Analysis of spatial temporal converters for all-optical communication links

Transcription

1 Aalysis of spatial temporal overters for all-optial ommuiatio liks Da M. Marom, Pag-Che Su, ad Yeshaiahu Faima We aalyze parallel-to-serial trasmitters ad serial-to-parallel reeivers that use ultrashort optial pulses to irease the badwidth of a fiber-opti ommuiatio lik. This method relies o real-time holographi material for oversio of iformatio betwee spatial ad temporal frequeies. The aalysis reveals that the temporal output of the pulses will osist of hirped pulses, whih has bee verified experimetally. Whe the sigal pulses are trasmitted alog with a referee pulse, the distortios of the reeived sigal, aused by dispersio ad other fators i the fiber, are aeled beause of the phase-ojugatio property of the reeiver. This self-refereig sheme simplifies the reeiver struture ad esures perfet timig for the serial-to-parallel oversio Optial Soiety of Ameria OCIS odes: , , , , Itrodutio Ultrashort pulse shapig by Fourier sythesis i the temporal frequey domai has bee extesively ivestigated i the past deade. 1, Spetral holography has expaded the oept of holography to sigals i the time domai. The reordig of the iterferee of a sigal pulse ad a referee pulse permits later readout by a ultrashort pulse ad orrelatio ad ovolutio of ultrafast sigals i the optial domai. 3,4 Spetral holography has utilized the duality betwee the temporal ad the spatial frequeies to overt iformatio from pulsed optial sigals to w ad vie versa. The iterferee of spatial frequey iformatio o a hologram a be read out i time by a ultrashort pulse s spetral frequey ompoets for spae-to-time oversio. 5 The iterferee of temporal frequey ompoets of opropagatig optial pulses a be reorded as a spetral hologram ad read out by diffratio of a w optial wave for time-to-spae oversio. 6 Based o these tehiques, trasmissio of spatial iformatio by meas of a temporal hael has bee performed 7 : A sequee of w poit soures has bee The authors are with the Departmet of Eletrial ad Computer Egieerig, Uiversity of Califoria, Sa Diego, La Jolla, Califoria Reeived 18 August 1997; revised mausript reeived 8 Jauary $ Optial Soiety of Ameria overted to a paket of pulses with a spae-to-time setup parallel-to-serial trasmitter ad overted bak to the spatial domai with a time-to-spae setup serial-to-parallel reeiver. Other methods of performig time-to-spae ad spae-to-time oversios by use of exitoi ad seod-order oliearities have bee reported as well The aalysis of a spetral deompositio system used i these experimets usually assumes a liear agular dispersio devie, the paraxial approximatio, ad a short optial pulse with a badwidth that is small ompared with the optial arrier frequey. I this paper a parallel-to-serial trasmitter ad a serial-to-parallel reeiver are aalyzed without the limitig assumptio that the badwidth of the pulse is small ompared with the optial arrier frequey. The results of this aalysis, preseted i Setio, show that the pulses at the output of the trasmitter are hirped, with the hirp ireasig for shorter pulses or a greater time delay relative to the iidet pulse. The performae of the four-wave-mixig 4WM reeiver is show aalytially to be uaffeted by the hirped pulses whe these pulses are trasmitted through a idealized distortioless medium for data reovery i the spatial domai. I pratie, these sigal pulses will be trasmitted through a optial fiber that possesses dispersio ad radom phase variatios due to ambiet oditios, thereby affetig the perfet sigal reovery i the spatial domai at the reeiver. To overome the fiber dispersio effet, we sed the referee pulse alog with the data pulses, as desribed i Setio 3. Thus the 858 APPLIED OPTICS Vol. 37, No May 1998

2 Fig. 1. Struture of the parallel-to-serial trasmitter. w writig beams emaatig from poit soures separated by propagate from right to left, oupled i by a beam splitter dashed lie. A iidet short pulse propagates from left to right, with the spetrum spatially dispersed at plae by a gratig i plae 1, where a real-time 4WM proess mixes the w iterferee with referee soure ot show ad the pulsed spetrum. phase distortio of the hael, sampled by the referee pulse, is ompesated for at the reeiver by meas of the phase-ojugatio proess, whe iterfered with by the data pulses. A estimate of the aggregate bit-rate apaity of this trasmissio method is arried out i Setio 4 for Gaussia iput pulses. I Setio 5 we verify experimetally that the output pulses are hirped, as predited by the aalysis, by performig their ross-orrelatio measuremets with a trasform-limited pulse. We olude with a summary i Setio 6.. Aalysis of a Commuiatio Lik Based o Ultrashort Pulses The aalysis of the parallel-to-serial trasmitter ad serial-to-parallel reeiver is arried out i the mixed temporal spatial frequey domai. A real-time hologram by 4WM of w spatial frequeies ad temporal frequeies from ultrashort pulsed waves is used. The trasfer of iformatio from the spatial domai to the temporal domai ad vie versa is aomplished through the iterhageability of the spatial frequeies ad the temporal frequeies iformatio for sigals i spatial ad temporal domais, respetively. The parallel-to-serial trasmitter is based o real-time 4WM i the frequey domai. The temporal frequey spetrum of a iidet ultrashort pulse is modulated by a sequee of liear phase futios reated by the iterferee of a sigal sequee of w poit soures ad a referee poit soure from the spatial hael see Fig. 1. The resultat temporal hael output is a sequee of ultrashort pulses shifted i the time domai aordig to the slope of the orrespodig liear phase futio from the spatial frequey domai. Further details of the operatio are disussed i Subsetio.A. The iformatio-arryig pulses ad the referee ultrashort pulse are trasmitted to the reeiver for detetio. The serial-to-parallel reeiver, show i Fig., trasforms the sigal iformatio bak to the spatial hael by a seod 4WM proess. The iterferee Fig.. Struture of the serial-to-parallel reeiver. The iomig data pulses are show i gray, the referee pulse i blak. Realtime hologram at plae trasfers iterferee iformatio of the pulses to a readout w beam. The output w sigal is oupled out by a beam splitter dashed lie ad is foused oto a detetor array at output plae 3. of the iformatio-arryig ultrashort pulses ad a referee pulse i the temporal frequey domai, provided by the spetral deompositio waves, reates liear phase futios that modulate a readout w beam by 4WM iteratio. The resultat spatial hael output is a sequee of plae waves propagatig i diretios aordig to the liear phase futios obtaied from the iterferee of the spetrally deomposed temporal pulses. A Fourier trasform fouses the spatial hael output to a sequee of foused spots orrespodig to the trasmitted sigal. The diffrated w sigal is read out with a liear detetor array. I this aalysis we use a trasform-limited optial short pulse st, desribed by st pt t 0 exp j 0 t, (1) where pt is the temporal evelope futio of the pulse, t 0 is the time loatio of the pulse peak, ad 0 is the optial arrier frequey. The pulse s spetrum is foud by a Fourier trasform of Eq. 1, whih yields S P 0 expj 0 t 0, () where P is the Fourier trasform of the evelope futio pt. Next we desribe ad aalyze the parallel-to-serial trasmitter for a iidet pulse spetrum of the form of Eq.. A. Temporal Output of the Parallel-to-Serial Trasmitter To evaluate the system frequey respose, osider a moohromati plae wave of frequey iidet upo the first refletig gratig at a iliatio agle see plae 1 i Fig. 1. The gratig is arraged to diffrat the arrier frequey ompoet 0 ito the diretio of the optial axis of the system. Thus the diffrated optial field of frequey is a plae 10 May 1998 Vol. 37, No. 14 APPLIED OPTICS 859

3 wave propagatig at a frequey-depedet agle with respet to the optial axis, give by s 1 x; expj 0 xwx, (3) where si, wx is the pupil futio of the refletig gratig, ad is the speed of light i vauum. The pupil futio wx is defied by either the field distributio of the light o the gratig or the physial size of the gratig surfae, whihever is smaller. The field i plae is determied by the spatial Fourier trasform of the field i plae 1, yieldig s f x ; Wf x 0, (4) where W f x is the spatial Fourier trasform of wx, f x x F is the spatial frequey, F is the les foal legth, ad x is the Cartesia oordiate i plae. This results i a off-axis spot, i whih the spot loatio depeds o the temporal frequey ad the spot size depeds o the size of the iput pupil. Equatio 4 shows that, as the temporal frequey of the iidet wave deviates from the frequey 0, its spatial Fourier trasform is shifted i the spatial frequey domai f x. Furthermore, if a short optial pulse is itrodued ito the system, it will be spatially dispersed i the Fourier trasform plae, where eah spetral ompoet will oupy a width determied by the futio W f x, shifted by the frequeydepedet iremet 0. Let a 4WM rystal be plaed i the Fourier plae of the trasmitter to futio as a real-time hologram, eablig real-time iterhage betwee the spatial frequey ad the temporal frequey domais. 5 The hologram otais iformatio o the iterferee of the spatial Fourier trasform of a referee poit soure ad a sequee of equally spaed mutually oheret poit soures i whih eah poit orrespods to a sigle bit of data see Fig. 1. The hologram serves as a liear phase filter with trasmittae t f w A exp jf w, (5) where A is the biary amplitude of the th bit i the spatial data array either 1 or 0, is the spatial separatio betwee adjaet poit soures i the data array, ad f w w x F is the spatial frequey for a w writig beam of optial frequey w used i the 4WM arragemet. To use the hologram trasmittae futio to modulate the short pulse spetrum by the orrespodig data, the spatial frequey f w oordiate of Eq. 5 has to be overted to the frequey f x of the temporal hael see Eq. 4. By equatig the Cartesia oordiate x o the same Fourier plae see Fig. 1, we obtai the relatio f x f w w, yieldig Eq. 5 i the form t f x A expj w x f. (6) The ratio w will aout for the differee i diffratio of the broad spetrum of a short pulse o a ostat spatial frequey gratig reorded by a w frequey soure. The field behid the hologram i plae is s f x ; s f x ; t f x, with s f x ; ad t f x from Eqs. 4 ad 6, respetively. A spatial Fourier trasform of the field s f x ; yields the optial field i plae 3: s 3 x; A wx w expj 0 x w, (7) where the mius sig i wx idiates image iversio. The field i plae 3 is diffrated by the seod refletig gratig, yieldig the output field propagatig i the z diretio: s 3 x; A wx w, (8) expj 0 w where a oordiate rotatio of degrees from x, z to x, z is performed. The ew pupil futio wx is a projetio of the pupil futio wx oto the x oordiate. Equatio 8 represets the trasfer futio of the system for ay iput frequey. The output field propagates i the z diretio idepedetly of the frequey variable. To omplete the aalysis, osider itroduig ito the iput of the system a short optial pulse whose orrespodig spetrum is desribed by Eq.. The system output is determied by the iverse temporal Fourier trasform deoted by F 1 of the produt of Eqs. ad 8: s 0 x; t F 1( P 0expj 0 t 0 A w x w exp j 0 w ). (9) Equatio 9 represets the omplete spatial temporal output of the trasmitter. Next we attempt to simplify the oliearity of the frequey respose aused by the 1 term whe the futio P 0 is badlimited with a badwidth of. Let frequey be replaed with 0, 860 APPLIED OPTICS Vol. 37, No May 1998

4 with 0, suh that the followig approximatio holds: O (10) The assumptio that 1 a be aurately approximated by the right-had side of Eq. 10 will be valid for badwidths as large as 10% of the optial arrier orrespodig to a 1% error i the limit. This liearizatio approah is the ovelty of this aalysis, sie the previous solutios simplified the oliearity by replaig with 0. 7 The previous aalysis results are valid for relatively log ultrashort pulses hudreds of femtoseods ad loger time duratio. For shorter pulses, we derive our ew result obtaied from Eq. 9 with the approximatio of Eq. 10, whih yields s 0 x; t F 1( Pexpjt 0 A wx expj j 0 ), (11) where we defie w 0. The results of Eq. 11 show that our ew approximatio itrodued two additioal terms that are ot preset i the previous aalysis. The third term i the aperture futio w will give rise to a frequey-depedet aperture shift, ad the seod term i the imagiary expoet imposes a hirp futio aross the badwidth of the pulse. To obtai a aalyti form for the temporal output, we assume, as a example, a trasform-limited Gaussia pulse shape with time ostat ad its orrespodig Fourier trasform pair give by 0 pt exp t FT N P exp. (1) The output of the trasmitter with the Gaussia shape iput pulse yields s 0 x; t A exp j 0 t exp expj wx 0 jt d, (13) 0exp where t t t 0 is the time delay of the th pulse or bit eoded. The first term i the itegrad of Eq. 13 is a Gaussia pulse spetrum therefore badlimited, the seod term is a quadrati phase term, ad the third term aouts for the iput Fig. 3. Time-domai output simulatio of parallel-to-serial trasmitter. The simulatio is show for a pulse duratio of 90 fs, 1.55-m wavelegth, ad 0.3 mm. aperture, whih has some frequey-depedet shifts. We ext argue that the iput aperture depedee o the frequey a be egleted, sie the edge effet will be isigifiat ompared with the size of the aperture, whih is muh smaller tha the spetrum of the pulse that sets the limits of itegratio. Therefore the frequey depedee of the aperture is egleted ad the aperture futio is take out of the itegrad, as it is o loger frequey depedet. A similar justifiatio does ot apply to the quadrati phase term. The quadrati phase results i a frequey hirp imposed o the etire pulse spetrum. After the frequey depedee of the aperture is egleted, Eq. 13 a be diretly itegrated ad the output is s 0 x; t 1 A exp j 0 twx 4 1 expj ta1 exp t 1 j 1, (14) where we defie the dimesioless parameter 0. Equatio 14 desribes a sequee of temporal pulses, eah with a aperture positio shifted by, ad a temporal separatio of, as a be see i the output of Fig. 1, after plae 3. The peak power of the pulses drops ad the duratio beomes broader as ireases, preservig the eergy i eah bit pulse. The phase of the pulses is also varyig from pulse to pulse. Sie all pulses propagate i the same diretio, they a effiietly be oupled ito a fiber with a ouplig les. After isertio ito a sigle-mode fiber, the pulse sequee will lose its spatial iformatio, retaiig oly the temporal properties pulse separatio ad broadeig. We assume that all -shifted pupil output pulses are ompletely oupled ito the fiber. The umber of pulses that a be ompletely oupled is the limitig parameter used i alulatios of bit-rate apaity i Setio 4. A simulated temporal output for the itesity of a sequee of suh pulses i a sigle mode fiber with A i 1 is show i Fig. 3. The iformatioarryig pulses broade at the expese of lowerig 10 May 1998 Vol. 37, No. 14 APPLIED OPTICS 861

5 their peak amplitude as their distae from the eter pulse ireases. B. Perfet Reostrutio of a Serial-to-Parallel Reeiver The aalysis of the detetio proess at the reeiver show i Fig. is arried out for the iterferee of a sigle pulse from the sequee of pulses defied by Eq. 11 ad a referee pulse. The reorded iterferee patter is assumed to orrespod to oly the iterferee of the data pulse with the referee pulse without itersigal iterferee iterferee betwee differet data pulses i the paket. This assumptio is valid as log as the referee pulse power is muh stroger tha eah idividual data pulse i the paket. As a osequee of this assumptio, we a use the superpositio priiple ad aalyze the iterferee for a sigle data pulse ad the itrodue the etire sequee defied by the summatio. The th reeived data pulse from Eq. 11 is defied by s t F 1Pexpjt 0 A expj j 0, (15) where we have dropped the spatial iformatio i the pulse, as it will be lost after pulse trasmissio through a sigle-mode fiber. Note that, at this poit, we assume that the fiber does ot itrodue ay distortios, a limitatio that will be mitigated i Setio 3. We therefore model the fiber hael as a perfet hael with a flat frequey respose over the badwidth of the pulse. Durig the detetio, the above data pulse of Eq. 15 iterferes at the reeiver with the referee pulse: rt ptexp j 0 t, (16) whih has the same pulse evelope as the origial short pulse used i the trasmitter. The time separatio betwee the referee pulse ad the data pulse is ow t 0 beause of the ombiatio of the two liear phases i Eq. 15, where t 0 is the time separatio betwee the eter of the referee pulse ad the eter of the data paket, ad the seod term,, is the time shift of the th data pulse relative to the eter of the data paket see Fig.. The orrespodig spetra of the data pulse ad the referee pulse are the Fourier trasforms of Eqs. 15 ad 16, S Pexpjt 0 A expj j 0, (17) R P, (18) respetively, where we use 0. Both pulses are itrodued ito the reeiver, where they are both spetrally deomposed by the iput gratig ombied with a spatial Fourier trasform les, agai produig the spatially dispersed spetrum o the Fourier plae plae. Followig a idetial derivatio for the system respose as was doe for the trasmitter, the system respose is similar to Eq. 4: W f x ; Wf x, (19) where agai, f x 0 x. F The system frequey respose Eq. 19 multiplies the sigal ad referee pulses spetra Eqs. 17 ad 18, respetively to provide the spatially dispersed spetra o the Fourier plae plae i Fig.. The output sigal resultig from 4WM at the reeiver is determied by the itesity variatios from the iterferee of the temporal sigals spetra, whih i tur diffrat a w beam. The itesity, give by RW f x ; S W f x ;, will have four ompoets that a be separated spatially whe a displaemet is itrodued betwee the sigal ad referee i the y diretio to provide the desired holographi sigal: RS *W f x ; A P expjt 0 j W f x ;. (0) 0 If we assume that there is o spetral depedey i the hologram withi the pulse badwidth, the reorded holographi sigal at every loatio alog x is give by H x RS *W f x ; d (1) by substitutio of the expressio for f x,asitisa futio of ad x. A preise derivatio of the reorded holographi sigal is provided i Appedix A, ad a simplified approah is give below. To simplify evaluatio of the itegral i Eq. 1, we approximate the Fourier-trasformed aperture by a Dira delta futio, W f x ; f x ;, whih yields H x A P expjt 0 j 0, alog with the substitutio () x 0 x F, (3) whih is the relatio betwee the Cartesia oordiate x ad the pulse spetrum derived from the itegratio over the Dira delta futio. Equatio 86 APPLIED OPTICS Vol. 37, No May 1998

6 3 a be further simplified by a seod approximatio, similar to that made i the trasmitter aalysis, x 0 1 F 1 F x x 0 F 1 x F F O x F x 0 F 0 x 0x F OF x 3 OF x 3, (4), (5) as xf 1. Whe the above expressios are substituted for, the spatial variatios take the fial form of H x A x 0 P 0 F F x expj x 0 F F expj 0 t 0 x F x. (6a) The first term i Eq. 6a desribes the power spetrum evelope of the pulse badwidth, the seod term is the liear phase sigal that arries the thbit iformatio, ad the third term aouts for the additioal liear phase of the separatio t 0 ad a distortio term i the form of a quadrati phase. The power spetrum evelope of the reorded hologram serves as a apodizatio for the readout beam. Its effet is to irease the spot size o the diffrated sigal, whih will limit the umber of bits that a be trasmitted by this sheme. This issue is disussed further i Setio 4. The liear phase is a futio of, whih orrespods to a uique phase futio for eah data pulse i the paket trasmitted. The distortio term depeds o the time separatio of the referee pulse ad the eter of the data paket t 0 ; thus, to elimiate this distortio term, the timig of the referee pulse must be set suh that t 0 0. Fially, a w plae wave iteratig with the realtime spetral hologram will geerate a diffratio patter that oiides with the origial trasmitted data. The approximatio made i the derivatio of Eq. 6a, i.e., W f x ; f x ;, resulted i some loss of iformatio regardig the temporal overlap of pulses i the iput pupil futio. We reitrodue this lost iformatio by addig the temporal fator K t to the equatio, whih yields H x; t A x 0 P 0 F F x expj x 0 F F expj 0 t 0 x F x K t. (6b) The temporal fator is foud i the preise aalysis of Appedix A ad is give i Eq. A7 as K t ww t t t 0. The temporal fator determies how log the iterferee patter would exist betwee a referee pulse ad a pulse separated i time by t 0. Similar to the example i Subsetio.A, osider agai a trasform-limited Gaussia pulse shape with time ostat. Whe the spetrum of the Gaussia pulse is substituted ito Eq. 6b, the spatiotemporal sigal is H x; t A exp 0 F x expj x 0 F F K t. (7) expjt x F x The spatial itesity variatios i the rystal are read out by a w beam i a 4WM ofiguratio, with a temporal evelope determied by K t. The readout w beam of wavelegth r diffrats from the itesity-idued spatial variatios, ad after a Fourier trasform results i the output field o plae 3 i Fig. assumig o distortio by settig t 0 0: hx ; t h x ; t x A r exp 4 0 K t, (8) 0 whih is a olletio of Gaussia spots separated i spae. The loatio of the sigal of eah data poit is at x r w, where w is the w wavelegth used to write the phase futio i the trasmitter. The eergy at eah diffrated sigal defied by itesity times time is determied by itegratio of the temporal fator, whih is equivalet to the autoorrelatio futio of the pupil futio evaluated at a lag of. As ireases, less eergy is available for detetio at the output. 3. Dispersio Compesatio Property of Serial-to-Parallel Reeiver The preedig aalysis treated the trasmissio medium as a perfet hael. The trasmitted sigals did ot experiee ay distortio betwee the trasmitter ad the reeiver, where the data are iterfered with by a referee pulse for detetio. If the data were orrupted by the trasmissio, ad a loal referee pulse would be iterfered with for detetio at the reeiver, the deteted data would be distorted. However, whe the referee pulse is set over the same optial fiber alog with the data, both sigals experiee the same distortio effet, ad, by the phase-ojugatio proess iheretly preset i the 4WM tehique, the readout beam ompesates for the distortio. 10 May 1998 Vol. 37, No. 14 APPLIED OPTICS 863

7 Let the sigal st ad the referee pulse rt propagate i a fiber. The fiber itrodues distortios to the pulses beause of dispersio, absorptio, satterig, mehaial stress, ad temperature flutuatios, all of whih a be modeled as aexp j, where a ad exp j aout for the trasmittae ad the aquired phase, respetively. This distortio model exludes oliear, itesity-depedet pheomea ad impliitly assumes a time-ivariat respose o the time sale of paket time duratio. The aquired phase is aeled by the ojugatio of the sigal ad referee pulses i the 4WM proess with the holographi sigal of iterest: Raexp jwx; Saexp jwx; * RS*a Wx;. (9) This result is early idetial to that of the ideal hael, illustratig the dispersio ompesatio property of this trasmissio sheme. The aquired phase term has bee ompesated for by the phase ojugatio. The effet of the frequey seletivity i the fiber trasmittae serves to ehae further the apodizatio of the eergy profile, similar to the effet of the power spetrum disussed i Subsetio.B. This would affet the diffrated orders spot size. A additioal beefit of trasmittig the referee pulse alog with the data is the self-refereig i the reeiver. Sie the referee pulse does ot have to be geerated i the reeiver, o timig errors a our, ad with proper aligmet, the t 0 0 oditio a be easily satisfied, guarateeig proper syhroizatio of the system. Sie the referee pulse a be made muh stroger tha the data pulses, it a be also used i diret detetio for sigalig the reeiver iruitry to aquire the data from the detetor array, thus syhroizig the reeiver detetio iruitry to the trasmitted data. The umber of data bits that a be trasmitted i a sigle paket is disussed i Setio Bit-Rate Capaity of Trasmissio System Several system assumptios have to be made i order to alulate the aggregate bit-rate apaity of this trasmissio lik. The apaity, i bits per seod, is the umber of bits that a be eoded oto a sigle ultrashort pulse multiplied by the repetitio rate of the system. I our model we assume a eter wavelegth of 1.55 m for low atteuatio ad ew allfiber ultrashort laser soures at this wavelegth. 1 The trasmissio medium is a sigle-mode fiber so that modal dispersio a be egleted. Two riteria have to be satisfied to ouple the output data pulses effiietly ito a sigle-mode fiber see Fig. 4: 1 The output beams foused spot sizes have to be omparable with the mode field diameter of the optial fiber, ad the foused beams have to arrive from diretios that lie withi the N.A. of the fiber. The ouplig-les foal legth F is hose Fig. 4. Couplig output pulses of trasmitter ito a optial fiber. The output pulses, of the size of wx, are delayed by ad laterally shifted by. Les F ouples the pulses ito a fiber, where the lateral shift iformatio is lost. aordig to the spot-size riteria, suh that eah output beam width, determied by the projetio of the iput pupil wx, is foused to the mode field diameter of the fiber. For a hose fousig les F, the umber of haels that a be olleted M is foud i paraxial approximatio by simple trigoometry: M FN.A., (30) where we assume that the fiber ouplig behaves as a hard-lipped futio i.e., the light will be ompletely oupled if it is withi the N.A.. For determiig the miimum spatial shift of the pulses that will maximize the apaity, the reeiver output spatial haels eed to be separated eough to reate low ross talk i the detetor array. As disussed i Setio 3, a plae wave illumiatig the reorded iterferee patter i the reeiver will diffrat to differet loatios. The spot size of the diffrated sigals ad their separatio will determie. The allowable ross-talk power is hose to be 0 db ad is geerated maily from adjaet haels. We assume that the temporal fator atteuatio plays a egligible role i ross-talk evaluatio, as the atteuatio differee betwee adjaet pulses is small whe the pupil futio is desiged to ilude may pulses i.e., temporal legth of pupil futio is Fig. 5. Itesity output of diffrated data poits. Spots are displaed by x 0, a distae that guaratees a sigal-to-ross-talk ratio of 0 db. The shaded area orrespods to light itesity gathered by a detetor of width D. 864 APPLIED OPTICS Vol. 37, No May 1998

8 large ompared with pulse separatio,. For our example of a Gaussia pulse iput, with egligible trasmissio losses ad with t 0 0, the diffrated output sigal is give by Eq. 8, whih is a olletio of Gaussia spots separated i spae see Fig. 5. The spatial loatio of the sigal of the th data pulse is give by x r w. The sigal-to-rosstalk ratio SXR is defied by the ratio of the sigal power itegrated o a detetor of width D to that of the ross-talk power o the same detetor from the adjaet haels. For Gaussia sigals the SXR is give by SXR erf 0 D w r erf 0 erf 0 r D w D r exp 1 w 0 erf 0 r D, (31) whih is a mootoially dereasig futio of the detetor width D. Whe is umerially solved to obtai the speified SXR of 0 db, with a trasmissio wavelegth m, readout wavelegth r 1.55 m, writig wavelegth w 0.5 m, 0.6, pulse duratio 90 fs for a time ostat of 54 fs, ad detetor width D 50 m, the value foud is 190 m. The separatio of the spatial haels i the trasmitter,, is withi the pratial rage of spatial light modulator tehology. Isertig this value ito Eq. 30, after overtig to, we fid that the umber of pulses that a be eoded ad reliably deteted i a paket is M 31. For this alulatio we assume a stadard sigle-mode fiber with a mode field diameter of 9.3 m, a N.A. of 0.11, ad a 100-mm foal-legth les that a effiietly ouple the beams. For a laser repetitio rate of 100 MHz, the aggregate data rate is the pulse repetitio rate times M, whih is 3.1 Gbitss for the example assumed. Notig that a trasform-limited Gaussia pulse of 90-fs duratio has a badwidth of 4.9 THz, we are learly ot utilizig the badwidth effiietly. Yet this sheme still leaves ample room for time-divisio multiplexig, as the paket duratio M 36.5 ps is muh shorter tha the paket repetitio time 1100 MHz 10 s. Whe other users are itrodued ito a time-divisio-multiplexig etwork eviromet, eah with aess to a 36-ps widow withi a 10-s frame, with guard bads betwee pakets of half a paket s duratio, approximately 185 users a share the fiber lik, for a total aggregate rate of Gbitss. The data rate derived above is by o meas a boud o the possible rate. A larger fousig les F, alog with a larger pupil futio o the gratigs, wavelegth variatios, ad other parameters, a irease the rate further. A detailed desig must ot oflit with the assumptio made, i.e., that the eergy i the bits is eough for 4WM yet ot high eough to stimulate oliear pheomea i the fiber. 5. Experimetal Results of Temporal Output Measuremets Previous experimets have verified time-to-spae oversios, 5 spae-to-time oversios, 6 ad a omplete spae time spae trasmissio. 7 The aalysis i Subsetio.A presets a ew result i the temporal output of spae-to-time oversios Eq. 14, whih predits hirpig of the output pulses, i whih the amout of hirp depeds o the slope of the liear phase futio ad the pulse badwidth. The atiipated hirpig of the output pulses from the parallel-to-serial trasmitter were verified experimetally by ross-orrelatio measuremets see Fig. 6. I this experimet, a short pulse from a Coheret Mira 900 laser m ad 93 fs, assumig a Gaussia pulse as desribed by Eq. 1 is split by a beam splitter ito two arms. Oe arm iludes a pulse shaper that simulates the trasmitter, ad the seod is a equal-legth delay arm. Table 1. Calulated ad Experimetal Results of the FWHM for 0-, 8-, ad 39-lpmm Gratigs Frequey lpmm Calulated FWHM Measured FWHM Stadard Deviatio of Measuremet Error Fig. 6. Shemati of experimet setup. G s, dispersio gratigs; f g, Rohi gratig NA NA May 1998 Vol. 37, No. 14 APPLIED OPTICS 865

9 The pulse shaper has 600-lpmm gratigs lpmm is lie pairs per millimeter i a optial 4F setup. The liear phase futios i the spetral plae were geerated by amplitude Rohi gratigs, whose Fourier series deompositio reveals that the 0 ad the 1 orders are predomiat. Therefore the output sigal for eah gratig osisted of three strog pulses, the origial trasform-limited pulse ad the two sigal pulses, oe preedig ad oe trailig the trasform-limited pulse. I this experimet we did ot osider higher-order diffrated beams. Oe of these pulses is up hirped, ad the seod is dow hirped. This output was aalyzed by a itesity ross orrelatio Irad Auto-orrelator 514-BX modified for ross-orrelatio measuremets with the origial trasform-limited pulse i the delay arm. The FWHM value was used as a basis for ompariso betwee the alulated ad the measured results. The odiffrated sigal was used to measure the FWHM of the trasform-limited pulse ad to obtai the value of. Eah measuremet was take five times ad averaged. Three differet Rohi gratigs of varyig frequeies, i.e., 0, 8, ad 39 lpmm, were used as the spetral filters i the pulse shaper. Experimetal results are summarized i Table 1, ad the superimposed traes of all the measuremets are show i Fig. 7. Eah rossorrelatio trae was idividually saled for a peak itesity of 1, as eah orrelatio was a differet measuremet that required a adjustmet of the gai i the detetor. Some variatios i the waveforms of the measured ad the theoretial pulses are show. I geeral, the measured autoorrelatio urve diverges slightly from the Gaussia profile. This divergee a be aouted for by the fat that the Gaussia model does ot fit preisely the pulse shape, ad other pulse-shape models ould provide a better fit e.g., hyperboli seat. Furthermore, the ross orrelatios reveal a asymmetry osistetly to oe side of the data. This a be attributed to third-order dispersio through glass elemets suh as the leses used i the pulse-shapig devie. From the tabulated results, we see that the alulated ad the measured values are i exellet agreemet. The error ad the stadard deviatio of the measuremets are of the same magitude, exept for the lowest-frequey gratig, for whih the alulated FWHM is 8% higher. This experimet proves that the output pulses are hirped, as expeted by the theoretial aalysis of Subsetio.A. Fig. 7. Plot of the ross-orrelatio traes. The solid urves are the measured urves, ad the dashed urves are the theoretial urves. 6. Summary A aalysis of a trasmissio method based o parallel-to-serial trasmitters ad serial-to-parallel reeivers has bee preseted. The trasmitted sigal is i a format of data pakets, osistig of a sequee of pulses, eah represetig a bit with ON OFF keyig. We show that the pulses i the paket have differet amouts of hirp, depedig o the pulse loatio i the paket. However, this hirpig pheomeo, predited ad experimetally measured by this aalysis, does ot affet the reostrutio ability of our reeiver, beause the serial data are overted bak to parallel haels by our oheret detetio sheme based o 4WM at the reeiver. Furthermore, whe the referee pulse is trasmitted alog with the data o the same optial fiber, the effets of aumulated phase distortios are selfompesated for by the phase-ojugatio proess iheret i the 4WM tehique. Additioally, this method provides self-refereig, as o referee pulse has to be geerated i the reeiver. This may solve timig errors that our whe a loal referee pulse has to be supplied at the reeiver, whih has bee show to ause distortio of the reostruted sigal. This sheme, however, relies o real-time holographi material, whih presets a hallege as well. A omplete trasmissio system has bee experimetally ostruted by use of ferroeletri photorefratives, as a proof of oept. 7 This group of materials requires a reordig time of the order of seods, learly elimiatig them from pratial systems. Curret researh o semiodutor multiplequatum-well photorefratives is produig material with a muh faster respose time. 13,14 Yet realisti implemetatios will udoubtedly eed to rely o eletroi 3 oliearities for almost istataeous respose times i a 4WM ofiguratio. To aess suh oliearities, high-power beams are required, whih oflit with the requiremet that the trasmitted beams be of low eough power ot to stimulate oliearities i the optial fiber. Oe possible solutio to this problem is to amplify the optial sigal ear the reeiver suh that the trasmissio would ot iur distortios yet the reeived sigal would be strog eough to exite 3 oliearities. Aother solutio may be to hirp the pulses itetioally to lower their peak powers for trasmissio i a fiber ad to reompress or dehirp at the reeiver, ireasig the istataeous power before the oliear 4WM proess. 866 APPLIED OPTICS Vol. 37, No May 1998

10 Appedix A I this appedix we rederive the perfet reostrutio reeiver Setio.B i the temporal domai, as opposed to the temporal frequey domai approah utilized above. Usig this approah, we fid the time fator K t that ditates how log the iterferee patter o the real-time holographi material will exist. We begi with the field distributio i the mixed spatiotemporal domai of the referee pulse. The field is give as the produt of Eqs. 18 ad Eq. 19: U r f x ; P 0 Wf x 0, (A1) where f x x F is frequey depedet. To fid the temporal variatios, we perform a iverse temporal Fourier trasform: U r x; t 1 P 0 W x F 0 exp jtd, (A) where we have swithed to the real oordiate x. To simplify the Fourier itegral of Eq. A, we ote that the pupil futio W is badlimited relative to the pulse spetrum, ad we a therefore pull the spetral futio P out of the itegral ad substitute for the value 0 the loatio depedee aordig to the pupil futio. This simplifiatio results i U r x; t 1 P 0x x F 0 exp jtd. W x F (A3) Solvig this itegral is trivial at this poit, ad after its solutio we simplify the relatio of the Cartesia oordiate x i the pulse spetrum by Eq. 4, yieldig U r x; t P 0x 0 F F x expj 0 t1 x F x 1 x F F w t 1 x F. (A4) Equatio A4 is reogized as the field desriptio of spetrally dispersed waves, whih have a fiite time of existee, defied by the pupil futio, ad the propagatio diretio is time varyig. The field distributio for the sigal pulse a be foud i a similar maer, resultig i U s x; t A P 0x 0 F F x expj 0 t1 x F x F expj 0 t 0 x F x expj 0 1 x F F t0 x Fwt 1 x F. (A5) The itesity distributio o the real-time holographi material is give by U s x; t U r x; t, ad the holographi sigal of iterest U s *x; tu r x; t is F U s *x; tu r x; t A 0x P 0 F x where 1 F x expj 0 expj 0 t 0 x F x K x; t, w K x; t t wt t0 1 F x w t 1 x F w t t0 F x F (A6) K t. (A7) The result derived i this Appedix, Eq. A6, is similar to the result i Eq. 6a, apart from the temporal variatio fator K t ad a early ostat fator term. The temporal fator multiplies the output iterferee sigal by a futio that measures the time at whih both the sigal ad the referee pulses overlap o the iput pupil futio. As the temporal separatio of the two pulses ireases i.e., t 0 is larger, a real-time 4WM output sigal will exist for a shorter time, ad therefore less eergy will reah the detetor. This study was supported i part by the Foused Researh Iitiative of the Ballisti Missile Defese Orgaizatio, the U.S. Air Fore Offie of Sietifi 10 May 1998 Vol. 37, No. 14 APPLIED OPTICS 867

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