Integrated All-Optical Fast Fourier Transform: Design and Sensitivity Analysis

Int egr at ed) All - Opti cal( F as t(F ouri er(T r ans f orm:(Des ign( and$Sensitivity$Anal ysis " H ANI" N EJADRIAHI , " 1 " D AVID" H ILLER K USS , " 2 " J ONATHAN " K. " G EORGE , " 1 " V OLKER " J. " S ORGER 1 * " ! "#$%&'(#)(*+, *-.#/(&0/%.*%) 1*2+'$3(#&*-)40)##& 0)45*6#+&4#*7%890 )4(+)*:)0;#&80(< 5*=> >*?? )1 * @(A*B+&(9C#8(5*7%890)4(+)5*"A2A *?>>D?5*:@E * 2 Huawei(Technologi es(D ü sseldorf(GmbH,(Optical(&(Quantu m(La borato ry,(Riesstrasse(25 - C3,(80992(Munich,(Germany ( F2+&$+)10)4*%3(9+&G* 8+&4#&H4C3A#13 * ! The! f ast! Fourier! t ransform! (FFT)! is! a! useful ! and! prevalent! algorit hm! in! sig nal! processing.! It! characterizes ! the! spectral ! components! of! a! sig nal,! or ! is! used! in! combination! with! other! operations! to! perform! more! complex! comp utations! such! as! filtering,! co nvolution ,! and ! correlation.! Digi tal ! FFTs! are! lim ited! in ! speed! by! the! necessity! of! moving! charge! within! logic! gates.! A n! analog! temporal! FFT ! in! fib er! optics ! has ! been ! dem onstrated ! with ! high est ! data! bandwidth .! How ever,! th e! implem entation! with! discret e! fiber! optic! FFT! components! is! bulky .! Here ,! we! present! and! analyze ! a! design! of! an! optical! FFT! in! Silicon! photonics! and! evaluate ! its! perform ance! w ith! respe ct! to! variations! in! phase! and ! amplitude.! We! d iscuss! the! im pact! of! the! deployed! devices! on! the! FFT ’s! tr ansfer! function! quality! as! defined! by! th e! transmission! output! power ! as!a! function! of! frequency,!detuning! phase,! optical! delay ,!and! loss .! The$ fast$ Fou rier$ tran sform$ (FFT)$ has $ a pplications$ rang ing $ fro m$ signal$ filtering,$ to $ orthogonal$ frequency$ division$ multiplexing$ (OFDM)$ transmission$ [ 1,$ 2 ] .$ While$ digital$ electronic$ FFT $ impleme ntations$ are$ lim ited$ by$ the$ physical$ charg ing$ of$ wires$ and$ the $ necessity$ of$ driv ing$ logic$ gates$ to$ perform$ the$ arithmetic$ oper ations$ digitally ,$ optics$ natural ly $ enab les$ Fourier$ transform $ functionality ;$ for$ in stance$ a$ spatial$ F ourier$ transform $ when$ propagating$ through$ a$ lens $ [3 ] ,$ an d$ tempo ral$ FFT $ via$ Cooley - Tukey$ butterfly$ patter n$ [ 4,$ 5 ].$ S patial$ optical$ Fourier$ transforms $ are$ gen erally$ challenged$ by$ bulky$ free - space$ optics,$ how ever $ recent$ advances$ in $ m eta - surfaces$ show ing $ flat - lens$ functionality$ [ 6 ]$ may$e nable$ more$co mpact$f orms .$A n$efficient $ tempo ral$FFT $ has$ been$ realized$ in$ fiber$ optics$ treating$ the$ two$ operations , $ addition$ and$ multipli catio n , $ by$ phase - cascaded$ delayed$ interferom e ters $ and$ sampling$ the$ output$ with$ electro - optic$ modulators $ [ 4 ].$ In$ theory,$ such$ a$ syste m$ is$ entirely$ passive , $ apart$ from$ the$ sampl ers ,$ and$ the$ FFT$ rate$ simply $ depends$ on$ the$ physical$ propagation $ delay s $ o f$the$optical$signal $ in$the$FFT$structure .$ As$demonstr ated,$ it$ allows$ for$ wavelength$ division$ m ultiplexing$ (WDM)$ showing$ tens $ of$Tbit/s$of $ process ing $ bandwidth $ [ 2 ]. $ Here$we$sho w$a$design$o f$an$ B* =$4$ optical $FFT$(OFFT)$ integrated$ in$S ilicon$ph otoni c s $ for$si gnal$ modulation$frequency$of$ ,* s $ = $ 10 $ GHz .