STATISTICAL PROPERTIES ANALYSIS OF Er 3+ -DOPED Ti:LiNbO 3 M -MODE STRAIGHT WAVEGUIDE AMPLIFIERS

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1 Journal of Optolctronics and Advancd Matrials Vol. 6 No. March 4 p STATISTICAL PROPERTIES ANALYSIS OF Er 3+ -DOPED Ti:LiNbO 3 M -MODE STRAIGHT WAVEGUIDE AMPLIFIERS N. N. Puscas * Physics Dpartmnt Univrsity ''Polithnica'' of Bucharst Splaiul Indpndnti Bucharst Romania W rport a thortical analysis of som statistical paramtrs which charactriz th singl and doubl pass Er 3+ -dopd Ti:LiNbO 3 M -mod straight wavguids. For th drivation and th valuation of th Fano factor th statistical fluctuation and th spontanous mission factor w usd a quasi-two-lvl modl in th small gain approximation and th unsaturatd rgim. Th simulations show th volution of ths paramtrs undr various pump rgims and wavguid lngths. Th obtaind rsults can b usd for th dsign of complx rar arth dopd intgratd circuits. (Rcivd May 8 3; accptd Fbruary 7 4) Kywords: Er 3+ :Ti:LiNbO 3 M -mod straight wavguids Fano factor Statistical fluctuation Spontanous mission factor. Introduction Th thortical and xprimntal studis of th gain spctral distribution nois spctral distribution and th output statistics in a systm whr cohrnt and chaotic (thrmal) filds ar suprimposd play an important rol in th fabrication of intgratd amplifirs with low nois and high optical gain [-5]. In th last dcad svral paprs prsnting th bhaviour of Er 3+ -dopd Ti:LiNbO 3 optical wavguids hav bn publishd in th last yars [4-]. Basd on th modls prsntd in [59] and [] in this papr w valuat th output statistics of M mod straight Er 3+ -dopd Ti:LiNbO 3 amplifirs lik: th Fano factor th statistical fluctuation and th spontanous mission factor. This analysis is dtrmind by th fact that oftn in th wavguids is xcitd not only th fundamntal mod but also othr high ordr (M ) mods which influnc not only th output gain nois figur signal-to-nois ratio but also th statistical proprtis of th wavguid. Th simulations of th output statistics of th amplifir ar prformd undr th small gain approximation for z propagating crystals pumpd nar 484 nm for various pumping rgims and wavguid lngths. Th rsults thrby computd ar compard to thos calculatd for singl mod straight wavguids. Th papr is organizd as follows. Sction is dvotd to th basic quations usd to valuat th Fano factor th statistical fluctuation and th spontanous mission factor which charactriz th output statistics of th M -mod wavguids. In Sction 3 th discussion of th computd rsults is prsntd whil th conclusion of this work is outlind in Sction 4.. Basic quations In ordr to valuat th output statistics in straight M -mod Er 3+ -dopd Ti:LiNbO 3 wavguid amplifirs w us a quasi-two-lvl modl undr th small gain approximation in th wavlngth rang.44 µm < λ <.64 µm [59]. * Corrsponding author: pnt@physics.physics.pub.ro

