor ompliatd to analyz. 5 irstly, using Eq. (6), with a LOI gain G 8 0 and a bam splittr rfltivity 0 0.5, on obtains and sat sat h LOI signal is thrfor saturatd at th rlaxation frquny.. and non-saturatd for shift frqunis with: Sondly, th LOI SN shows an anti-rsonan phnomnon for th longst gration tims and a rsonan phnomnon for th shortst gration tims. In ig. 3(b), th anti-rsonan phnomnon orvd for 00 µs and 600 µs oms simply from th saturation of th LOI signal. h SN rsonan phnomnon orvd for 6 µs 0.08 and 0 µs 0.06 an b xplaind by th analytial rsults shown on ig. (), with nvrthlss a rsonan hight lss important du to th saturation of th LOI signal. 8

19 inally for. and, th following ondition is satisfid: and in agrmnt with Eq. (8) on an orv on ig.3(b) that th LOI SN is always smallr (by a fator approximatly givn by ) than th LOI shot nois limit valus givn in abl. a) G µs S LOI /N Lasr b) G Normalizd frquny 5 ( 8 0 / 333 ) µs S LOI /N Lasr Normalizd frquny ( / ) ig. 3. SN ( S N ) of a lass-b lasr ( P 60 mw ; 356 khz, LOI Lasr vrsus th normalizd shift-frquny ( ) ) for diffrnt valus of th lok-in gration tim: 9

20 ( ) 600 µs, () 00 µs, ( ) 60 µs, ( ) 0 µs, () 6 µs. Uppr graph 4 G 0 and 4 µs ( 0, µs ( 0, r. ) r ); Lowr graph: 8 0 G and 6 µs 0 µs 60 µs 00 µs 600 µs SN abl. LOI SN [Eq. (3)] obtaind with th lasr put powr P 60 mw 7 p 3. 0 photons / s at 064 nm ), a targt fftiv rfltivity splittr rfltivity (i and a bam 4. AUODYNE IMAGING A. rans LOI signal Suppos now that du to th sanning of th lasr bam on th targt undr invstigation, th targt proprtis suddnly hang at a tim t=0 (for xampl at th dg of th targt) with: r xp,,, j for 0 t and r j for t 0. Undr ths onditions, th xp,,, lasr put powr modulation is omposd of a stationary signal, for t 0 : p t,, r p G ( ) p os t (3a) 0,,, and for t 0, of th sum of a stationary and of a trans signal: p t,, r p G ( ) p os t 0,,, t xp G ( ) C p os t (3b) 0

21 whr for a lass-b lasr ( ), w assum that th trans signal is an osillating signal ntrd at th rlaxation frquny nsuring th ontinuity of th signal and of its first drivativ.. h onstants C and an b dtrmind by os C os os (4a),,,,, sin C sin C os sin,,, (4b) or a lass-b lasr( ), Eqs. (4) shows that, whatvr th xprimntal onditions, a good approximation of th amplitud of th trans osillations (i.. th ordr of magnitud of C ) is givn by: C os,,,,,, (5a) and thrfor C,,,, (5b) If,,, Eqs. (5) show that th amplitud of th trans signal ( C ) and of th stationary signal (, ) ar approximatly qual ( C, ), whil in th ontrary as, th trans signal is highr than th stationary signal,, C,,.

22 In th most gnral as,, and, ar of th sam ordr of magnitud (,, ) and th trans signal is at maximum of th sam ordr of magnitud than th stationary signal ( 0 C, ). At th put of th lok-in amplifir, using an gration tim and a first ordr filtr, th trans LOI signal at th dmodulation frquny an b xprss in th tim domain by: t 0, C, xp G ( ) C p t xp j t dt (6) LOI with th trans shap: t xp os t, and whr th avraging shap t t xp orrsponds simply to th ourir ransform of th first ordr filtr, dfind prviously by Eq. (9b). or (i.. no frquny mismath), on obtains: LOI, C, G C p. (7a) or idntially by roduing th LOI signal givn by Eq. (7): LOI, C, S, LOI, C, (7b)