$ We$exp lore $it s$per for mance $by$an aly zing $the $qual ity $of$t he$t rans fer $ function $ and$ perform$ a$ component - ba sed$ sensitivit y$ analysis$ to$ understand$ its$ operation$ limitations.$ By$ relating$ the$ physical$ behavior$ of $ th e$ com ponent $ level$ to$ the$ tra nsfer$ function,$ we $ study$ the$ resilience$ of$ the$ OFFTs$ with$ respect$ to$ fab rication$ imperfectio ns$ and$ therm al$ effects .$ These $ two$ effec ts$ lead$ to $ interferom eter$ im balances$ due$ to$ optical$ loss$ variat ions,$ p hase$ detuning$ of$ the$ interferome ters,$ and$ even$ temporal$ delay$ variations .$ The$u nreliabilit y $ in$ fabrica tion$ originat es $ from$ material$ selection$ and$ processing$ conditions,$ wafer$ dicing,$ bo nding$ techniques ,$ etc.$ W hile$ absolute$ delay$ and$ losses$ are$ fixed$ after$ fabrication,$ the$ introduce d$ phas e$ uncertainties $ can$ be$ compensated$ with$ phase$ shifter,$ here $ through$ heaters ,$ to$ ensure$ the$proper$ function ality$of$ the$optical$filtering.$ $ The$ FFT$ is$ performed$ by$ the$ Cooley - Turkey$ method$ where $ signal $ addition$ is$ realized$ by$ integrated $ 2x2$ directional$ coup lers,$ while$ phase$ multipli catio n$ is$ implement ed$ by$ a$ relative$ phase$ difference $ [4,$ 5] .$ Th e$ resultin g$ butterflies $ are$ all$ passive$ components,$ b ut$ may$ req uire$ active$ tuning$ for$ ph ase$ alignment$ against$drift$as $discussed$below$ [ Fig.$1 (a)] . $ Following$ the$ signa l$ path$ from $ generatio n$ (this$ could$ be$ off - chip)$to$ the$OE$co n version$at$the$samples,$the$si gnal$enters$the$FFT$ via$ grating$coup lers$and$ is$ 2:1$ fanned - out$in$ two$ stages$ [ Fig.$ 1 ( b )] .$ The$ signal$ to $ be $ processed$ can$be$ generated $ either$ON$or$OFF - chip.$ Interest ingly,$ the$ FFT$ dela y$ decreases$ linearly $ with$ increas ing$ modu l ation$ rate,$ a$ unique$ feature$ of$ optics.$ The$ on - chip$ portion $ of$ the$ OFFT $ consists$of$cascaded$delayed$interferometers$and$passive$ components$ such$ as$ directional$ co uplers,$ y$ branch es ,$ straig ht$ an d$ spiral$wa veguides , $ and$operate s $ on$time$domain$ signals.$ We$opted $ for$ the$ silicon$ on$ insulator $ ( SOI ) $ platform $ due$ to$ its$high$ refractive$ index $ difference , $ stabl e $ fabrication$p rocess , $ and$l ow$ cost $[ 7 ]. $ Whi le$ this$ work$ is$ a$ theoretica l$ ana lysis$ o f$ this$ on - chi p$ OFFT,$ we$ briefly$ discuss$ our$ tapeouted$ process$ (chip$delivery$ pending),$ since$ it$ the$ design - to - reality$ discrepancies$ are$ of$ interest$ here.$ All$ passive$ design,$ the$ on - chip$ OFFT$ inc luding$ a ll$ passive $ compo nents$ an d$ heaters$ for$ phase$ tunability$ are $ fabricat ed$ at$ IME ’s$ Silicon$ Photoncis$ general - purpose$ fabrication l. $ process .$ An $ extra$ oxide$ cladding$ is $ deposited$ to$ reduce $ propagation$ los s .$ To$ take$ advantage$ of$ the$ thermo - optic$ effects$ in$ sili con$ photonics,$ metall izat ion$ and$ sel ectiv e$ oxide$ rel ease$ are $ selected $ to$ create$ the$ active$components$needed$ for$tuning$the$ refractive $index$ of$silicon$ as$a$function$of$ temperature$(resistive$he ating$process). $ $ $ Fig.$ 1 .$ ( a)$ Cooly - Tukey$ Butterfly$schematic$ of$ OFFT$ ( b )$ Optical$ B* I* J* FFT$ on$ chip$ with$ heaters$ and$ modulation$ on - chip$ (ideal$ design)$ with$ a$ waveguide$ width$ of $ 0.5$ 𝜇𝑚 $ and$ the$ heaters$ width$ were$ 8$ 𝜇𝑚 $ to$ enable$ the$ heaters$ ca rry$ twice$ the$ level$ of$ current$ and$ the$ resistance$ will$ be$ about$ half$ the$ pre vious$ v alue. $ λ signal $ =$ 1550 $ nm. $ OFFT$ substrat e$ uses$ a$ silicon$ thickness$ of$ 220$ nm$ with$ buried $ oxide$(BOX)$thickness$o f$2 𝜇𝑚 . $ $ The$ length$ of$ the$ M ZI$ is$ determined$ by$ the$ system$ frequency.$ For $ the$ system$ frequency ,$ , 8 ,$ the $ delay,$ K ,$ of$ waveguides $ in side$ the$ MZIs $ is$ determined$ by$ 𝑓 ! = 10 𝐺𝐻𝑧 $-$ where$the$ eff ecti ve$ ind ex$ is : $$ 𝑛 !"" = 2 . 5 ,$$ the$tim e$delay$bec omes $ 𝑇 !"#$% = 1 / 𝑓 ! = 10 ! !" 𝑠 $ and$ the$ physical$length,$ $ 𝑑 !" = 𝑐 / 𝑛 𝑇 !"#$% = 12 𝑚𝑚 .$ F or$the$ 10 $ GHz $ sampling$ f requency ,$ t he$ MZIs$in$ the$ first$ stage$ O FFT,$ with $ T/2$ has $ a$ length$ of$ 6 $ mm$ and$ for$ T/4$ is$ 3 $ mm. $ The$ length$ of$ the$ shorter $ arms$ of $ the$ M ZI$ for$ the$ first$ stage$ is$ 500$ 𝜇𝑚 an d$ for$ the$ second$ stage$is$ 440 $ 𝜇𝑚 .$ $ The$ MZI$ must$ compensate$ for$ phase$ shifts$ created$ by$ fabrication$ and$ tempera ture$ variance.$ We$ selected$ the $ short$ arm$ of$ the$ MZI $ heater - tunable$ over$ a $ 𝜋 / 2 $ relative$ phase$ shift $ a nd$ we$ ca n$ use$ the$ following$ calculation$ for$ the$ needed$ temperatu re$ change . $ $ ! ! !"" !" ≈ !" !" $ and$ for$Silicon $ !" !" = 1 . 9 × 10 ! ! 𝐾 ! ! ,$ giving : $ ∆ 𝜙 = ! ! = ! ! ! !" !" ∆ 𝑇𝐿 𝟏 For$ the$ second $ MZI$ with $ 𝐿 = 500 𝜇𝑚 ! the$ temperatu re$ change$ required$ (solving$ for $ ∆ 𝑇 )$ is $ 4. 2$ K. $ Thus,$ we$ find$ that$ temperature$is$a$c ritical$para m ete r$that$ m ust$b e$ maintained .$ To$ do$ so,$ we$ integrate $ resistive$ heaters $ on$ one$ of$ the$ MZI$ arms$ such$ that$ with$ the$ change$ in$ temperature,$ the$ desired$ refractive$ ind ex $ cha n ge $ prov id e s $ the$ relative$ phase$ shift$ (in$ this$ case 𝜋 / 2 )! that$ is$ required$ to$ perform$ the$ FFT. $ Heaters$ tune$ effective$ index$ of$ the$ Silicon$ w a vegu ide $ via$ the$ k now n $ thermo - optic$ effect .$ Interestingly,$ higher$ modulating$ rates$ dictate$ a$ smaller$ waveg uide$ length$ difference ,$ henc e $ requiring$ a$ higher$ temp eratu re $ for$ detu n in g . $ S olving$ for$ ∆ 𝑇 $ we$ obtain$ a$ temperat ure$ change$ of $ 4. 2$ K$ to$ create$ a$ 𝜋 / 2 $ phase$ shift. $ Thus, $ heaters $ are$ capable$ of$ producing$ the$ necessary$phase$shi ft$to$compensate$for$fabr ication$ variance $ with$ the$appro priate$ control$ and$fee dback .$ W e$ selected$ a$ minimum $ wave guide$bending $r adius$ of$ 50$ 𝜇𝑚 $ to$ keep$ the$ radiative$ bending $ losses$ low . $ This$ results$ in$ a$ delay - line$ spiral$ area$ of$ 3. 9 x10 - 3 $ mm 2 $ for$ K .$T he$ total$ area$ of$ th is$ B $ =$ 4$ OFFT$ is$ 0.0 12 $ $ mm 2 .$ To$ make$ this$ design$ more$ compact$ the$ sampling$ in$ the$ ac tive$ design$ can$ be$ done$ on - chip$ using$ MZI $ modulato rs$ following$ Ref. $ [8]. $ Th is$incre ases$the$ total$ OFFT $ area$ by$about$ 60%$ to$ 0.019 $ mm 2 . $ Sampling$ is$required $t o$ obtain$ the$frequency$ components$ o f$ the$ transfer$ function $ of$ the$ FF T,$ performed$ here$ via$ electro - optic$ modula tors$ ( EOM ) .$ The$ OFFT$ design$ allows$ placing$ the $ EOM s $ either$ before$ or$ after$ the$ FFT$ butterflies.$ However,$ i n$ order$ to$ minimi ze $ detuning $ phase,$delay,$and$optical$loss$ differ ences$acr oss$ the$ four$ m odulators ,$ the$ EOMs$ can $ appear$ at$ the$ end$ of$ the$ last$ stage$ of$ OFFT$ to$ ensure$ synchronous$ sampling$ (F ig.$ 2 ). $ Here$ we$ opted$ for$ Michelson$ modulator s$ rather$ than $ MZIs$ due$ to$ their$ smaller$ (about$ 50% ) $ 𝑣 ! 𝐿 ! $ when$ operat ed$ in$ DC$ voltag es. $ This$ is$ especially$ rewarding$ in$ optical$ circuitry$ for$ minimizing$ power .$ To$ avoid$ power$ mismatch$ loss$ from$ the$ difference$ in$ waveguide$ length$ of$the$ca scaded$MZ Is,$ ‘waivy’ $ wavegu ide$ bends$ are $ added$to $ the$ sh orter$ arm$ of$ the$ interferom eters$ to$ compensate $ fo r$ the$ power$loss$at$ the$output$of$t he$couplers.$ $ T he$ transfer$ function$ of$ the$ entire$ system$ was $ studied $ to$ obtain$ performance$ characteristi cs .$ The$ FFT $ separates $ freque ncy $ contributions$ of$ the$ temporal$ input$ signal.$ Since$ the$ system$ frequency$ is $ 10 $ GHz,$ the $ fre quency$ spa cing $ of $ the$ OFFT$ output$ ch annels$ are$ approximately $ 10 $ GHz ,$ but$ the $ exact $ locati on$ of$ the$ probe$frequency$ for$which$ the$maxi mum$ tran smissi on$is$obtained $ for$d ifferent$ outputs $ is$a$ function$ of$tim e$dela y$and $ can$vary$ across$ the$ outputs .$ For $ e ve ry$ transmission$ peak$ at$ a$ specific$ frequency$ the$ contributions$ from$ the$ neig hboring$ channels$ is $ either $ mini mal$ or$zero$ [ Fig.$3 ( a ) ] . $$ Fig.$ 2 .$ ( a)$ S ensitivity$ a na lysis$ t ests$ setup$ o f $ the$ OFFT’s$ first$ stage$ interferom eter$ in $ term s$ of $ phase$ sweep$ from$ π/2$ ±$ π/2$ with$ incremen ts$ of$ π/100. $ ( b)$ Time $ delay$ from$ 12.5$ ±12.5$ p s$ (T/4$ ±T/4)$ p s$with$increments$of$0.5$ p s$ corresponding$to$an$optical$loss$ from$ 6.2 5$ ± $ 6.25$ dB$ with$ increments$ of$ 0.5$ dB$ for$ the$ first$ stage$ Mach - Zehnder$interf erometer. $ $ We $ are$ intere s ted$ in$ the$ FFTs$ performance$ with$ detuning$ critical $ parameters$ such$ as$ phase,$ delay,$ and$ loss$ fro m$ their$ design$ components. $ Understandi ng$ the$ phase$ sensitivi ty$ and$ variati ons$ across$devices$can$be$critical$as$has$been$seen$during$development$ o f $ ring - resonators $ [ 9 ]. $ This$ is$ obtained$ by$ d efining$ performance$ parameters,$and$ analyzing$ the$ transfer$function$ and$ the$ extinction$ ratio$ of$ the$ cascaded$ interferometers . $ The$ discrete$ nature$ of$ the$ OFFT$ yields$ intrinsic$ quantiz ation$ errors$ in$ frequency$ and$ sampling$ artifacts$ that$ a re$ a$ function$ o f$ the$ phase . $ First,$w e$ s weep $ the$ pha se $ (0$ to$ π ) ,$ t ime$ de lay$ and $ loss$ in$ p articular$ that$ of$ th e$ lower$ ar m$ of$ the$ interferomet er$ in$ the$ first $ stage$ of$ th e$ OFFT $ (Fig.$ 2 ) $ a nd$ analyze$ the$ change$ in$ the$ transfer$ functions $ of$ the$ output . $ This$ location$ is$ particularly$ impactful$ on$ the$ transfer$ function, $ since$ it$ has$ the$ highest$ oscillation$ and$ narrow$ spacing$ in$ the$ frequency$ domain $[ 5 ].$Our$ob servable$is$ the$fre quency$detu ning$of$ the$ maximum$ point$ of$ the$ transfer$ function,$ w hich$ for$ instance$for$ output$port$ L ? $ appears$near$6.8 $ GHz$[Fig. $3(a)].$At$the$ideal$case$ for$ vanishing$ phase$ detuning ,$ the$ entire$ powe r$ exits$ to$ the$ secon d$ branch$ L ? $ of$the$ top$interferometer$ due$to$ the$additional$ phase,$i.e.$ the$relative$phase$difference$ in$b idirectional$coup ler.$ $ $ Fig.$ 3.$ (a)$ Frequency$ Sensitivity$ Analysis$ on$ the$ transmission$ power$ (transfer$ function)$ of$ OFFT$ at$ ideal$ phase$ (b)$ Phase$ Sensitivity$Analysis$ on$the$ transmission$ power$at$ probe$ frequency$ of$6.78$GHz. $ With $ phase $ detuning$away$f rom$ the$ ideal$ design$ point $ however ,$ the $ energy$of$the$output $ transfer$function s$ leaks$to$ th e$neighboring$ ports$ [ Fig.$ 3( b )] ; $f or$ instan ce , $ the$ power$ of$ the $ first$ interferometer$ shifts$to$ the$top$ of$the$ second $ stage$lower$ interferometer$ exiting $ of$ at$ the$ nex t$ frequency$ prob e$ (16.78 $ GHz) , $ since$ each$ output$ bin$ in$ the $ frequ ency$ dom ain$ has$ a$ 10 $ GHz $ spacing$ given$ by$ the$ system $ frequency.$ By$ sweeping$ the$ frequency,$ we$ can$find$ points$ at$ which$ th e$ OFFT$ has $ full$ transm ission ,$ these$ points$ are$ defin ed$ by $ time$ delay .$ Hence$ the$ effective$ impact$ of$ the$ phase$ detunin g$ is$ a$ lateral$ frequency$ sh ift.$ In$ the$ ideal$ case$ of$ no$ addition al$ phase $ and$ at$ different$probe$ frequencies $ (frequency$ at$ which$ each$ OFFT$ output$ has$ a$ maximum$ transmission$ in$ power) ,$ full$ tr ansmission$ can$ be$ a chieved . $ H owever$ sweeping$ the$ ph ase,$ changes$ the$ outpu t$ amplitudes$ respectively$ whe re $ the$ m aximum$ tra nsmission$ decreases$ until$ it$ vanished $ and$ the$ next$ full$ transmission$ from$ another$ output$ port$ is$ achieved ,$ which $ is$ a$ cyclic$ behavior. $ Note$ that$ the$ value$ of$ 6.78$ GHz$ is$ not$ significant$ as$ it$ can$ b e$ shifted$ by$ detuning$the$delay$ lines .$ However , $ to$determine$ the$overa ll$quality$ of$ this$ cascaded$ system$ of$ interferometers,$ we $ set$ a $ th reshold $ for$ the$ phase - detuned$ power$ ratio$ (i.e.$ freq uency$ power$ leakage)$ betwe en $ target - to - neighboring$ output $ ports ,$ while $ the$ threshold$ selection$ is$ arbitrary$ and$ applicati on$ dependent ,$ we$ here$ chose$ - 20 $ dB$ channel - to - channel$ separation$ typical$ for$ telecommunica tion $ applications$ [ 5 ] .$ Given$ this$ threshold,$ the$ maximum $ phase $ t oleran ce$ is$ < $ 0.2$ radians$ to$ ensure$ acceptable$ spectral$ leakage,$ i.e . $ channel$ cr osstal k $ [ Fig. $ 4( a )] .$ Physically$ this$ range$ correspond s$ to$a $ small$( 0.54 $ K) $ temperatu re$change$ that$the$ wavegui de$ index$ can$ tolerate $ to$ keep $ within$ the$ - 20 $ dB $ attenuation$ threshold . $ C onsequently, $ the$ phase$ control$ m ust$ be$ pr ecise$ and$ the$ required$ temp erature$ difference$ needs$ c areful$ environment$ con trol$ to$ be$ ach ieved.$ One$ potenti al $ approach$ is$ to$ place$ the$ OFFT$ chip$ in$ an$ ambient$ chamber$ w ith$ temperature$ isolation$ such$ that$ the$ heat$ co uld$ be$ tr ansported$ to$ only$ the$ specific$ areas$ as$ desired . $ Alternative ly,$ control$ loops$ and$ temperature$ stabilization$ of$ th e$ chip$ cou ld$ also$ be$ employed.$ Indeed,$ we$ observe$ a$n on - linear$ phase$error ,$ which$ is$ likely$ d u e$to$ nature$ of$ cascaded$ interferometers $ and$ their$ phase$ sensitivity$ with$ respect $ to$ ph ysical$ delay$ lines.