2 64 N. N. Puscas In our modl w considrd that th wavguid is pumpd by a radiation having a Gaussian distribution of th fild using an optical fibr. For th dtrmination of th optical fild distribution in th wavguid w calculatd first th rfractiv indx profils using th Fick s diffusion law [3]. Aftr that th optical mod filds wr calculatd numrically for both TE and TM polarisation using th ffctiv indx mthod prsntd in paprs [34]. For th singl and th doubl pass configuration of th optical wavguid amplifir having th lngth L th corrsponding quations for th Fano factor and th statistical fluctuation ar []: and f ( L ) f L ( z ) ( L ) = + G( L ) G( z ) () dz = + G ( ) ( ) dz G L n L ( ) ( ) ( ) ( ) ( z ) = G L R L + G L G( z ) L ( ) ( z ) L G( z ) / () L dz + ( ) ( ) ( ) + z G z dz G L (3) ( ) = R ( L) G ( L ) n( ) + L L ( ) ( ) ( ) ( ) ( z ) z G z dz + R L G L G( z ) dz / (4) whil th xprssion of th spontanous mission factor for singl pass configuration is givn by: In Eqs. ()-(5) G n sp ( L ) rprsnts th gain function at th z sction G( z ) ( z ) ' ( z ) ' ( ) L =. (5) dz G G z ( z ) = xp [ ( z ) ( z ) ] a z a dz (6) = = ( z ) = σ ( ) N ( x y z) i( x y) a dxdy ( z ) = σ ( ) N ( x y z) i( x y)dxdy N N ar th populations of th ground and th xcitd lvls σ ( ) σ ( ) z mission cross sctions rspctivly and ( ) a A A a ar th absorption and n is th photon man valu. In th abov mntiond quations th polarization subscript m has bn omittd for simplicity [5]. P + + Taking into account th rlations btwn th forward P ( z) P ( ) G( z) ( ) = P ( z) G( z) powrs w calculatd th gain ( z) (7) = and backward G for th singl pass (mirror rflctivitis in

3 Statistical proprtis analysis of Er 3+ - dopd Ti:LiNbO 3 M - mod straight wavguid amplifirs 65 z = L RP ( L) = RS ( L) = ) and th doubl pass ( RP ( L) RS ( L) = R optical amplifir. Th xprssions for th output signal gain ar: for th singl pass and P S G + ( L ) = P ( L) / P ( ) S S = ) configuration of th linar (8) ( ) / P ( ) = G ( L ) R( L) S (9) for th doubl pass configurations. In th drivation of Eqs. ()-(5) w considrd that th normalizd mod intnsity profil i ( x y) in th M mod straight Er 3+ -dopd Ti:LiNbO 3 wavguid amplifirs is not uniform ovr its transvrsal sction [] and also that th input signals ar charactrizd by Poissons statistics n n( ) n( ) (cohrnt light) P ( ) = for which is wll known that σ ( ) = n( ) n. In linar rgim n! stimulatd mission by itslf dtrmins changs in th light statistics in addition to thos introducd by spontanous mission which ar charactrizd by Bos-Einstin statistics. In Eqs. ()-(3) th factor + G dz L ( ) ( z ) L G( z ) / charactrizs th dviation of th output statistics from Poisson. It is wll known that for Poisson statistics th Fano factor is = Bos Einstin statistics = n + / f and = ( + / n ) [] [5]. / / f and = ( n ) = ( n ) whil for 3. Simulation rsults and discussion Solving th systm of coupld first ordr diffrntial quations for th populations and th optical powrs (pump signal and amplifid spontanous mission ASE) by numrical mthods (Rung-Kutta formula-4 th ordr 4 stags) with an itrativ procdur w obtaind th gain for singl and doubl pass configuration and th Fano factor th statistical fluctuation and th spontanous mission factor (Eqs. ()-(5)) [5] []. In our simulations th spontanous mission spctrum is dividd into slots which corrsponds to a wavlngth rsolution λ = nm in th rgion nm. In our modl th lowst four ordr mod intnsity profils i ( x y) for both TE and TM polarisations ar calculatd numrically with th mthod dscribd in papr [] whil th dopant 6 3 distribution d ( x y) (with a surfac concntration of about m a diffusion dpth of µ m and of 5. µ m in width (dfind at /) is approximatd by an rfc but also by Gaussian or constant functions in dpth (lablld and 3 rspctivly) and a Gaussian function in width. Ths diffrnt Er concntration profils corrspond to various diffusion conditions (non complt or complt diffusion or doping during th crystal growth). Th computr simulations for th valuation of th statistical proprtis of Er 3+ -dopd LiNbO 3 wavguid amplifirs hav bn prformd using th paramtrs found in th litratur [9]. W assumd a 484 nm pump and a signal at λ =53 nm having µ W input powr. In computing th profil of th optical mods w usd th following paramtrs charactristic of Ti wavguids: 7 m for wavguid width. m for Ti thicknss hours for diffusion tim. In our simulations w usd for TE and TM pump (p) and signal (s) absorption (a) and mission () cross sctions th valus prsntd in paprs [9] and []. Th wavguid lngth is R = assumd to b L = 5.4 cm. whn not spcifid; th pump and signal input rflctivity always ( ) whil th output is R ( L) = or ( L) =. 98 R for th singl and doubl pass configuration rspctivly.