23 ig. 4 shows th normalizd ( / C ) ratio btwn th stationary and th trans LOI, signals ( S ) vrsus th normalizd shift frquny ( LOI LOI ) for diffrnt valus of th lok-in gration tim ( ) ompard to th trans tim ( ). a) 000 S LOI / LOI 00 0 b) ) Normalizd frquny ( / ) ig. 4. Normalizd ( C, ) ratio btwn th stationary and th trans LOI signals ( S ) LOI LOI vrsus th normalizd shift frquny ( ) for diffrnt valus of th lok-in gration tim ( ) ompard to th trans tim ( ): a) 0, b) ) 0. h lasr is a lass-b lasr with 4. B. Imaging with a fast rspons tim lasr ( ) 3

24 With a fast rspons tim lasr and with C, (orrsponding to th most gnral as), Eq. (7b) givs: S LOI (8), C,,, LOI whih an b rdu to th following inquality: S LOI, C, (9),, LOI ig. 4(a) onfirms that by using a lasr with a fast rspons tim, th stationary LOI signal is gnrally highr than th trans LOI signal, whatvr th working frquny is. C. Imaging with a slow rspons tim lasr ( ) With a slow rspons tim lasr and with C, (orrsponding to th most gnral as), Eq. (7b) boms: S LOI, C,, LOI, (30), and two diffrnt ass nd to b analyzd. irstly, if w work nar th rsonan frquny, with, Eq. (30) rdus to: S LOI, C, (3).,, LOI 4

25 ig. 4() onfirms that by using a lasr with a slow rspons tim, th stationary LOI signal and th trans signal ar of th sam ordr of magnitud nar th rlaxation frquny: Sondly, if w work far away from th rsonan frquny, ( ), Eq. (30) givs: S LOI, C, (3),, LOI and th trans dynamis an b ignord ompard to th stationary dynamis. ig. 4() onfirms that by using a lasr with a slow rspons tim, th stationary LOI signal is highr than th trans signal whn working far way th rlaxation frquny At this po on an also noti that by using a lasr with a slow rspons tim ( ), th minimization of th trans signal far away from th rsonan is mad to th dtrimnt of th LOI gain. Indd for, th valu of usabl LOI gain [Eq. (3)] is limitd by th following inquality: G ( ) G (33) inally, w onlud this stion by rminding that th trans signal an b ngltd ompard to th stationary signal if on uss a lasr with a fast rspons tim ( ) whatvr th working frquny is, and also with slow rspons tim lasr ( ) if th working frquny is far away nough from th rlaxation frquny ( ). In a prvious xprimntal papr [9] w hav shown that th t working frquny alld, is 5

26 th frquny at whih th lasr quantum nois is qual to th dttor nois lvl. In th mntiond papr, th lasr tim rspons is 0 s, th lasr rlaxation frquny is. 8 MHz and w hav xprimntally dtrmind 6 MHz,. Undr ths onditions, th trans dynamis an b ngltd if th following inquality is vrifid:, 0. µ s. hrfor on an us µs to obtain high spd LOI 0 imaging with any prturbation from th trans signal ( S,, 53 LOI,, LOI, ompatibl with our starting hypothsis ( ). ). Not also that ths xprimntal onditions ar also D. Numrial simulation In th most gnral situation, th LOI SN is givn by th ratio btwn th LOI signal dividd by th dttor nois plus th lasr quantum nois plus th trans flutuation. Hr, our main objtiv is to ompar th lasr quantum nois with th trans flutuations whih an bom important for short gration tims (i.. fast imaging ondition). W hav thrfor ngltd th dttor nois in th urrnt numrial study. o show th fft of th trans dynamis on LOI imaging, w hav ompard D sans xtratd from th masurd put powr modulation. hs sans hav bn obtaind from th numrial gration of th st of Eqs. () (with no avraging). h targt undr invstigation is a symmtri rfltivity pyramid omposd of four lvls whih allows th orvation of th trans dynamis ffts during th san in th as of an fftiv rfltivity inras (,, ) or dras (,, ). or th urrnt numrial study, w hav hosn vry low valus of th fftiv rfltivity to study th LOI snsitivity undr 6

