Photonic temporal integrator for all-optical computing

Transcription

1 Phoonic emporal inegraor for all-opical compuing Radan Slavík 1,* Yongwoo Park 2, Nicolas Ayoe 3, Serge Douce 3, Tae-Jung Ahn 2,4, Sophie LaRochelle 3, and José Azaña 2 1 Insiue of Phoonics and Elecronics AS CR, v. v. i., Chaberská 57, Praha 8, Czech Republic *Corresponding auhor: slavik@ufe.cz 2. Insiu Naional de la Recherche Scienifique (INRS), Monréal, Québec, Canada, H5A 1K6 3. Cenre d opique, phoonique e laser (COPL), Universié Laval Québec G1V 0A6, Canada 4. Now wih Dep. of Phoonic Engineering, Chosun Universiy, 375 Seosuk-dong, Gwangju, Korea Absrac: We repor he firs experimenal realizaion of an all-opical emporal inegraor. The inegraor is implemened using an all-fiber acive (gain-assised) filer based on superimposed fiber Bragg graings made in an Er-Yb co-doped opical fiber ha behaves like an opical capacior. Funcionaliy of his device was esed by inegraing differen opical pulses, wih ime duraion down o 60 ps, and by inegraion of wo consecuive pulses ha had differen relaive phases, separaed by up o 1 ns. The poenial of he developed device for implemening all-opical compuing sysems for solving ordinary differenial equaions was also experimenally esed Opical Sociey of America OCIS codes: ( ) Analog opical signal processing; ( ) All-opical devices; ( ) Fiber opics componens References and links 1. L. Venema, Phoonics Technologies, Naure Insigh 424, No (2003). 2. C. K. Madsen, D. Dragoman, and J. Azaña (ediors), Special Issue on Signal Analysis Tools for Opical Signal Processing, EURASIP J. Appl. Signal Proc. 10, (2005). 3. J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cinconi (ediors), Special Issue on Opical Signal Processing, IEEE/OSA J. Lighwave Technol. 24, (2006). 4. N. Q. Ngo and L. N. Binh, Opical realizaion of Newon-Coes-Based Inegraors for Dark Solion Generaion, IEEE/OSA J. Lighwave Technol. 24, (2006). 5. N. Q. Ngo, Opical inegraor for opical dark-solion deecion and pulse shaping, Appl. Op. 45, (2006). 6. N. Q. Ngo, Design of an opical emporal inegraor based on a phase-shifed fiber Bragg graing in ransmission, Op. Le. 32, (2007). 7. J. Azaña, Proposal of a uniform fiber Bragg graing as an ulrafas all-opical inegraor, Op. Le. 33, 4-6 (2008). 8. M. A. Preciado and M. A. Muriel, Ulrafas all-opical inegraor based on a fiber Bragg graing: proposal and design, Op. Le. 33, (2008). 9. M. T. Hill, H. J. S. Dorren, T. De Vries, X. J. M. Leijens, J. H. Den Besen, B. Smalbrugge, Y.-S. Oel, H. Binsma, G.-D. Khoe, and M. K. Smi, A fas low-power opical memory based on coupled micro-ring lasers, Naure 432, (2004). 10. M. Kulishov and J. Azaña, Long-period fiber graings as ulrafas opical differeniaors, Op. Le. 30, (2005). 11. R. Slavík, Y. Park, M. Kulishov, R. Morandoi, and J. Azaña, "Ulrafas all-opical differeniaors," Op. Express 14, (2006), hp:// 12. P. Yao, F. Zeng, and Q. Wang, Phoonic generaion of Ulra-Wideband signals, J. Lighwave Technol. 25, (2007). 13. J. Xu, X. Zhang, J. Dong, D. Liu, and D. Huang, High-speed all-opical differeniaor based on a semiconducor opical amplifier and an opical filer, Op. Le. 32, (2007). 14. Y. Park, M. Kulishov, R. Slavík, and J. Azaña, Picosecond and sub-picosecond fla-op pulse generaion using uniform long-period fiber graing, Op. Express, 14, (2006), hp:// (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18202

2 15. R. Slavík, L. K. Oxenløwe, M. Galili, H. C. H. Mulvad, Y. Park, J.Azaña, and P. Jeppesen, Demuliplexing of 320 and 640 Gbi/s OTDM daa using ulrashor fla-op pulses, IEEE Phoon. Technol. Le. 19, (2007). 16. F. Li, Y. Park, and J. Azaña, Complee emporal pulse characerizaion based on phase reconsrucion using opical ulrafas differeniaion (PROUD), Op. Le. 32, (2007). 17. F. Li, Y. Park, and J.Azaña, Precise and simple group delay measuremen of dispersive devices based on ulrafas opical differeniaion, in Proc. of OFC/NFOEC 08, Paper OWD5, A. V. Oppenheim, A. S. Willsky, and S. H. Nawab, Signals and Sysems, 2 nd ed. (Prenice Hall, Upper Saddle River, NJ, USA 1996). 19. K. Ogaa, Modern Conrol Engineering, 4 h ed. (Prenice Hall, Upper Saddle River, NJ, USA 2001). 20. G.F. Simmons Differenial Equaions wih Applicaions and Hisorical Noes, 2 nd ed. (McGraw-Hill, New York, USA 1991). 21. C. K. Madsen and J. H. Zhao, Opical Filer Design and Analysis: A Signal Processing Approach (John Wiley & Sons, New York, USA 1999). 22. G. Brochu, S. LaRochelle, and R. Slavík, Modeling and experimenal demonsraion of ulracompac muliwavelengh disribued Fabry-Pero fiber lasers, J. Lighwave Technol. 23, (2005). 23. Y. Barbarin, S. Ananahanasarn, E. A. J. M. Bene, Y. S. Oei, M. K. Smi, and R. Nozel, 1.55 µm range InAs-InP (100) quanum-do Fabry-Pero and ring lasers using narrow deeply eched ridge waveguides, IEEE Phoon. Technol. Le. 18, (2006). 24. R. Kashyap, Fiber Bragg Graings (Academic Press, San Diego, 1999). 25. G. E. Town, K. Sugden, J. A. R. Williams, I. Bennion, and S. B. Poole, Wide-band Fabry-Pero-like filers in opical fiber, IEEE Phoon. Technol. Le. 7, (1995). 1. Inroducion All-opical circuis for compuing, informaion processing, and neworking can overcome he speed limiaions of elecronic-based sysems [1-3]. However, in phoonics, here are sill no equivalens o some fundamenal signal processing devices ha form basic building blocks in elecronic circuis, e.g. inegraor. Thus, he design and demonsraion of hese fundamenal phoonic devices is a necessary firs sep owards he realizaion of all-opical processing circuis. Alhough several schemes for performing phoonic inegraion have been previously proposed [4-8], a main challenge associaed wih heir experimenal implemenaion is ha an advanced ligh-wave sorage (capacior-like) elemen [9] is necessary. A phoonic emporal inegraor is a device ha performs he cumulaive ime inegral of he complex emporal envelope of an inpu arbirary opical waveform [4-8]. Applicaions of he phoonic inegraor are in many fields; many of hem can be found in direc analogy wih applicaions of a phoonic differeniaor, which is is signal processing counerpar. The phoonic emporal differeniaor has been recenly demonsraed [10,11] and has already proved o be very useful in a wide range of applicaions in ulrafas opical pulse processing [11-13], shaping [14,15], and merology [16,17]. Noneheless, he imporance of he opical inegraor would go beyond hese applicaions. For example, in direc analogy wih he elecronic-domain developmens [18,19], a phoonic emporal inegraor is a key elemen for implemening all-opical analog compuing sysems. Analog compuers have been shown o have a far superior performance, e.g. in erms of speed, as compared wih convenional digial compuers for a number of specific compuing asks, paricularly for real-ime solving of scienific and engineering problems ha can be described by differenial equaions [18,20]. I is well known [18,19] ha hese problems can be solved direcly in he analog domain in real ime using a suiable combinaion of inegraors, adders and mulipliers. Because differeniaors have significanly worse performance in erms of high-frequency noise, he use of inegraors is srongly preferred [18,19]. The possibiliy of realizing hese compuaions allopically ranslaes ino a poenial for speeds several orders of magniude higher han wih convenional elecronic sysems [1-3]. This perspecive is paricularly aracive when he soluion is required in real-ime, a speeds far beyond he reach of digial elecronic compuers, for immediae analysis or processing, e.g. in conrol/feedback applicaions requiring operaion in picosecond ime scales. (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18203

3 From basic signal processing heory [18] i follows ha a emporal inegraor can be implemened using a linear filering device wih a emporal impulse response h() proporional o he so-called uni sep funcion u () : h( ) u( ); u() = 0 for < 0, u() = 1 for 0. (1) where is he ime variable. Physically, his requires he use of a srucure capable of soring an incoming ime-varying waveform (e.g. elecric field inensiy) wih an oupu being a coninuous signal proporional o he sum of he oal sored field a each insan of ime. In elecronics, his funcionaliy is provided by a capacior, which sores he elecric charge proporionally o he sum of he incoming elecric field. The inegraed signal is hen direcly proporional o he volage measured a he capacior. The same principle canno be direcly ransferred ino phoonics since complee sopping of phoons would be necessary. In he specral domain, he ransfer funcion H(ω) of an ideal phoonic inegraor is inversely-proporional o he base-band frequency (ω-ω 0 ), i.e. H( ω) =I{ h( ) } 1 j( ω ω0) + π δ( ω ω0), where I is he Fourier ransform operaor, δ is he Dirac dela-funcion, ω is he opical frequency and ω 0 is he carrier frequency of he signal o be processed [4-6]. This implies ha he ransmission should be > 1 in he proximiy of ω 0 and, ideally, i should become infinie a ω 0. The firs suggesions of a phoonic inegraor [4,5] were based on a general feedbackbased phoonic filers, in which a gain elemen placed in he feedback loop made i possible o obain ransmission > 1 in he viciniy of he resonance (in pracice, he ransmission would no reach infiniy a he resonance frequency due o he gain sauraion of he considered gain medium). Alhough represening a significan sep forward, being he firs descripions of a device ha inegraes phoons, hese original proposals did no evaluae any pracical consrains in erms of required level of amplificaion versus achievable processing speed. Subsequenly, hree very ineresing designs ha already included discussions of pracical limiaions and rade-offs were proposed [6-8]. All of hem, however, considered only passive resonan srucures ha do no allow o fulfill he condiion of exac inegraion, as heir ransmission is limied o values 1. This has wo main drawbacks. Firs, he energeic efficiency of he inegraion process is exremely low, for example an energeic efficiency of 0.1% is obained in [6] (for processing error of 2%) and 0.3% in [7]. Secondly, he inegraion of long and narrow-bandwidh pulses suffers from high processing error as he filer ransfer funcion around he resonan frequency ( 1) differs significanly from ha of an ideal inegraor ( ). The firs design [6] is based on a phase-shifed fiber Bragg graing (FBG), which physically forms a resonan caviy in which ligh resonaes beween wo FBG reflecors ha have relaive π-phase shif. The phase shif ensures ha he consrucive resonance condiion occurs a he cener of he FBG reflecion bandwidh. To obain an accepable processing error, numerical analysis ha did no consider propagaion loss [6] showed ha each FBG reflecor should provide a refleciviy of a leas 99.99%. Obviously, his would be exremely challenging o achieve in pracice, even considering oher echnologies of preparing he wo-reflecor resonaor, e.g., Fabry-Pero (FP) resonaor. The second suggesed configuraion [7] is formed by a single uniform FBG operaing in reflecion ha provides a square-like impulse emporal response. A feaure of his configuraion ha may be disadvanageous for some applicaions is ha i provides he inegraed signal wo imes: an ideal inegraor provides a sep-like impulse response (1) and hus he oupu of his inegraor is he signal inegral (processed by he rising sep of he square-like impulse response) followed by ime-inversed inegral of he ime-invered signal (processed by he falling sep of he square-like impulse response). The las proposed scheme is based on a specially-apodized FBG operaing in reflecion ha provides a decreasing-exponenial emporal impulse response [8]. Alhough his eliminaes he presence of an undesired waveform ouside he device s (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18204

4 operaion ime window presen in he previous scheme, i is expeced o be challenging o fabricae, as he designed graing modulaion deph varies significanly over a very shor lengh, requiring a very high level of phooinduced refracive index change a he very beginning of he FBG. In our work, a emporal inegraor wih noably relaxed fabricaion consrains is suggesed and fabricaed by incorporaing an acive medium ino a resonan caviy configuraion similar o ha described in [6]. The good performance of he prepared inegraors allowed us o demonsrae heir poenial for all-opical compuing applicaions, paricularly for real ime solving of differenial equaions of ineres o a wide variey of engineering and scienific problems [20]. 2. Principle of operaion A schemaic of he idea for implemening he phoonic emporal inegraor is shown in Fig. 1. Le us assume a general FP caviy composed of wo idenical mirrors, each characerized by a field refleciviy r defined as he raio of he refleced and inciden field ampliudes, r 1, separaed by a disance L. The ne gain in he caviy medium, defined as he round-rip field ampliude gain ha excludes loss due o he mirrors, is given by γ wih γ < 1 for loss and γ > 1 for gain. I can be easily proved [21] ha for such FP caviy, he emporal impulse response is, wihin a cerain fracion of he FP free specral range: where k ( 1/ T) ln( r 2 γ ) ( k) u( ) h( ) exp, (2) = and T is he round-rip propagaion ime in he FP caviy ( T = 2Ln c, wih n being he caviy refracive index and c being he speed of ligh in vacuum). Simply, he signal sored in a FP caviy is leaking ou following an exponenial ime variaion. Comparing he impulse response of he FP caviy, Eq. (2), wih ha of an ideal inegraor, Eq. (1), we infer ha he FP caviy would behave as a emporal inegraor when 2 r γ = 1. This condiion, known also as he lasing hreshold condiion [22], means ha he loss associaed wih he reflecions in he FP mirrors are perfecly compensaed for by he ne gain in he caviy. A his condiion, he field of an incoming ulrashor impulse will be sored and subsequenly delivered from he FP caviy wih a consan flow (Fig.1), exacly as required for opical emporal inegraion. I should be noed ha here is a fundamenal limiaion in erms of he fases emporal feaure of he inpu waveform ha can be processed wih a resonan caviy-based opical inegraor; his limiaion is given by he specral range over which FP caviy provides he emporal impulse response given by (2). For pulses shorer han he round-rip propagaion ime in he caviy, T, he signal would be released in he form of discree impulses, emporally spaced by T. In pracice, only emporal feaures longer han 5T will be inegraed wih small-enough processing error [6], i.e. he inegraor processing bandwidh is limied o 1/5 of he inegraor free specral range. (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18205

5 Inpu 980/1550 coupler All-opical inegraor Oupu x () y() = x( τ) dτ τ = Pump in (Er-Yb fiber wih superimposed FBGs) Transfer funcion of he all-opical inegraor Power ransmission, db Baseband frequency, GHz Measured Calculaed Ideal Fig. 1. Concep diagram of he proposed phoonic emporal inegraor. The inegraor is implemened using wo superimposed fiber Bragg graings (acing as a resonan caviy) permanenly phoo-inscribed in an Er-Yb co-doped opical fiber ha provides opical gain. The gain level is conrolled via power of he opical pump (980-nm laser diode). The inse shows he measured (circles) and numerically calculaed (solid, blue curve) inegraor specral ransfer funcion. For comparison, he specral ransfer funcion of an ideal inegraor is also shown (solid, red curve). In addiion of he already-menioned significanly increased energeic efficiency of he inegraion process, he use of an acive filering configuraion provides an imporan advanage wih he possibiliy of adjusing he filering characerisics hrough uning of he opical pumping. This laer feaure is paricularly ineresing for solving differenial equaions as will be discussed in a few examples presened below. A disadvanage of using he gain medium is he presence of sponaneous emission, which generaes noise. 3. Implemenaion and fabricaion Differen echnological approaches could be considered for implemening he proposed inegraor concep [9,22,23]. Our implemenaion is based on opical waveguide (all-fiber) echnology [22]. Here, he wo FP mirrors can be realized using wo FBGs creaed by a periodic change of he refracive index along he direcion of ligh propagaion wihin a single-mode opical fiber [24]. The lengh of he wo FBG-made FP mirrors limis he minimum achievable FP caviy lengh and hus also he caviy round-rip ime T ha deermines he inegraor processing speed. To ge T<10 ps wih corresponding processing bandwidh of ens of GHz, we need a mirror spacing of 1 mm or less when considering a caviy wih refracive index of 1.45 corresponding o silica opical fibers. Such shor FBGs are (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18206

6 difficul o make wih he required level of refleciviy. This limiaion, however, can be overcome by using spaially overlapped (superimposed) FBGs ha are slighly chirped (he graing period is varied along he propagaion axis in he fibre core) [25], forming a disribued resonan caviy wih a lengh no limied by he desired round-rip ime T. Such srucure made in an acive fiber was already repored o reach he laser hreshold condiion [22]. Deailed descripion and heoreical analysis of his srucure can be found in [22]. The wo superimposed chirped FBGs were phooinscribed ino a specialy high-gain (~40 db/m) fiber co-doped wih Er and Yb wih phoosensiive inner cladding (ErYb-302, INO, Canada) loaded wih Deuerium. Graing wriing was done by scanning a chirped phase mask [24] (period chirp of 0.5 nm/cm) wih a laser beam from a frequency-doubled Argon ion cw laser (244 nm and power of 50 mw) [24]. Each FBG was fabricaed by a single phase mask exposure. Beween he wo exposures, he phase mask was moved wih respec o he fiber by he required effecive caviy lengh. An 8-cm long acive fiber secion was spliced wih passive single-mode fibers. We prepared various samples of an opical inegraor design wih he round-rip propagaion ime T = 10 ps ha required a 1-mm shif of he phase mask beween he wo consecuive fiber exposures. The FBGs were ypically 3-4 cm long depending on he used FBG apodizaion profile ha was of anh shape. We analyzed he srucures heoreically using he heoreical model developed in ref. [22] and found ha he 2 FBGs refleciviies necessary o fulfill he lasing hreshold condiion, r γ = 1, were r=99.9% corresponding o a power ransmission of -27 db. Obviously, using fibers wih even higher acive ions doping level (resuling in higher ne round-rip gain γ) would furher reduce he refleciviy needed. For shorer round-rip imes, he ne round-rip gain γ would be smaller, as i is given by he produc of he fiber gain facor and he caviy lengh. Consequenly, he FBGs refleciviies would need o be increased or a fiber wih a higher gain facor should be used. Alernaive echnologies, such as ring resonaors in InP [9] wih diameers down o 10 µm, would allow he realizaion of round-rip imes as fas as abou 0.3 ps corresponding o processing speeds up o 650 GHz. Furhermore hese InP rings have ypical gain around 30 db/mm ha is hree orders of magniude higher han in he used acive fiber. The heoreically expeced and measured specral filering responses for one of he realized emporal inegraors are shown in Fig. 1 ogeher wih he filering ransfer funcion of an ideal inegraor. The specral ransfer funcion was measured using an Opical Vecor Analyzer (OVA from LUNA Technologies, U.S.A.). As he specialy fiber had a slighly ellipical core, he realized componen was slighly birefringen. Thus, he experimen was se in such a way ha he processed ligh propagaed hrough he componen wih is polarizaion aligned along one of he principal axes of he inegraor. Clearly, he resoluion of he OVA (1.5 pm) was insufficien o measure properly he cenral par of he ransmission peak, which had a full widh a half maximum (FWHM), according o he simulaed daa, of 0.2 pm. Despie his fac, we were sill able o measure abou 9 db of amplificaion while he offresonance amplificaion was measured o be 4 db. I is believed ha considerably higher values of he peak amplificaion could have been measured using an insrumen wih sufficienly high specral resoluion (he heoreically-prediced value was 23 db as shown in he inse of Fig. 1). From he specral filering responses shown in Fig. 1, we esimaed ha he heoreical and experimenal bandwidh over which he implemened inegraor can operae was abou 20 GHz, which corresponds o emporal feaures of approximaely 5T = 50 ps. The observable sligh oscillaions in he experimenal daa are believed o be arifacs of he measuremen mehod used for he device characerizaion. 4. Experimenal resuls: inegraor properies To evaluae he performance of he developed inegraor, we carried ou wo basic experimens. The firs one demonsraes he basic inegraion propery, i.e. evaluaion of a ime cumulaive inegral of a simple opical pulse. The second one demonsraes he coheren operaion of he inegraor, i.e., ha he device operaes on boh he ampliude and he phase of he processed signal. In boh experimens, he sligh exponenial decay of he inegraed (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18207

7 signal was caused by pumping slighly below he lasing hreshold condiion ( r 2 γ = 1). The decay ime τ was ypically 5 ns. The inpu power was low enough o preven gain sauraion ha would oherwise affec he gain facor γ and hus break he inegraion condiion of 2 r γ = 1. A polarizaion conroller was placed in fron of he inegraor o eliminae he effec of he sligh phooinduced birefringence of he realized inegraors. The se-up for he firs experimen is shown in Fig. 2. Firs, we prepared individual pulses wih a FWHM ime widh of 140 and 60 ps, respecively. These pulses were generaed by modulaing a 5-mW CW unable laser uned o he inegraor resonance wavelengh hrough a Mach-Zehnder inensiy modulaor. The modulaor was driven by an elecric pulse generaor (Model 3600 Impulse generaor, Picosecond Pulse Lab. Inc., Boulder, CO, U.S.A) wih a repeiion rae of 200 MHz and a pulse widh (FWHM) of 70 ps. To obain 60-ps and 140-ps opical pulses, 35-GHz and 2.5-GHz 3-dB bandwidh modulaors were used, respecively. The repeiion rae was furher reduced o 2 MHz using a square-like gaing from an elecric programmable waveform generaor (Model AWG710B, Tekronix Inc., U.S.A.). In our experimen, he carrier frequency of he inpu signal was se o mach he cenral frequency of he inegraor. However, in pracice, he cenral frequency of he inegraor (resonan caviy) could be uned, e.g., via emperaure conrol. The expeced and measured resuls in erms of power inensiy are shown in Fig. 3. The inpu and oupu signals from he inegraor were capured by a 20-GHz bandwidh phoodiode and viewed wih a sampling oscilloscope. For comparison, we numerically calculaed he ime inegral of he measured inpu pulse fields. The inensiy of he resuling inegraed signal (square of he numerically obained ime inegral) is also given in he same plo showing an excellen agreemen beween he calculaed and measured ime inegrals. These resuls confirm ha he inegraor is fully capable of processing emporal feaures as fas as 60 ps FWHM, corresponding o a full signal bandwidh over 20 GHz. CW ligh source Opical Modulaion Opical inegraor Measuremen PC Sampling Oscilloscope TL ISO IMOD PC Amp OS Volage Bias PPG or AWG pump Inegraor Elecric Triggering Fig. 2. Experimenal seup for he inegraion of pulses generaed using elecro-opically modulaed signal. TL: Tunable laser, ISO: opical isolaor, PC: polarizaion conroller, IMOD: Opical inensiy modulaor, PPG: Picosecond elecric pulse generaor wih 70 ps FWHM ime widh, AWG: Elecric arbirary waveform generaor wih 500 MHz bandwidh, Pump: 980-nm semiconducor pump laser, Amp: Erbium-doped fiber amplifier, OS: Opical sampler (phooreceiver). (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18208

8 (a) 140 ps (FWHM) inpu pulse Inensiy, a.u. (b) 60 ps (FWHM) inpu pulse Inpu pulse Opical inegraion Numerical inegraion of he inpu pulse Fig. 3. Experimenal resuls demonsraing ime-domain inegraion of a single opical Gaussian pulse for wo differen inpu pulse FWHM ime widhs ((a) 140 ps and (b) 60 ps). The emporal opical inensiy of he inpu pulse (orange curve) and he inegraor oupu (green curve) are capured using a 20-GHz phooreceiver. For comparison, he square of he numerically calculaed ime cumulaive inegral of he measured inpu pulse field (square roo of he measured emporal inensiy profile) is also shown (yellow curve). In he second experimen, we used a double-pulse waveform composed of wo replicas of he same opical pulse as an inpu signal. For iner-pulse relaive phases fixed o zero, inphase pulses, we would expec he ime cumulaive inegral o look like wo seps wih each corresponding o he inegral of one pulse, separaed by he pulses relaive ime delay. However, for iner-pulse relaive phases fixed o π, ou-of-phase pulses, he second sep should have he opposie direcion, forming hus a square-like emporal waveform, he lengh of which would be given by he relaive ime delay. This phenomenon is schemaically shown in Fig. 4(a). The se-up for implemenaion of his experimen is shown in Fig. 4(b). The individual opical pulses were generaed by a passively mode-locked fiber laser (Priel Inc., Naperville, IL, U.S.A.), operaing a a repeiion rae of 16.7 MHz, followed by a 0.4-nm Gaussian-shape band-pass opical filer, which resuled in individual pulses wih a FWHM ime widh of 14 ps. The pulse wavelengh was uned o coincide wih he inegraor resonance wavelengh. The pulse replicas were obained using a Michelson inerferomeer adjused o achieve differen iner-pulse delays by coarse moving mirror alignmen and differen iner-pulse relaive phases by fine alignmen of he moving mirror. The relaive phase was fixed o eiher zero (in-phase pulses) or π (ou-of-phase pulses). For he shores ime delay (170 ps), we confirmed he relaive phase of he wo pulses by measuring he opical specrum of he double-pulse srucure. For in-phase pulses, he specrum had a maximum a is cenre, whereas for he ou-of-phase pulses, here was a minimum. The wo (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18209

9 pulse replicas were polarized along one of he inegraor s birefringence polarizaion axis by adjusing a polarizaion conroller. The inegraed oupu was amplified by an Er-doped fiber amplifier and deeced by a 20-GHz phoodiode and a sampling oscilloscope. a) INPUT, x () OUTPUT, y() = x( τ) dτ In-phase double pulse (field envelope) Ou-of-phase double pulse (field envelope) ( ) b) Pulse source Pulse shaper (inerferomeer) Opical inegraor Measuremen FFL BPF PC L M B L M PC pump Inegraor Amp OS Elecric Triggering Sampling Oscilloscope FFL : Femosecond fiber laser (16.7 MHz pulse repeiion) BPF : Opical bandpass filer PC : Polarizaion conroller L: Collimaing lens B: Beam splier M : Mirror Pump: Opical pump inpu from a Laser Diode operaing a 980 nm wavelengh Inegraor: Er-Yb doped fiber wih superimposed FBGs Amp : Erbium-doped fiber amplifier OS: Opical sampler (phoo-receiver) : To indicae he opical pulse propagaion direcion Inensiy, a.u. Ou-of-phase double pulse In-phase double pulse Inegraed oupu specrum Wavelengh, nm Fig. 4. (a) Diagram showing all-opical inegraion of wo consecuive opical pulses wih differen relaive phases. For relaive phases of 0 (in-phase he field ampliudes are of he same sign, red curves) and π (ou-of-phase he field ampliudes are of opposie signs, blue curves), he ime inegral is expeced o be a double sep-rising waveform, and a fla-op waveform, respecively. (b) Experimenal seup for he double pulse inegraion. 14-ps pulses are generaed from he FFL followed by an opical bandpass filer (0.4 nm 3dB-bandwidh). Time-delayed pulse replicas are made by using a fiber-coupled Michelson inerferomeer. For he shores ime delay (170 ps), we confirmed he relaive phase of he wo pulses measuring he opical specrum of he double-pulse srucure, which is shown in he inse: for in-phase pulses (red line), he specrum has a maximum a he inegraor cenral frequency, while for he ou-of-phase pulses (blue line), here is a minimum. The oupu specrum is shown as green line. (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18210

10 The resuls of his experimen are shown in Fig. 5. As expeced, he inegraor simply summed up he area under he wo waveforms in ime when he wo pulses had no phase difference, resuling in wo consecuive seps respecively corresponding o he inegraion of he leading pulse and he subsequenly added inegraion of he railing pulse. The inegral of he wo idenical pulses is wo imes he inegral of he single pulse, which is four imes in inensiy consequenly, he measured second sep in he inegral should be hree imes higher han he firs one. In conras, when he pulses were ou of phase, he second opical pulse compensaed for he cumulaive ime inegral of he firs pulse, leading o a square-like oupu ime profile wih a duraion fixed by he inpu iner-pulse delay. In boh experimens, he sligh exponenial decay of he inegraed signal was caused by a sligh ne round-rip loss wihin he caviy. The observed spikes are believed o be arifacs caused by he limied bandwidh of he phooreceiver wih respec o he bandwidh of he used pulses. Neverheless, hese resuls confirm ha he developed inegraor operaes on he complex emporal envelope (ampliude and phase) of he opical signals o be processed. (a) Double inpu pulse 170 ps (b) Opical inegraion (In-phase) (c) Opical inegraion (Ou-of-phase) Inensiy, a.u ps ps Fig. 5. Experimenal resuls demonsraing ime-domain inegraion of double opical pulses. (a) Time-domain opical inensiy of he inpu signal - wo 14-ps consecuive opical pulse wih various iner-pulse delays (170 ps, 500 ps, and 1000ps). The inegraed oupu depends on he relaive phase of he wo pulses: (b) inegraion for in-phase pulses; (c) inegraion for ou-ofphase pulses. Finally, we performed an experimen ha combines boh basic properies shown above and should serve as an example of an arbirary waveform inegraion. In his experimen, we processed wo consecuive fla-op pulses se in-phase and ou-of-phase. The experimenal seup is shown in Fig. 2 an elecric arbirary waveform generaor (AWG) wih 500 MHz bandwidh ha was programmed o produce he desired waveform (wo fla-op pulses of 1 and 2 ns duraion, respecively, se in-phase or ou-of-phase) was used o drive he modulaor. The opical inegraion of a single fla-op pulse should resul in a simple rising ramp, duraion of which corresponds o he inpu pulse duraion. The resuls of he carried-ou experimens are shown in Fig. 6: he lef panel shows he resuls when he wo pulses are in phase, while he righ panel shows he ou-of-phase seing. The inpu elecric waveform is shown firs (blue), followed by he inpu opical inensiy waveform (red). Finally, he oupu opical inensiy waveform (yellow) is shown. As anicipaed for he in-phase seing, wo summed rising ramps are presen a he oupu. These wo ramps are disinguished by heir differen (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18211

11 ime widhs; ~2 ns and ~1 ns, respecively. The ou-of-phase oupu is formed by a posiive slope ramp wih a duraion corresponding o he firs fla-op pulse (~2 ns) followed by a negaive slope ramp wih duraion corresponding o he second opposie-phase fla-op pulse (~1 ns). Elecrical inpu pulse Elecrical inpu pulse Inensiy, a.u. Opical inpu pulse Double ramp pulse Opical inegraion of inpu pulse Opical inpu pulse Triangular pulse Opical inegraion of inpu pulse Fig. 6. Experimenal resuls for he opical inegraion of wo fla-op pulses se in-phase and ou-of-phase. Lef: in-phase; Righ: ou-of-phase. The inpu elecric waveform: blue; he modulaed opical inensiy waveform: red, and he oupu opical inensiy waveform: yellow. 5. Experimenal resuls: all-opical compuing of differenial equaions To give a simple applicaion example of he developed phoonic emporal inegraor, Fig. 7(a) shows an inegraor-based compuaion sysem [18] used for solving he following consancoefficien firs-order linear differenial equaion: dy() + ky( ) = x ( ) (3) d where x() represens he inpu signal, y() is he equaion soluion (oupu signal), and k represens a complex consan of an arbirary value. The simple differenial equaion Eq. (3) acually models a wide variey of basic engineering sysems and physical phenomena [20], including problems of moion subjec o acceleraion inpus and fricional forces, emperaure diffusion processes, response of differen RC circuis, populaion dynamics in biology and economy, ec. As anicipaed, he key elemen in he resuling compuaion sysem is a emporal inegraor. Ineresingly, he phoonic device demonsraed here is governed by he general differenial equaion Eq. (3), wih k being a real-valued consan, 2 k = ( 1/ T) ln( r γ ). This is associaed wih he fac ha our inegraor design iself is based on a feedback configuraion (FP resonan caviy) such as ha of he compuing sysem required for solving Eq. (3), see schemaic in Fig. 7(a). As demonsraed above, he device behaves as a emporal inegraor when i is operaed a he exac lasing hreshold condiions (k = 0). Moreover, he demonsraed device is capable of solving Eq. (3) when i is operaed eiher slighly below he lasing hreshold condiion (k > 0) or slighly above he hreshold condiion (k < 0). Figs. 7(b)-(c) represen he soluion of Eq. (3) as compued experimenally wih our phoonic device for wo differen inpu signals, i.e. an ulrashor emporal impulse (Fig. 7(b)) direcly generaed from our mode-locked fiber laser, and a consan exciaion over (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18212

12 a limied emporal window (2.9 ns, Fig. 7(c)) obained by modulaing a cw laser ligh wih an elecrical square-like pulse prepared using he previously-specified AWG. In he wo cases, he differenial equaion was experimenally solved for differen posiive values of he variable k, which have been varied by simply adjusing he opical pumping power in he acive medium. For comparison, Eq. (3) was also numerically solved for he wo considered inpu waveforms and various values of k. In he firs experimen (resuls shown in Fig. 7(b)), he inpu ulrashor pulse can be considered as a emporal dela funcion, x() = δ(), and he sysem response can be analyically calculaed from Eq. (3) resuling in he equaion anicipaed for a general FP caviy, y( ) = h( ) = exp( k) u( ). The inensiy of his analyical emporal impulse response is depiced in Fig. 7(b) for he differen evaluaed values of he consan k, showing an excellen agreemen wih he experimenal resuls. The general soluion of Eq. (3) is obained as he numerical convoluion of he considered inpu signal (e.g. fla-op waveform used in he second experimen) wih he analyical emporal impulse response h() defined above. The inensiy of he numerically obained soluions of Eq. (3) using he squared-roo of he measured fla-op waveform as he inpu signal are ploed in Fig. 7(c) for he wo differen evaluaed k values, confirming again he excellen agreemen beween he heoreical and experimenal soluions. The excellen agreemen beween he experimenal and numerical soluions of Eq. (3) proves he poenial of he demonsraed phoonic emporal inegraor for all-opical compuing of differenial equaions. (a) Inegraion operaor Firs-order differenial equaion x () ( ) y( ) = x( τ) dτ dy() + κ y() = x() d κ τ = Implemenaion x () ( ) y( ) = x( τ) dτ κ y( τ) dτ τ= τ= 1.0 (b) Inpu pulse, 2 x () (c) Inpu pulse, 2 x () Inensiy, y() 2 [a.u.] Consan ' κ ' value 1.3 ns ns ns ns -1 Numerical soluion Inensiy, y() 2 [a.u.] Inensiy, a.u Consan ' κ ' value 10.0 ns ns -1 Numerical soluion Fig. 7. (a) Schemaic diagram of an inegraor-based opical compuing sysem designed for solving he firs-order linear ordinary differenial equaion (ODE) defined in he figure. The wo graphs a he boom show he experimenal (solid curves) and numerical (circles) soluions of he ODE for wo differen inpu opical signals: (b) an inpu ulrashor emporal impulse (FWHM ime-widh = 60 ps) and (c) a consan exciaion over a limied emporal window (2.9-ns long square-like pulse). In each case, he ODE is solved for differen posiive values of he parameer k. (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18213

13 6. Conclusions We have repored he firs experimenal demonsraion of a phoonic emporal inegraor capable of performing he ime-domain cumulaive inegral of an inpu arbirary opical waveform. We have shown ha his fundamenal signal processing funcionaliy can be implemened in he opical domain using an acive resonan caviy operaed a he exac lasing hreshold condiion. This developmen is of paricular ineres for he implemenaion of ulrahigh-speed all-opical compuers since he ime inegraor is a key device in an analog compuing sysem devoed o solving differenial equaions in real ime. This imporan feaure has been illusraed by successfully solving a basic, general differenial equaion using he developed experimenal inegraor. As an addiional advanage, he phoonic inegraor demonsraed here is based on a compac and robus waveguide (all-fiber) implemenaion, which could be readily incorporaed in fuure inegraed phoonic circuis. Acknowledgmens This research was suppored in par by he Naural Sciences and Engineering Research Council of Canada (NSERC), by he Gran Agency of AS of he Czech Republic (conrac no. KJB ), by he Czech Science Foundaion (conrac no. GA102/07/0999) and he Canada Research Chair in Opical fibre communicaions and componens. We hank o Prof. M. Rochee and Prof. L. Chen from McGill Universiy for lending us he picoseconds pulse generaor and he unable narrow-bandpass filer. (C) 2008 OSA 27 Ocober 2008 / Vol. 16, No. 22 / OPTICS EXPRESS 18214