4 66 N. N. Puscas Scattring losss α =3.7 dbm for TE α =4.8 dbm for TM and spontanous mission liftim τ =.6 ms hav bn assumd. W limit our prsntation to th TE pump and signal polarization; similar rsults can b obtaind for th othr cass. Psignal ( z) Th pak valus of th small signal gain G ( z) = ln for th doubl pass Psignal ( ) configuration G ( ) for th input pump powr of mw (high pump rgim) at λ =53 nm for th rfc (.55 db) Gaussian (5.57 db) and constant (8.3 db) dopant profils in th cas of M -mod opration ar smallr than in th singl on of about 4.5 db for Gaussian profil in dpth [5]. Th pak valus of th nois figur for th rfc (5.58 db) Gaussian (6.3 db) and constant (6.4) dopant profils in th cas of M -mod opration ar gratr than thos corrsponding to th singl on of about 5.3 db for Gaussian profil in dpth [5]. Fig. shows th wavlngth dpndnc of th Fano factor f ( λ) for th doubl pass P = mw. As can b sn th bhaviour of th configuration (Eq. (3)) for input pump powr ( ) Fano factor is dtrmind also in this cas by th absorption and mission cross sctions of Er 3+ - dopd LiNbO 3 [59]. Th valus of th Fano factor for Gaussian (97.45) and constant (67.55) dopant profils in th cas of M -mod opration ar gratr than thos corrsponding to th singl mod opration which is about 9 for Gaussian profil in dpth but that for rfc profil (3.5) is smallr rspctivly [5] Fano Factor W avl ngt h (m ) Fig.. Spctral bhaviour of th Fano factor in th doubl pass configuration for input pump powr: P()= mw in th cas of rfc (curv ) Gaussian (curv ) and constant (curv 3) dopant profils in dpth TE polarization and L=5.4 cm for th wavguid lngth. x -6 Th spctral bhaviour of th statistical fluctuation in doubl pass configuration (Eq. (4)) is prsntd in Fig. whil Fig. 3 shows th spontanous mission factor spctra in singl pass configuration (Eq. (5)). Th valus of th last two paramtrs ar gratr in th M -mod opration than in th singl on [5]. For a high input pump powr th spontanous mission factor is lss than ovr most of th 6-nm spctral rang considrd in th figur (i.. for Gaussian profil of th dopant in th spctral rangs.45 µ m.48 µ m and.6 µ m.65 µ m which corrspond to th amplification of an quivalnt input nois of on photon pr unit frquncy.

5 Statistical proprtis analysis of Er 3+ - dopd Ti:LiNbO 3 M - mod straight wavguid amplifirs 67 Fig.. Th spctral dpndnc of th statistical fluctuation as in Fig.. Th M -mod opration in comparison with th singl on dtrmins th diminution of th gain and th nhancmnt of th nois figur bcaus th ovrlap intgral btwn th population of th xcitd lvl and th normalizd intnsity fild profil is smallr in th cas of M -mod opration than in th cas of th singl on (i... tims in th cas of Gaussian profil of th dopant for a wavguid having 5.4 cm lngth and for an input pump powr of mw). Also th M -mod opration affcts th output statistics diminishing th cohrnt proprtis of th light. Fig. 3. Th spontanous mission factor spctra in singl pass configuration for th input pump powr of mw in th cas of rfc (curv ) Gaussian (curv ) and constant (curv 3) dopant profils in dpth TE polarization and L = 5.4 cm for th wavguid lngth. In Figs. 4 a) b) 5 a) b) th dpndncs of th Fano factor (a) and th statistical fluctuation (b) in doubl pass configuration TE polarization for a signal at λ =53 nm vrsus th wavguid P = mw in th cas of rfc (curv ) Gaussian (curv ) and lngth for input pump powr ( )

6 68 N. N. Puscas constant (curv 3) dopant profils in dpth (Fig. 4) and vrsus th pump powr rspctivly for an amplifir lngth L =5.4 cm (Fig. 5) ar prsntd. a) b) Fig. 4 a) b). Th Fano factor a) and th statistical fluctuation b) vs th wavguid lngth. Th Fano factor incrass both th incrasing of th wavguid lngth and th pump powr (Figs. 4 5 a)) whil th statistical fluctuation incrass with th wavguid lngth and dcrass whn th pump powr is augmntd (Figs. 4 5 b)). Thrfor it sms that th amplifir lngth is rsponsibl for th rduction of th Poissonian aspct of th amplifid light. This is not th cas for shortr wavguid lngths but highr pumping lvls for which th statistical proprtis of th light ar roughly maintaind. A consqunc concrns th dsign of complx structurs whn th cohrnc of th amplifid light must b consrvd. If th miniaturization of th intgratd dvics hav to b considrd it is prfrabl to choos short lngths and high lvl pumpings rathr than long arms and low pumping rgims. 5 Fano Factor P um p powr (m W ) a) b) Fig. 5 a). Th Fano factor a) and th statistical fluctuation b) vs th pump powr.

7 Statistical proprtis analysis of Er 3+ - dopd Ti:LiNbO 3 M - mod straight wavguid amplifirs Conclusions In th small gain approximation an original analysis of th output nois statistical proprtis of a M -mod Er 3+ -dopd LiNbO 3 straight wavguid amplifir has bn prsntd. Th simulations concrn th Fano factor th statistical fluctuation and spontanous mission factor. Th study dmonstrats that in calculating th Fano factor in th singl and doubl pass configurations of th amplifir th Poissonian photon statistics ar maintaind for pump powrs lowr than mw and wavguid lngths smallr than 5 cm. Ths paramtrs ar of importanc whn th cohrnc of th amplifid light is an issu for th dopd structur undr invstigation. For xampl w hav shown that th cohrnc of th output amplifid signal is mor snsitiv to th wavguid lngth than to th pumping powr. Th thortical rsults of this simulation charactriz th Er 3+ -dopd LiNbO 3 wavguid amplifir from th point of viw of nois statistical proprtis and can b usd in th bttr undrstanding of th amplification procss. Rfrncs [] E. Dsurvir Erbium-Dopd Fibr Amplifirs Nw York: Wily (994). [] P. Diamnt M. C. Tich IEEE J. Quant. Elctron (99). [3] A. Yariv Optics Lttrs 5 64 (99). [4] E. L. Goldstin M. C. Tich IEEE J. Quant. Elctron (989). [5] N. N. Puscas D. Scarano R. Girardi I. Montrosst Opt. and Quant. Elctronics (997). [6] I. I. Mansour F. Caccaval Applid Optics (996). [7] M. Majd B. Schüpprt K. Ptrmann IEEE Photonics Tchnology Lttrs 5 86 (993). [8] R. Tavlykav K. Kücklhaus E. Vogs J. of Optical Communications 5 7 (994). [9] M. Dinand W. Sohlr IEEE J. Quant. Elctron 3 67 (994). [] N. N. Puscas J. Optolctron. Adv. Matr. 4 9 (). [] D. Marcus Light Transmission Optics Nw York: Van Nostrand (97). [] I. Turk I. Martink R. Stránsky Optical Enginring (). [3] N. N. Puscas N. B. Wacogn A. Ducariu B. Grapp Opt. and Quant. Elctronics 3 (). [4] G. B. Hockr W. K. Burns Applid Optics 6 3 (977).