27 Modulation ontrast (%) Modulation ontrast (%) modulation ontrast (%) Modulation ontrast (%) ultimat onditions whr th trans dynamis an b of th sam ordr of magnitud than th lasr quantum nois. ig. 5 shows th numrial rsults obtaind with th two lasrs alrady studid in th stion 3 of th prsnt manusript. o visually sparat th trans dynamis ffts from th nois ffts on th LOI imags, th solid urvs ar numrially ralizd with lasr quantum nois (i.. with th Langvin nois fors), whil th urvs with irls ombin both ffts (trans signal and quantum nois). a ), 0 s 5 b).5, 0 s Pixl numbr Pixl numbr ), 0 s d ).5, 0 s Pixl numbr Pixl numbr ig. 5. Numrial D san obtaind from th masurd lasr put powr MC of a LOI st-up, whn th lasr is sannd on a symmtri rfltivity pyramid omposd of 4 lvls. Exprimntal onditions: 7 P 60 mw (i.. p 3. 0 photons / s at 064 nm ), 356 khz, 0 µs. Lvl : (pixls -0 & 6-70), 0 ; lvl : (pixls -0 & 5-60), 4 0 ; lvl 3: (pixls -30 & 4-7

28 50), 0 0 ; lvl 4: (pixls 3-40), bottom row: op row: 4 G 0 with µs ; G 8 0 with 330 µs ;; lft olumn: ; right olumn:. 5. Curvs with irls ( ): rsults with lasr quantum nois; Solid urvs ( ): rsults with lasr quantum nois. 4 or th lasr having th lowst valu of th LOI gain ( G 0 4 ), i.. th shortst rspons tim ( µs 4 ), th numrial rsults shown on igs. 5(a) and 5(b) ar in good agrmnt with th analytial prditions of th stion 3. Indd, th trans dynamis flutuations ar always smallr than th lasr quantum nois flutuations, whih onfirms th fat that by using a lasr with a fast rspons tim ( 0 µs ), th trans dynamis is ngligibl whatvr th working frquny is. Not also that whn th fftiv rfltivity is multiplid by a fator 00 (lvl n to lvl n 4), th MC inrass by a fator 0, whil whn th fftiv rfltivity is multiplid by a fator 4 (lvl n 3 to lvl n 4) th MC inrass by a fator. Morovr, in abl, th MC and th SN, numrially dtrmind using th urvs with th irls of igs. 5(a) and 5(b) and analytially alulatd [from Eqs. (5) and ()], ar vry los. On an noti that th good agrmnt oms from th fat that for this lasr, th optial 0 fdbak lvl is blow th saturation lvl of th lasr ( , 4 sat ) and baus th trans flutuations ar ngligibl in th amount of nois. With this lasr, on also orvs that by inrasing th shift frquny, th valus of th MC is smallr but that th SN rmains approximatly unhangd, whih onfirms again th fat that by using a lasr with a fast tim rspons ( ) th LOI SN is approximatly frquny indpndnt, in agrmnt with Eq. (3). 8

29 0 MC(%) SN. 5 MC(%) SN 0.8 (0) () 0. (0) () 4 x 0. (.0). (.9) 0.6 (0.5).5 (.6) x (0.) 0. (9.7).7 (.6) 0. (8.) 4 x (0.5) 4. (9.4) 5. (5.) 8.7 (6.3) abl. MC and SN of th LOI imags [igs. 5(a) and 5(b)] obtaind with th lasr having th lowr valu LOI gain ( G 0 4 ). Numrial rsults ar in bold whil th analytial rsults (Eq.) ar writtn in italis btwn parnthss. In th numrial rsults th nois is omposd of th trans flutuation plus th lasr quantum nois whil in th analytial rsults, th nois orrsponds only to th lasr quantum nois. 0 MC(%) SN. 5 MC(%) SN. (0) () 0.8 (0) () 4 x (66.7).8 (7.4). (0.5) (0.4) x (833.3) 3. (37.).8 (.7) 4 (.) 4 x (667.0) 4.6 (74.3) 5. (5.4) 5 (4.) abl 3. MC and SN of th LOI imags [igs. 5() and 5(d)] obtaind with th lasr having th highr valu LOI gain ( G ). Numrial rsults ar in bold whil th analytial rsults [Eq.()] ar writtn in italis btwn parnthss. In th numrial rsults th nois is omposd of th trans flutuation plus th lasr quantum nois whil in th analytial rsults, th nois orrsponds only to th lasr quantum nois. 5 Now, if th valu of th LOI gain ( G 8 0 ) is inrasd, by inrasing th lasr rspons tim ( 330 µs ), on an orv [ompar igs. 5() and 5(a)] that th MC is highr but that th SN is lowr with th lasr having th highst LOI gain (i.. th longst rspons tim). h dgradation of th SN oms from th high valu of th trans LOI signal [s Eq. (3)], haratrizd on ig. 5() by trans dynami flutuations as high as th th lasr quantum nois flutuations. h dgradation of th LOI SN also oms from th saturation of th LOI signal of this ovrsnsitiv lasr 9

30 ( sat ). h saturation fft is also visibl in abl 3 whr for both SN and MC, th numrial rsults [obtaind from ig. 5()] ar always lowr than th analytial ons, obtaind from Eq. (5) for th MC and from Eq. () for th alulation of th SN [4]. At this po on an also noti that du to th vry high valu of th trans LOI signal (linkd to th slow rspons tim of th lasr: 0.06 ) and du to th saturation of th LOI 5 signal (indud by th high valu of th LOI gain: G 8 0 ), th LOI signal is hardly prturbd. In ths onditions, th orvation at th rlaxation frquny, of a LOI SN highr than th LOI shot nois limit [s Eq. () and ig. ()] is unfortunatly unobtainabl (i.. xprimntally unorvabl) in th imaging ondition [5]. o dras th trans dynamis ffts and to avoid th lasr dynami saturation orvd on ig. 5(), on nds to work far away from th rsonan frquny. If w now ompar igs. 5() and 5(d), on an orv that for. 5 th MC is lowr but that th LOI SN has inrasd (th pyramid is roughly distinguishabl). In ig. 5(d) on an also orv that th SN is now limitd by th lasr quantum nois (irls flutuations) whih is highr than th LOI trans flutuations (solid urv flutuations). In agrmnt with Eq. (5), on an also orv on ig. 5(d) that th trans dynamis fft is muh mor important for a drasing stp (,, ) than for an inrasing stp (,, ) if w look at th solid urv. If w now look at th urv with th irl, on an also orvd that this dissymmtri fft is ompltly hid by th lasr quantum nois. 30

31 inally, th omparison of igs 5(b) and 5(d) shows that far away from th rsonan, th two lasrs hav approximatly th sam gain ( G (.5 ) 0 3 ) and thrfor th sam MC [Eq. (5)]. On an also larly orv that for th sam gration tim ( 0 µs ), th t LOI imag (th t SN) is larly obtaind whn using th lasr with th lowst rspons tim allowing th following ondition ( ) and thrfor a shot nois limitd LOI dttion [s Eq. (3)]. inally for fast imaging, w nd a tim valu of as short as possibl and to b shot nois limitd w nd to us a lasr with a fast rspons tim ( ). or LOI imaging, th t lasr is thrfor a lass-b lasr with th shortst possibl valu of allowing th us of rlativly short gration tim. In pratial LOI xprimnts, to b shot nois limitd, th lasr quantum nois nds to b just abov th dttor nois. So finally, this is th dttor nois lvl whih dtrmins th lowst possibl valu of th LOI gain [8,9] and thrfor th shortst possibl valu of th lasr rspons tim. or this partiular tim, th lasr usd in th LOI xprimnt is optimizd and allows to obtain imag as fast as possibl with a shot nois limitd dttion. In a prvious papr [8], w hav shown that for a dttion nois lvl haratrizd by a nois quivalnt powr: NEP ( W Hz ). h optimum valu for l LOI gain is givn by: G opt ( ) NEP h (34) r opt p whih allows to dtrmin th optimum valu of th lasr rspons tim : 3

32 NEP h (35) r, opt opt p or xampl, for a lasr with an put powr 7 P 60 mw ( p 3. 0 photons / s at 064 nm ), a avity damping rat s and for a stup with a bam splittr 9 rfltivity 0. 5 and a nois quivalnt powr NEP 6 0 W Hz, on obtains: G ( ) 57 and finally 3 ns, opt. opt hrfor w an tak: µs, as th minimum aquisition tim ompatibl, opt with th shot-nois ondition ( ). o b ompatibl with our initial hypothsis ( ), w also tak a fator 0 and th lasr rlaxation frquny nds to as high as: 0 40 MHz, opt. inally, for high spd imaging ombind to shot-nois limitd dttion, a lasr diod with:, and ns G 5 0 ( 5 0 s, s, r= ) sms to b an rsting 5 9 andidat ompard to mirohip lasr with 00 µs and G 5 0 ( 5 0 s, s, r=). Howvr th us of a lasr diod with a rlaxation frquny. 5 GHz (ompar to 800 k Hz for a solid stat mirohip lasr) rquirs th us an ltro-optis modulator to gnrat th frquny shift (instad of an aousto optis modulator) and abov all rquirs a rapid ltroni dttion (with a gigahrtz bandwidth) whih is thnially muh mor ompliatd to arry than an ltroni stup with a mgahrtz bandwidth. 3

33 Evn if a lasr diod sms to b an rsting andidat for LOI xprimnts, th thortial and xprimntal study nds to b don to vrify this possibility. Indd th lasr rat quations usd in this papr ar not ompltly orrt to dsrib th dynamial bhavior of a lasr diod. Indd, our modling dosn t tak o aount of th phas amplitud oupling of th lasr ltri fild (i.. th Hnry fator) ourring insid th lasr avity of a lasr diod and also dosn t tak o aount of th optial fdbak tim dlay whih annot b ngltd for a lasr diod with a lasr rlaxation frquny in th gigahrtz rang. Indd, for GHz and for an optial fdbak tim dlay 0 ns (whih orrspondst to lasr-targt distan of.5m), on obtaind 0. hrfor vn if th lasr diod sms to b a good andidat, th omparison btwn a lasr diod and a solid-stat lasr for autodyn rfromtry annot b mad so dirtly and thrfor nds to b mad arfully. 5. CONCLUSION In a LOI stup, th bating btwn th rfrn bam and th signal bam taks pla insid th lasr avity and thrfor th lasr fulfills simultanously th rols of mittr and dttor of photons. In ths onditions, th lasr rlaxation osillations play a lading rol both in th lasr quantum nois whih dtrmins th SN and in th lasr trans dynamis whih dtrmins th rspons tim of th LOI stup. In th prsnt study, w hav thortially ompard th stationary LOI SN and th LOI rspons tim of two lasrs having th sam put powr, th sam rlaxation frquny, but having two diffrnt valus of th LOI gain indud by two diffrnt valus of th lasr rspons tim. 33

34 irstly, w hav dtrmind that th stationary LOI SN is frquny indpndnt and abov all shot nois limitd whn th rspons tim of th lasr is shortr than th lok-in gration tim ( ). Invrsly, for a slow rspons tim lasr ( ), th stationary LOI SN is frquny dpndnt and is highr than th LOI shot-nois limit at th lasr rlaxation frquny and smallr than th LOI shot-nois limit far away from th lasr rlaxation frquny. Sondly, w hav shown that th trans LOI signal an b ngltd ompard to th stationary LOI signal ithr by using a lasr with a fast rspons tim ( ) or by working with a slow rspons tim lasr ( ), if th frquny shift is far away from th rlaxation frquny. hrfor to obtain a shot nois limitd dttion with any prturbation from th lasr trans dynamis w nd to work undr th ondition:. hrough this whol study, w hav numrially onfirmd that for a fixd gration tim ( ), th t LOI imags (imags with th t SN) ar always obtaind whn using th lasr with th lowst LOI gain, (i.. th shortst lasr tim rspons ) and that th dttion is shot nois limitd if th following ondition: is satisfid. inally for fast imaging, w nd a tim valu of as short as possibl whras to b shot nois limitd w nd to us a lasr with a fast rspons tim ( ). or LOI imaging th t lasr is thrfor a lass-b lasr with th shortst possibl valu of rlativly short gration tim, allowing th us of. hrfor, for high spd imaging ombind with a shotnois limitd dttion, a lasr diod with a vry short rspons tim (in th nanosond rang) and a vry high valu of th rlaxation frquny (in th gigahrtz rang) sms to b an 34

35 rsting andidat ompard to mirohip lasr with a rspons tim of svral tns of mirosond and a rlaxation frquny in th mgahrtz rang. 35

36 EEENCES.. Yoshizawa, ditor, Handbook of optial mtrology: Prinipls and Appliations (CC Prss, 009).. K. Otsuka, Effts of xtrnal prturbations on LiNdP 4 0 Lasrs, IEEE J. Quantum Eltron., QE-5, (979). 3. K. Otsuka, Slf-Mixing hin-sli Solids-Stat Lasr Mtrology, Snsors, (0). 4. K. Otsuka, Highly snsitiv masurmnt of Dopplr-shift with a mirohip solid-stat lasr, Jpn. J. Appl. Phys. 3, L546 L548 (99). 5. S. Okamoto, H. akda, and. Kannari, Ultrahighly snsitiv lasr-dopplr vloity mtr with a diod-pumpd Nd:YVO4 mirohip lasr, v. Si. Instrum. 66, (995). 6.. Kawai, Y. Asakawa, K. Otsuka, Ultrahigh-Snsitivity Slf-Mixing Lasr Dopplr Vloimtry with Lasr-Diod-Pumpd Mirohip LiNdP 4 O Lasrs, IEEE Photonis hnology Ltt., (999). 7. S. Suddo,. Ohtomo, Y. akahasvhi,. Oishi, K. Otsuka, Dtrmination of vloity of slf-mobil phytoplankton using a slf thin-sli solid stat lasr, Appl. Opt. 48, (009). 8. K. Otsuka, K. Ab, J.Y. Ko, and.s. Lim, al-tim nanomtr vibration masurmnt with slf-mixing mirohip solid-stat lasr, Opt. Ltt. 7, (00). 9. V. Muzt, E.Laot, O. Hugon, Y. Gaillard, Exprimntal omparison of sharography and lasr optial fdbak imaging for rak dttion in onrt struturs, Pro. SPIE 5856, (005). 36

37 0. E. Laot, and O. Hugon, Phas-snsitiv lasr dttion by frquny-shiftd optial fdbak, Phys. v. A 70, (004).. H. Gills, S. Girard, M. Laroh, and A. Blaroui, Nar-fild amplitud and phas masurmnts using htrodyn optial fdbak on solid-stat lasrs, Opt. Ltt. 33, -3 (008).. S. Blaiz, B. Bérnguir, I. Stéfanon, A. Bruyant, G. Lrondl, P. oyr, O. Hugon, O. Jaquin, and E. Laot, Phas snsitiv optial nar-fild mapping using frquny-shiftd lasr optial fdbak rfromtry, Opt. Exprss 6, (008). 3. E. Laot,. Day, and. Stokl, Lasr optial fdbak tomography, Opt. Ltt. 4, (999). 4. A. Witomski, E. Laot, O. Hugon, and O. Jaquin, Synthti aprtur lasr optial fdbak imaging using galvanomtri sanning, Opt. Ltt. 3, (006). 5. O. Hugon, I.A. Paun, C. iard, B. van dr Sandn, E. Laot, O. Jaquin, A. Witomski, Cll imaging by ohrnt baksattring mirosopy using frquny shiftd optial fdbak in a mirohip lasr, Ultramirosopy 08, (008). 6. O. Hugon,. Joud, E. Laot, O. Jaquin, H. Guillt d Chatllus, Cohrnt mirosopy by lasr optial fdbak imaging (LOI) thniqu, Ultramirosopy (0) doi: 0.06/j.ultrami E. Laot,. Day, and. Stokl, Cohrnt lasr dttion by frquny-shiftd optial fdbak, Phys. v. A 64, (00). 8. E. Laot, O. Jaquin, G. oussly, O. Hugon, H. Guillt d Chatllus, Comparativ study of autodyn and htrodyn lasr rfromtry for imaging, J. Opt. So. Am. A 7, (00). 37

38 9. O. Jaquin, E. Laot, W. Glastr, O. Hugon, H. Guillt d Chatllus, Expérimtal omparison of autodyn and htrodyn lasr rfromtry using an Nd:YVO4 mirohip lasr, J. Opt. So. Am. A 8, (0). 0. J.J. Zaykowski and A. Mooradian, Singl frquny mirohip Nd lasrs, Opt. Ltt., 4-6 (989).. M.I. Kolobov, L. Davidovih, E. Giaobino, and C. abr, ol of pumping statistis and dynamis of atomi polarization in quantum flutuations of lasr sours, Phys. v. A 47, (993).. A. Bramati, J.P. Hrmir, V. Jost, E. Giaobino, L. ulbrt, E. Molva, and J.J. Aubrt, Effts of pump flutuations on nsity nois of Nd:YVO4 mirohip lasrs, Eur. Phys. J. D. 6, 53-5 (999). 3. Y.I. Khanin, Prinipls of lasr dynamis, (Elsvir, 995). 4. h saturation fft orvd numrially an t b obtaind analytially, baus Eq. () was obtaind aftr a linarization of th st of Eqs. (), i.. far away from th saturation onditions. 5. o orv th SN amplifiation prdits by Eq. (), on ould imagin drasing th LOI gain by drasing th lasr tim rspons, but in this as, th amplifiation fft whih is proportional to lasr tim rspons thn falls to vry small valu hids by th lasr quantum flutuations. 38