$ T o$ probe $ the$ effects$ of$ phase$ detuning $ errors$ and$ distortions$ in$ the$ signal $ fu rther ,$ we $ study$ the$ difference$ in$ the$ transmission$ power$ as$ a$ function$ of$ phase $ ( 𝑃 !"#$%!%&'() ) $$ [Fig.$ 4] .$ Thus , $ the $ 𝑆𝑁𝑅 = ( ! !"# ! ! !"#$%!%&'() ) ! !"#$%!%&'() $ is$ obtained$ by$ taking$ the$ difference$ in$ the$ transmission$ output$ power$ values$ relative$ to$ the$ ideal$ zero $ phase.$ The$ SNR$ indicates$ the$ degradation$ in$ the$ system$ as$ a$ function$ phase$ de tuning$ and$ determines$ the$ performance$ quality$ of$ the$ OFFT $ in$ resp onse$ to $ optical$ noise$ created$ from$ phase .$ Here$ we$ aim$ to$ understand $ the$ range$ at$w hich$the$ OFFT$ can$ be$opera ted$with$ minimal$ sacrifice$ in$ power$ and/or$ maximum$ stabil ity$ a nd$ qu ality.$ $ We$ define $ the$ power$mismatch$ratio n$and$figure$of $merit$(FOM)_as$f ollows: $ 𝑃 !"#!$%& ! !"#$% = 𝑃 !"# ! 𝑃 !"# ! 𝜙 # ( 𝟐 ) 𝐹𝑂𝑀 = 𝑆𝑁𝑅 𝑃 !"#!$%& ! !"#$% # ( 𝟑 ) The$idea$b ehind$the$definition$of$FOM$ is$that$the$sma ller$the$powe r$ mismat ch$ ratio$ ( deviation$ from $ unity ) ,$ and$ the$ higher$ the$ SNR$ in$ the$ syste m,$ the$ higher$ is$ the$ quality$ of$ the$ O FFT$ as$ a$ function$ o f$ detuning$ phase.$ As$ a$ result , $ the$ system$ has$ the $ highest$ FOM$ at$ the$ ide al$ zero$ phase$case$ as$expected$ for $ L ? $ since$ t he$power$ mismatch$ ratio$ betw een $ L ? * and *L > $ is$ close $ to$ 1,$ where $ th e$ FOM$ div erges .$ For$ the$ca se$of *L ! * and $ L M $ however,$$ their$SNR$values$are$ low ,$ despite$the$ p ower$ mismatch$ ratio$ being$ close$ to$ unity$ 1 ,$ since$ their$ transmission$ is$ minim al$ for$ fre quency$ contribution$ at$ 6.78$ GHz . .$ The $ maximum$ FOM$ for$ all$ four$ outputs$ aligns$ with$ the$ design$ probe $ frequenc y$ value$ for$ e ach$ freque ncy$ bin.$ As$ the$ phase$ is$ swept$ the$ FOM$ also$ decrea ses$ drastically$ for $ L ? $ an d even$ further$ for * L M * since$ the$ OFFT$ output$ is $ no$ longer$ at$ the$ probe$ frequency$ ( max.$ transmission) , $ as$a$result$SNR$decreases $ as$well .$ $ $ Fig.$ 4 . $ Degradation$ for$ - 20$ dB$ tolerance$ (a)$ and$ FOM$ (b)$ as$ a$ function$ of$ detun ing$ phase$ at$ 6.78$ GHz$ probe$ frequency - $ where$ L ? $ has$ the$ ideal$ FOM$ while$ L M* is$ m inimal$ leakage$ from $ L M* bin$ (similar$ to$ L >** and * L ! , $ not$shown). $ For$ perfect$ OFFT$ filtering$ and$ op timal$ transm ission,$ the$ output$ power$ of$ the$ cascaded$ MZI$ arms$ must$ match. $ This$ howeve r$ is$ challenging$ since$ in$ the$ OFFT$ design$ the$ MZI$ arms$ have$ different$ physical$ lengths$ (waveguides)$ hence$ in$ order$ to$ understand$ how$ the$d ifference$in$ length$chan ges$the$ q uality$of$the$OFFT$ output,$the$ MZI$ exti ncti on$ rati o $ (ER) $ was$ analyzed $ on$ the$ sweeping$ of$ the $ delay/additional$loss$in$ the$ first$ and$ corresp ondingly$ second $ stage$ of$ OFFT.$ Shown$ in$ [ Fig.$ 2 ( b )] .$ The$ ext inction$ ratio$ is$ define d$ as $ 𝐸𝑅 = ! !"# ! !"# 𝛾 !"## where P max * ( N '0) ) $ is$ the$ maxim um $ (mini mum) $ power$ at$the$ output $ of$the$ OFFT . $ Delay$lines$ corresp onding$to$ loss$ values$ were$ swept$ across$ the$ lower$ arm$ of$ the$ cascaded$ MZIs.$ Note$that$ in$the$ second$stage$the$ delay $ is$half$ of$ the$ first$ stage ,$ an d $ so$ is$ the $ loss. $ This$ is$ important$ for$ consistency $ and$ symmetry$ in$ the$ overall$ system$ d esign. $ As$ proven $ analytically $ by$ the$ ideal$ coupler ’s$ extinctio n$ ratio$ behavior $in $ the$ lower$ arm $[ Fig.$5 ( b )] ,$ the$ loss$ increases$ exponenti al $ with$ waveguide $ length$ as$ expected .$ However,$the$ l oss$ increases$ if$ the$ leng th$ imbalance$ increases.$ This$ is$ becaus e$ of$ the$ extra$ powe r$ mismatch $ between$ the $ MZI $ arms , $ impacting $ the$qu ality$of$the$ OFFT ’s$t ransfer$fu nction .$ Thus,$the$ aim$ is$to$max imize$ E R$ si milar$ to$modu lator s $ [ 10 ]$and$switch es$[ 11,$12 ] ,$ but$with$the$difference$of$improving$the$ power$mismatch$between$ the$MZI$arms$ rather$signal$on - off$rati o .$ $ $ Fig.$ 5 .$ (a)$ Extinction$ ratio $ of$ the$ O FFT$ full$ system$ as$ a$ function$ of$ physical$ optical$ loss$ from$ the $ spiral$ waveguides$ (delay$ lines)$ (b)$ Analytical $exponential $fit$( a )$based $ on$ an$ideal$coupler . $ The$ FFT$ data$ capacity , $ the$ nu mber$ of$ bits$ that$ can $ be$ propagated $ throug h$ the$ system ,$ depends$ on$ the$ modulation$ type ; $ a ssuming$ QAM$256$ for$ a$ high$SNR$channel $ wi th $ a$bandw idth$of$10$ GHz$ the$ upper$ bound$ for$ bandwidth$ is $ 80 $ G bps $ for$ a$ single$ OF FT$ channel$ and $ 320 $ G bps$ for$ B* I * J . $ Whil e$ we$ have$ ana l yzed$ the$ sensitivity $ and$ p erformance$ for$ B* = $ 4,$ it$ is$ in teresting $ to$ ask$ how $ larger$ sy stems$ scale.$ Increasing$ the$ num ber$ of$ samp les$ ( B ),$ our$ $ O FFT$ grows $ with $ O 𝑁 − 1 P* cascaded$ delayed$ interferometers$ and $ ?O 𝑁 − 1 P* couplers .$ U nlike$ an$ electronic $ FFT,$ w hich$ scal es$ with$ approximately 5 𝑁 log ! 𝑁 ,$the$o ptical$FFT $will$ne ed$to$co mpensate$ for$ inc reasing$ op tical$ losses$ with$ g reater$ optic al$ power $ [13] .$ Our$ FFT$ scaling$ a nalysis$ shows$ performance$ peaks$ for$ the$ OFFT$ on$ chip$ for$ small$ B $ which$ outperforms$ an$ electronic$ (NVIDIA$ P100$ GPU)$for$N $ < $ 200 $ [ 14 ]. $ $ Fig.$ 6 .$ Analysis$ of$ the$ optical $ FFT$ shows$ up$ to$ 3- orders$ of$ magnit ude$ higher $ performance$than$ a$ GPU$ (NVIDIA$P100)$ for$ B $< $ 200 $ using$ a$ figure$ of$ m erit$ of$ 1D$ FFT$ per$ second - Wat t - Area$ assuming $ insertion$los s :$ 0.9 $ dB $ (coupler),$ 3.5 $ dB $ (y - branch), $ 3.5 $ dB $ (modulator),$ 0. 7$ dB $ ( first$spir al $ delay$line,$then$with$linear$scal ing), $ photodetector$ power $ of$ 2.4 $ µW,$ and$ minimum$ optical$ pow er$ at$ the$photodetector$of$250 $ µW. $ In$ conclusio n,$ w e$ explored$ the$ design$ and$ operation$ sensitivity$ of$ a$ temporal$ B* I*J $ all - optical$fast$Fourier$transform$ (FFT)$on - chip $ based$ on$ silicon$ photonics .$ Our$ design$ based$ on$ cascaded$ interferom eters$ sh ows$ b oth$ pha s e$ and$ time$ delay $ (loss)$ sensitivity .$ We$ obtain$ the$ transf er$ funct ion$ of$ this$ photon ic$ function$ by$ mo nitoriing$ the$ frequency$ bin s$ at$ the$ ou tput$ p orts$ of$ the$ FFT $ sampled$ by$ e lectro - optic$ modulators.$ In$ our$ sensitivity$ analysis$ w e$ find$ that$ the$ thermal$ operat ing$ range$ is$ rather$ s mall$ (<1K)$ in$ order$ to$ adhere$ to$ teleco mmunication - relevant$ 0.2$ radians$ pha se$ threshold s$ w ith $ 20 $ dB$ tolerance$ in$ power$ loss.$ Control$ over$ this$ range,$ however,$ is$ possible$w ith$ thermal$ on - chip$ heaters$ on$ the$ MZI$ arms$ of$ the$ silicon$ wave guide$ sections.$ $ Unlike$ electronics,$ here$ the$ number$ of$ FFT$ data$ processed$ per$ second$ only$ depends$ on$ the$ time - of - flight$ of$ a$ photon$ through$ the$ milli meter $ short$ photoni c$ chip.$ As$ such,$ we$ find$ the$ performance$ (#FFT$ data$ per$ second ,$ pow er , $ and$ areal$ footpr int ) $$ outperforms$ state - of - the - art$ graphical$ processing$ units$ GPUs)$ by$ 3$ orders$ of$ magnit ude$ for$ B - scaling$ below$ 100.$ T aken$ together$ this$ temporal$ FFT$ shows$ how$ photonics$ enables$ data$ processi ng$ by$ simple$ routing$ light$ $ for$ an$ in - the - net work - computing,$ rat her$ than$ using$ photons$for$c lassical$trans ceiver$communication$i n$networks $ [15] . $ References " 1.$Schmogrow,$et$al.$Real - time$Nyquist$pulse$generation$beyond$100$ Gbit/s$and$its$re lation$to$OFDM,$Opt.$Express $ 20,$317 - 337$(2012) $ $ 2.$Hillerkuss,$D. $ et$al. $ 26$Tbi t$s−1 $ line - rate$super - channel$transmission$ utilizing$all - optical$f ast$Fourier$transform$processing. $ Nature$Photon. $ 5,$ 364 – 371$(2011). $ $ 3.$Goodman,$J.$W.$Introduction$ to$Fourier$Optics,$Roberts$&$Company$ (2005). $ $ 4.$Marhic,$M.E.$Discrete$ Fourier$transforms$ by$single - mode$st ar$ networks,$Opt.$Lett.$ 12(1),$63 – 65$(1987).$$ $$ $ 5.$Hillerkuss,$D.$et$ al.,$Simple$all - optical$FFT$scheme$enabling$Tbit/s$ real - time$signal$processing,$Opt.$Express $ 18 ,$93 2 4 - 9340$(2010). $ $ 6.$Zhan,$A.,$Colburn,$S.$ A.,$Trivedi,$R.,$Fryett,$T., $ Dodso n,$C.,$&$Majumdar,$ A.$(2016).$Low - Contrast$Diel ectric$Metasurface$Optics.$ACS$Photonics,$ 3,$209 - 214$(2016). $ $ 7.$Chrostwoski,$L.,$Hochberg, $M.$Silicon$Photonics$Design:$From$Devices$ to$Systems,$Cambridge$University$Press $$ (20 15 ). $ $ 8.$Patel,$D.,$et$al . $ High - speed$compact$silicon$photonic$M ichelson$ interferometric$mo d u la to r.$O p t.$E x p res s,$22 ( 22 ) ,$26 7 8 8$( 2 01 4 ). $ $ 9.$Peng,$Z.$Fattal,$D, $Fiorentino,$M.,$Beausoleil,$R.$CMOS - Compatible$ Microrin g$Modulators$ for$Nanophot onic$Interco nnect,$in $ IPR ,$ OSA$ Technical$D igest$(CD)$(OSA,$2010),$pap e r$IW A2. $ $ 10.$ Ma,$ Z.$ at$ al.$ Two - Dimensional$ Material - Based$ Mode$ Confinement$ Engineering$ in$ Electro - Optic$ Modulators," $ Q---* RA* +,* @#.A* K+$A* 0)* S3%)(A* -.#/(&A ,$23,$1,$1 - 8$(2017).$ $ $ 11.$ Ye,$ K.$ Liu,$ R.$ Soref,$ and$ V.$ J.$ Sorger, $ “A$ com pact$ plasmonic$ M OS - based$2x2$Switch”$Nanopho tonics,$4,$1,$pp.$261 - 268$(2015). $ $ 12.$Liu,$K.$et$al.$ Trench - coupler$based$Silicon$Mach - Zehnder$Thermo - optic$Switch$with$Flexible$Two - dimensional$Design, $Optics$Express.$24,$ 14,$15845 - 15853 $ (2016). $ $ 13. $ George,$J .$K. $ Towards$On - Chip$Optical$FFTs$for$ Convolutional$Neural$ Networks.$arXiv :$1708.09534 .$Accepted$at $ IE EE $Intern atio nal$ $ Conference$on$Rebooting$Computing$2017$(2017). $ $ 14.$NVIDIA.$(2016,$Oct$16).$ NVIDIA$TESLA$P100$GPU$Accelerator. $ $ 15.$ Narayana ,$ V.$K.$ et$a l. $ Mo rphoNoC:$Expl oring$the$De sign$Space$o f$a$ Configurable$Hybrid$NoC$using$Nanophotonics,$Microproce ssors$and$ Microsys tems, $ doi.org/10.1016/j.micpro.2017.03.006.$( 2017) . !