Mathematical Series for Analysis of Detuned Active Mode Locked Lasers

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1 Mathematical Series for Aalysis of Detued Active Mode Locked Lasers Huy Quoc Lam ac, P. Shum a, Le Nguye Bih b, Y. D. Gog c, Sogia Fu a a Network Techology Research etre, Nayag Techological Uiversity, #4-,5 Nayag Drive,Sigapore huylq@pmail.tu.edu.sg ; EPShum@tu.edu.sg b Electrical ad Electroic Egieerig Departmet, Moash Uiversity, Australia. le.guye.bih@eg.moash.edu.au c Istitute for Ifoomm Research, Heg Mui Keg Terrace, Sigapore gogyd@ir.a-star.edu.sg ABSTRAT We propose a series method to aalyze detued active mode-locked lasers. Usig this method, we ca aalyze ot oly oiseless lasers but also lasers modelled with amplified spotaeous emissio (ASE) oise. Moreover, the method also helps to study the trasiet evolutio of the pulse iside the cavity, which caot be doe usig classical aalysis. Keywords: laser, fiber laser, mode locked laser, detue, series, aalysis. INTRODUTION Geeratio of ultra-short optical pulses with multiple gigabits repetitio rate is critical for ultra-high bit rate optical commuicatios, particularly for the ext geeratio of terabits/sec, optical fiber systems. I additio, the field of optical packet switchig has gaied recogitio i recet years ad requires very short ad high peak power pulse geerators to provide all optical switchig. Active mode-locked fiber laser offers a potetial source of such pulse trai ad has bee researched extesively sice its first demostratio []. Steady state operatio of active mode-locked fiber lasers has bee aalyzed usig both time domai ad frequecy domai approach [, 3]. I [3], time domai approach was used. The authors assumed a Gaussia pulse solutio ad made approximatios to the lie shape ad modulatio fuctio to keep the pulse Gaussia. Usig self-cosistet coditio, the authors derived aalytic expressios for the pulsewidth ad badwidth of the mode-locked pulse. I [], the authors aalyzed the evolutio of the pulse i frequecy domai ad derived differetial equatios for forced mode lockig. The equatios had a Hermite-Gaussia solutio ad the mode-locked pulse characteristic was obtaied. It is oted that a cosiusoidal modulatio was used to acquire ijectio lockig. Furthermore, cotiuum spectrum approximatio of discrete frequecy spectrum ad assumptio of dese spectrum were made to derive the equatios. Detue i active mode-locked laser was also studied by the use of self cosistet coditio ad a assumptio of steady state Gaussia pulse shape [4]. The characteristic of the steady state Gaussia pulse whe the laser is detued has bee derived ad the shiftig of pulse positio has bee cofirmed. However, the papers show oly steady state solutio ad discuss othig about the trasiet build up process of the pulse iside the cavity. Moreover, Asychroous Spotaeous Emissio (ASE) oise, which caot be avoided i the gai medium of the laser, has bee igored i the models. I this paper, we employ a mathematical series istead of the selfcosistece coditio to trace the evolutio of a white oise travelig iside the cavity. The shape of the oise is featured by series S, Q, R, a, b, c ad r, which ca be easily calculated from the laser parameters. Usig this method, we observe that the oise is gradually shaped to steady state pulse uder the shapig effect of the filter ad modulator for every roud trip. Equatios for determiig the steady state pulse parameters are also derived usig these mathematical series ad show cosistet results with the above papers. Passive ompoets ad Fiber-based Devices IV, edited by Mig-Ju Li, Jiapig he, Satoki Kawaishi, Ia Hugh White, Proc. of SPIE Vol. 678, 6783D, (7) X/7/$8 doi:.7/ Proc. of SPIE Vol D- Dowloaded From: o /9/6 Terms of Use:

2 . SERIES ANALYSIS RbIE V E \JU34L Fig Active mode-locked laser model with ASE oise icluded The laser model cosists of a i-lie optical amplifier with a saturated gai of G, a additioal white oise ρ ASE with variace of σ N, a badpass optical filter ad a itesity modulator. The filter trasmissio fuctio T(ω) ad the Mach-Zehder modulator trasmissio fuctio M(t) ca be writte as ( ) exp( βω ) T ω =, () π M () t = cos ( b) mcos( ωmt) 4, () where β = l/ ω, ω is the filter 3 db badwidth, b accout for the bias poit of the modulator, b =.5 for the quadrature bias poit, m is modulatio idex, ω m is modulatio frequecy. For coveiece, M(t) ca be approximated by π ( ) exp( δ mt ) M t =Γ (3) i which δ m = mω m /4, Г accouts for the total loss of the laser cavity icludig isertio loss, modulator bias loss ad coupler loss. Startig from the white oise, we trace the evolutio of the oise for every roud trip. The oise after roud trips is βω jωtd δmt { } ( ) ( ) ρ ω = G F F ρ ω e e Γe (4) where t d = f m /Nf R is delay due to detuig f m of the modulatio frequecy f m from the N th harmoic of the cavity fudametal frequecy f R. After some mathematical simplificatio steps, we obtai: R S S t βω d Q jω t d 4β Q Q = N Γ G e e e (5) Q ( ) ( ) ρ ω σ where S =, Q = x+, R = x, x= 4 βδm S = S + Q, Q = xs + Q, R = xs + R The total sigal i the rig is sum of the st, d,, th roud trip oise ad is give as η ( ω) = ρ ( ω) (7) = (6) Proc. of SPIE Vol D- Dowloaded From: o /9/6 Terms of Use:

3 3. NOISELESS AMPLIFIER The sigal is formed from a white oise travelig the loop ifiitely without ay ASE added for every roud trip. Therefore, the steady state sigal is ( ) = lim ( ) η ω ρ ω e = e lim S S βω jω lim t Q d Q e βω jω td x+ 4x+ x x+ 4x+ x Usig iverse Fourier trasform, we obtai the Gaussia pulse shape i time domai as e (8) where η () t e = e x+ 4x+ x t td 8β x+ 4x+ x td α t (9) x + 4x+ x α = () 8β is the Gaussia pulse parameter. The Full Width at Half Maximum (FWHM) is l τ FWHM = () α The absece of t d i the pulse width equatio shows that the pulse width does ot chage with detuig. It is the same as that of exact tuig. However, whe the modulatio frequecy is detued, the pulse positio does ot pass the modulator at the peak of the modulatio trasfer fuctio but shifted by a amout of td ts =. () The power of the sigal after roud trips ( ΓG) + R σ td N βq ρ( ω) ω πβ SQ P = d e π = (3) 8 The gai required for lasig is determied usig the fiite power coditio ad is give as + G = GI = e β (4) Γ If the laser is mode locked without detue, substitutig t d = ito (9) ad (4), we obtai the pulse positio ad the required gai as t s =, ad G = G I = t d 4 +, respectively. This is cosistet with the results reported i [-4]. Γ Proc. of SPIE Vol D-3 Dowloaded From: o /9/6 Terms of Use:

4 a 5 5 b 5 5 c P x Roud trip Fig Evolutio of a, b, c r, P series i a mode-locked laser with frequecy detue of δf m =6x -6, oiseless amplifier It ca be see from (8) that the pulsewidth ad pulse positio are determied by the series a = S /Q. By followig the evolutio of a for every roud trip, we ca obtai the trasiet build up ad shapig of the pulse i the cavity. The evolutio of a i a mode-locked laser is show i Fig. The a icreases rapidly durig the first roud trips ad reaches its steady state value after about 47 rouds. This idicates that the pulse is strogly shaped to Gaussia shape due to strog filterig ad modulatio effect whe the pulse is differet to its steady state. The chage of a also idicates that the pulse positio is shifted from the ceter, t =, to its steady state value t s specified i (). Whe the pulse obtais its steady state, the pulse arrowig caused by the modulatio balaces out the pulse wideig effect of the filter. Therefore, the pulsewidth is uchaged. I additio, the modulatio also pushes (pulls) the pulse whe the pulse passes through the modulator at t s delayed (advaced) from the trasmissio peak. This pushig (pullig) effect balaces out the propagatio delay (advace) of the pulse after oe roud trip due to detue. Therefore, the pulse positio is kept uchaged. Usig (4), we obtai the power of the sigal after roud trip as Where ad σ N P = bc (5) 8πβ b ( + ) = (6) SQ c ( R / Q ) t d = exp β (7) Proc. of SPIE Vol D-4 Dowloaded From: o /9/6 Terms of Use:

5 alculatig the b ad c series, we obtai the chage of the sigal power after every roud trip. The chages of b, c, P series are plotted i Fig. The sigal power keeps uchaged after obtaiig its steady state value. 4. AMPLIFIER WITH ASE NOISE I practice, the oise is cotiuously added to the sigal for every roud trip passig the gai medium. This reduces the gai by a factor of r compared to the oiseless amplifier case ad hece the gai ad power ow become Where t d 4 + β G = GIr = e r Γ (8) σ N P = bcr 8πβ (9) r = r () ompared to (5), the idividual sigal power P i (9) has a additioal term r which reduces the power for every roud trip. It ca be see from Fig 3a that the sigal evolutio is similar to that of oiseless oe except the additioal series r. The sigal also quickly obtais its steady state after about roud trips. However, the log term evolutio of the sigal i Fig 3b shows that P is ot costat but expoetially decreases to zero due to the gai reduced factor r. a (a) 5 5 a (b) 5 b 5 5 b 5 c c 5 5 r P x Roud trip r P.5 5 x -6 5 Roud trip (x 4 ) Fig 3 Evolutio of a, b, c r, P series i a mode-locked laser with frequecy detue of δf m =6x -6 Proc. of SPIE Vol D-5 Dowloaded From: o /9/6 Terms of Use:

6 The trasiet build up of the pulse iside the cavity ca be explaied as followig. For every roud trip, a small amout ASE oise from the gai medium is added to the laser sigal (total sigal). The added oise is gradually shaped to the steady state Gaussia pulse sigal due to the filterig ad modulatio effect. After several shapig roud trips, it reaches the steady state shape of the sigal ad cotributes to the formig of the steady state Gaussia pulse sigal. However its power is ot kept costat but expoetially decreased ad therefore the total sigal power does ot icrease to ifiite although there is a additioal ASE oise power added for every roud trip. Fig 4 plots the evolutio of the series for a larger detue value, δf m = 8x -6. It is observed that the value of r is smaller tha that of δf m = 8x -6 ad hece the idividual sigal decreases more rapidly after reachig the steady state. This results i a decrease i the steady state Gaussia pulse sigal while the ewly added ASE oise icreases. Therefore, detue decreases the sigal to oise ratio of the laser. If the frequecy is detued so much that sigal power decreases to zero before reachig it steady state, the steady state Gaussia pulse is o loger obtai. This explais why the modulatio frequecy must be kept i a small lockig rage aroud the harmoic of the fudametal frequecy of the laser. a (a) 5 5 a (b) 5 b 5 5 b 5 c c 4 5 r P x Roud trip r P x -5 5 Roud trip (x 4 ) Fig 4 Evolutio of a, b, c r, P series i a mode-locked laser with frequecy detue of δf m =8x SIMULATION RESULTS Fig 5 shows the simulatio build up process of the pulse iside the cavity whe the modulatio frequecy is detued. Iitially, the pulse positio is shifted away from the trasmissio peak for every roudtrip due to detuig. After about roud trips, the pulse positio iclies to a costat which is the steady state value t s predicted by the theory. At this positio, the retimig effect of the modulatio completely compesates for the delay due to detuig ad hece the pulse positio is fixed or every roud trip afterward. Proc. of SPIE Vol D-6 Dowloaded From: o /9/6 Terms of Use:

7 w= =.3.. -, (a) H - - = w -. G I (o)/ - tju PA) lt*) U T (ps) -. I U bu IUU IbU U Roud trip Fig 5 Pulse build up i a active mode-locked laser with a detuig factor of 5x -6 ( f m = 5 khz) Pulse positio aalytic simulatio -4-4 Frequecy detue x -5 Gai aalytic simulatio -4-4 Frequecy detue x -5 Fig 6 Normalized pulse positio (a) ad gai (b) of the mode-locked laser whe detued The simulatio also cofirms that the pulse width is idepedet of detuig ad has the same value as that whe exact tuig. Variatio of the pulse positio ad the gai are show i Fig 6. Withi the calculated lockig rage, ±δf mmax = ±.3x -5, the simulatio results are i good agreemet with the aalytical results. As predicted i theory, the pulse positio liearly icreases while the required gai expoetially icreases with detuig. Whe the detue frequecy exceeds δf mmax, the theoretical calculatio based o the Gaussia approximatio is o loger valid. The pulse positio does ot icrease liearly with detuig but iclies to a costat of about T m /4. 6. ONLUSIONS Series method for aalyzig detued active mode-locked lasers has bee preseted. The advatage of this method is that it ca be used to aalyze the laser whe the ASE oise is take ito accout. Moreover, ot oly the steady state pulse but also the trasiet process ca be obtaied. By usig this series aalysis method, we obtai the same results with that usig classical aalysis method whe oiseless amplifier is used. Simulatio results show cosistecy with that predicted by our method. Proc. of SPIE Vol D-7 Dowloaded From: o /9/6 Terms of Use:

8 AKNOWLEDGEMENT This work is partially supported by the project M47439 of Agecy for Sciece, Techology ad Research (A*STAR), Sigapore, ad Ope Fud of Key Laboratory of Optical ommuicatio ad Lightwave Techologies, Beijig Uiversity of Posts ad Telecommuicatios, Miistry of Educatio, P. R. hia. REFERENES [] L. E. Hargrove, R. L. Fork, ad M. A. Pollack, "Lockig of He-Ne laser modes iduced by sychroous itracavity modulatio," Appl. Phys. Lett., vol. 5, 964. [] H. Haus, "A theory of forced mode lockig," IEEE J. Quatum Electro., vol., pp , 975. [3] D. Kuizega ad A. Siegma, "FM ad AM mode lockig of the homogeeous laser--part I: Theory," IEEE J. Quatum Electro., vol. 6, pp , 97. [4] Y. Li,. Lou, ad Y. Gao, "Theoretical study o a detued AM mode-locked laser," preseted at Lasers ad Electro-Optics Society 999 th Aual Meetig. LEOS '99. IEEE, Sa Fracisco, A, 999. Proc. of SPIE Vol D-8 Dowloaded From: o /9/6 Terms of Use: