JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 3, FEBRUARY 1, 2014 505 Wavelegth Lockig ad Thermally Stabilizig Microrig Resoators Usig Ditherig Sigals Kishore Padmaraju, Studet Member, IEEE, Dyla F. Loga, Takashi Shiraishi, Member, IEEE, Jaso J. Ackert, Adrew P. Kights, ad Kere Bergma, Fellow, IEEE, Fellow, OSA Abstract The badwidth bottleeck loomig for traditioal electroic itercoects has drive the cosideratio of optical commuicatios techologies as realized through the complemetary metal-oxide-semicoductor-compatible silico aophotoic platform. Withi the silico photoics platform, silico microrig resoators have received a great deal of attetio for their ability to implemet the critical fuctioalities of a o-chip optical etwork while offerig superior eergy-efficiecy ad small footprit characteristics. However, silico microrig-based structures have a large susceptibility to fabricatio errors ad chages i temperature. Itegrated heaters that provide local heatig of idividual microrigs offer a method to correct for these effects, but o largescale solutio has bee achieved to automate their tuig process. I this cotext, we preset the use of ditherig sigals as a broad method for automatic wavelegth tuig ad thermal stabilizatio of microrig resoators. We show that this techique ca be maifested i low-speed aalog ad digital circuitry, ledig credece to its ability to be scaled to a complete photoic itercoectio etwork. Idex Terms Frequecy locked loops, multi-processor itercoectio, optical itercoects, optical resoators. I. INTRODUCTION GROWING badwidth eeds are motivatig the replacemet of traditioally electroic liks with optical liks for applicatios as diverse as data ceters, supercomputers, embedded computig processor-memory itercoects, ad fiberoptic access etworks [1], [2]. For applicatios such as these, the silico photoics platform has received sigificat attetio because of its ability to deliver the ecessary badwidth, Mauscript received July 11, 2013; revised November 6, 2013; accepted December 6, 2013. Date of publicatio December 10, 2013; date of curret versio December 27, 2013. This work was supported i part by the Natioal Sciece Foudatio ad Semicoductor Research Corporatio uder Grat ECCS- 0903406 SRC Task 2001. This work was also supported by the Natural Scieces ad Egieerig Research Coucil of Caada. The work of K. Padmaraju was supported by a IBM/SRC PhD fellowship. K. Padmaraju ad K. Bergma are with the Departmet of Electrical Egieerig, Columbia Uiversity, New York, NY 10027 USA (e-mail: kpadmara@ee.columbia.edu; bergma@ee.columbia.edu). D. F. Loga is with the Raovus Ic., Ottawa, ON A6 K1A 0R6, Caada (e-mail: dyla@raovus.com). T. Shiraishi is with the Departmet of Electrical Egieerig, Columbia Uiversity, New York, NY 10027 USA, ad also with the Photoics Laboratory, Fujitsu Laboratories Ltd., Kaagawa 2430197, Japa (e-mail: ts2821@ columbia.edu). J. J. Ackert ad A. P. Kights are with the Departmet of Egieerig Physics, McMaster Uiversity, Hamilto, ON L8S 4L8, Caada (e-mail: ackertjj@ mcmaster.ca; akight@mcmaster.ca). Color versios of oe or more of the figures i this paper are available olie at http://ieeexplore.ieee.org. Digital Object Idetifier 10.1109/JLT.2013.2294564 Fig. 1. A optical lik composed of microrig-based devices. A wavelegth source (λ-source) is modulated by multiplexed microrig modulators (doped i a diode cofiguratio to eable fast carrier-iduced resoace shifts). A microrig switch ca the route the etire set of sigals appropriately before it is received by a demultiplexig microrig array. ad by leveragig its compatibility with complemetary metal-oxide-semicoductor (CMOS) fabricatio, at a potetial ecoomy of scale. I particular, silico microrig resoator based devices exhibit leadig metrics o size desity, eergy-efficiecy, ad ease of wavelegth-divisio-multiplexed (WDM) operatio [3]. Fig. 1 illustrates a portio of a evisioed microrig-based photoic etwork that would be used for trascribig electrical data sigals ito the optical domai, trasmittig ad routig them as ecessary, ad covertig the optical sigals back to the electrical domai at the termiatio of the lik. The begiig of the lik cosists of a multi-wavelegth laser source. These laser wavelegths are idividually modulated by cascaded microrig modulators i a multiplexed cofiguratio. The etire set of sigals ca the be routed as ecessary by microrig-based switches. Fially, they are received by a microrig array that demultiplexes the idividual sigals before receivig them o idepedet photodetectors. The basic cofiguratio of Fig. 1 is but oe of a myriad of proposed possibilities for photoic etworks eabled by microrigbased devices, with more complex etwork desigs fully leveragig the uique capabilities of microrigs [4], [5]. However, these proposed microrig-based photoic etworks curretly face severe challeges i the path towards commercial realizatio. Specifically, the relatively high thermo-optic coefficiet of silico combied with the wavelegth selectivity of microrig 0733-8724 2013 IEEE. Persoal use is permitted, but republicatio/redistributio requires IEEE permissio. See http://www.ieee.org/publicatios stadards/publicatios/rights/idex.html for more iformatio.
506 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 3, FEBRUARY 1, 2014 Fig. 2. A small dither sigal, applied thermally to the microrig resoator, results i a small modulatio of the optical sigal. resoators leds them susceptible to chages i temperature ad laser wavelegth. Additioally, fabricatio toleraces will likely result i microrig resoators that are iitially offset from their desiged operatig wavelegth. The domiat method of resolvig these problems is to use eergy-efficiet itegrated heaters to tue ad stabilize the microrig resoace to the laser wavelegth [6]. While demostratios of maual tuig validate the fuctioality of these heaters, for commercial implemetatios a eergy-efficiet ad scalable solutio to lock ad stabilize microrig resoators is required. There have bee several attempted solutios for wavelegth lockig ad thermally stabilizig microrig resoators [7] [11]. However, o prior demostrated system has satisfied all required criteria, that is, a system that is scalable, low-cost, eergyefficiet, immue to fluctuatios i optical power, compatible with WDM implemetatio, ad does ot require additioal photoic structures. Recogizig the limitatios of traditioal techiques i addressig the above listed criteria, we have pioeered a ovel method of lockig ad stabilizig microrig resoators. The uderlyig priciple of our method is to use ditherig sigals to break the symmetry of the microrig resoator [12]. For most applicatios, the miimum trasmissio poit of the microrig resoace is aliged with the laser wavelegth it is routig ad detectig. I this cofiguratio, it is difficult to lock the microrig resoace to the laser wavelegth by measurig just the optical power trasmissio, because the directio of resoace shift is ambiguous i relatio to the trasmissio of optical power. Historically, to overcome this, methods such as the Poud Drever Hall (PDH) techique have bee used to geerate ati-symmetric error sigals that ca be utilized by closed-loop feedback cotrollers [13]. However, the details of the PDH techique are ot suitable for the sceario of wavelegth lockig a microrig resoator, ad we look towards the use a ditherig sigal for the purpose of geeratig the desired ati-symmetric error sigal. Fig. 2 illustrates the ditherig mechaism (applied thermally), whereby a small modulatio is applied to the local temperature of the microrig i order to produce a small modulatio of the optical sigal. The geerated optical modulatio will either be i-, or out-of-phase with the drivig sigal, depedig o which side of the resoace the laser wavelegth is offset. By mixig the modulated optical sigal with the drivig ditherig sigal this iformatio ca be recovered as show i (1), where f D is the frequecy of the ditherig sigal, ad φ is the relative phase Fig. 3. A schematic of the device used i this experimet (ot to scale). The off-chip electroics iterfacig with the itegrated photoic elemets are show i the dashed box. Highlighted i the red box is the circuitry devoted to wavelegth lockig. (0 or π) of the modulated optical sigal cos(f D t) cos(f D t + φ) = 1 2 [cos(2f D t + φ)+cos(φ)]. (1) The higher harmoic ca be filtered, leavig the sig of the dc compoet {cos(φ)term} as a idicatio of the locatio of the resoace relative to the optical sigal. The ed product of this process is the desired ati-symmetric error sigal. II. DEVICE AND EXPERIMENTAL SETUP The device we used to demostrate our method is illustrated i Fig. 3, ad cosists of a 15-μm radius silico microrig resoator (o a SOI platform). A thi film titaium-based heater is situated directly above the microrig, separated from the microrigby1μm of oxide. The drop port of the microrig termiates i a defect-ehaced silico photodiode, eablig the moitorig of the optical power dropped ito the microrig [14]. The off-chip electroics implemetig the thermal ditherig system are show i the dashed box of Fig. 3. The electroics cosist of low-speed (<20-MHz badwidth) aalog ad digital ICs. The ditherig sigal is chose to be higher i frequecy tha the thermal fluctuatios it is moitorig. The optical sigal, modulated by the thermal ditherig, geerates a photocurret o the itegrated silico photodiode. The photocurret is coverted to a voltage usig a trasimpedace amplifier (TIA), ad the furthered amplified. A aalog multiplier IC (AD 633) is used to the mix the amplified sigal with the drivig ditherig sigal. A low-pass RC filter is used to suppress the ac compoet of the mixed product. The result of this process is the geeratio of a ati-symmetric error sigal
PADMARAJU et al.: WAVELENGTH LOCKING AND THERMALLY STABILIZING MICRORING RESONATORS USING DITHERING SIGNALS 507 Fig. 4. (a) The microrig resoace as it is subjected to thermal ditherig sigals of varyig magitude (simulatios i dashed). (b) The correspodig geerated error sigals (simulatios i dashed). that ca the be used for the purpose of iitializig (wavelegth lockig) ad thermally stabilizig the microrig resoator. (a) (b) A. Geerated Ati-Symmetric Error Sigal The use of the thermal ditherig sigal has the cosequece of reducig the extictio ratio of the microrig resoace. I Fig. 4(a), the simulated ad measured resoaces of the microrig resoator (Q of 14,000) are plotted for square-wave thermal ditherig sigals of magitude 0.1 ad 0.2 K. A larger thermal dither will result i a larger reductio of the extictio ratio. Fig. 4(a) shows the u-dithered resoace havig a origial extictio ratio of 17.3 db. For thermal ditherig of magitude 0.1 K ad 0.2 K the reductio i extictio ratio was measured to be 1.9 ad 4.8 db, respectively. Simulatios produced idetical results (Fig. 4(a)). While a larger thermal dither results i a larger reductio i extictio ratio, it has the advatage of producig a stroger error sigal. Fig. 4(b) plots the simulated ad measured waveforms of the geerated error sigal. The ati-symmetric respose of the error sigal [see Fig. 4(b)] clearly distiguishes betwee the red ad blue sides of the microrig resoace. Furthermore, the zero crossig of the Fig. 5. Simulatios of the geerated error sigal cotrast the use of a squarewave ditherig sigal versus the use of a siusoidal ditherig sigal. mootoic slope is located at the resoace miimum. Hece, a feedback cotroller ca easily stay locked to the zero crossig i order to lock the microrig resoace with the laser wavelegth. While a larger error sigal makes the system more robust agaist oise, we foud that the smaller 0.1 K ditherig sigal geerated a sufficiet error sigal for lockig ad stabilizig the microrig resoator. All further results described i this paper were doe usig a ditherig sigal of magitude 0.1 K. B. Optimizatio of the Error Sigal I our experimetal implemetatio we utilized a squarewave ditherig sigal rather tha the siusoidal ditherig sigal described i (1). I theory, the ditherig sigal ca be composed of ay periodic waveform. However, the square-wave offers the advatage of beig able to be sythesized easily i electroic circuitry. Additioally, the mixed product of (ideal) square-waves will produce a pure dc compoet ad oe of the higher harmoic compoets that are evetually filtered (see the Appedix). This has the cosequece of producig a larger error sigal for ditherig sigals of equivalet magitude. Fig. 5 graphs simulatios of the geerated error sigals whe usig a ideal square-wave ditherig sigal versus usig a ideal siusoidal ditherig sigal. As ca be see from the graph, the square-wave ditherig sigal geerates a error sigal that is twice as large i magitude. C. Immuity to Power Fluctuatios A sigificat advatage of the geerated error sigal is that it is relatively immue to chages i the optical power of the sigal. I future microrig-based optical etworks it is evisioed that optical paths ca be recofigured as ecessary to address dyamic badwidth allocatio requiremets. The isertio loss characteristics of optical paths will chage as they are re-provisioed, yieldig ucertaity i the optical power reachig ay give microrig resoator. As Fig. 6 shows the geerated error sigals will chage i magitude followig variatios i optical power. Additioally, the slope of the error sigal will also vary. However, a robust feedback cotroller (oe that is able to coted with the chage
508 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 3, FEBRUARY 1, 2014 Fially, it should be oted that the error sigal geerated from a modulated sigal will be smaller tha the error sigal geerated from a umodulated sigal. This is due to the broadeed spectrum of the modulated sigal. Our simulatios show that the error sigal geerated from a 10-Gbps NRZ sigal will be 20% smaller tha a error sigal geerated from a umodulated sigal (whe usig a microrig resoator with the prior discussed parameters). Fig. 6. The measured error sigals whe a 0.1 K thermal ditherig is applied ad the optical power of the laser ito the chip is varied. i slope of the error sigal) will be able to maitai the lockig betwee the microrig resoator ad laser wavelegth because the zero crossig of the error sigal remais costat. It should also be oted that, as demostrated, these error sigals ca be geerated with a relatively weak optical sigal reachig the photodetector. The fiber-to-fiber couplig loss of the chip was measured to be 20 db; assumig symmetric coupler ad waveguide losses, the power reachig the photodetector for the measured error sigals i Fig. 6 is the less tha 20 dbm. This figure covers the lowest optical powers that would be preset i a silico photoic lik due to the sesitivity limit o curret photodetectors [3]. D. Effect o Data Sigals For use i data applicatios, it is critical that the ditherig of the microrig resoace does ot egatively impact the itegrity of the optical data sigals. To test this, we simulated 10 Gb/s optical data sigals as they are routed from the iput port, through the microrig, ad ito the drop port of a dithered microrig resoator (a microrig i the demultiplexig cofiguratio). Fig. 7 shows the simulated eye patters of the received sigal whe a ditherig sigal of magitude of 0.1 ad 0.2 K was applied to the microrig, as well as whe o ditherig sigal was applied (as stated before, a 0.1 K ditherig magitude was sufficiet for experimetally lockig ad thermally stabilizig the microrig resoator). I these simulatios, we modeled the microrig resoator to have the same characteristics (extictio ratio ad Q-factor) as the microrig resoator we worked experimetally with [see Fig. 4(a)]. As expected, ad as evideced by the simulated eye patters, the ditherig has the effect of broadeig oly the 1 level of the optical sigal. However, this broadeig is miimal, resultig i eye closures of oly 0.3% ad 1.6% whe usig ditherig sigals of 0.1 ad 0.2 K, respectively. For small determiistic eye closures such as this, the power pealties ca be directly correlated as beig 0.01 ad 0.07 db, respectively. Power pealties of this magitude are well withi the optical lik budgets for microrig-based liks [3], ad hece will ot impede the use of the ditherig techique for data applicatios. III. WAVELENGTH LOCKING The first applicatio of the geerated error sigal is i the process of tuig the microrig resoator such that it is aliged i resoat wavelegth with the laser source. Deoted as wavelegth lockig, this is a critical fuctioality for ay give microrig-based platform as the lasers ad microrig resoators will be iitially offset i wavelegth due to fabricatio toleraces or chages i the ambiet temperature. The electroic circuitry devoted to the wavelegth lockig process is detailed i Fig. 3 (outlied by the red box). The fuctioality of this circuitry is succictly described i the state diagram of Fig. 8. A simple reset sigal is used to trigger the rampig of the voltage applied to the itegrated heater. As the microrig is tued to the laser wavelegth the error sigal will trip the system ito the hold state, i which the feedback cotroller is activated ad the microrig is locked ad stabilized agaist further drifts i temperature or laser wavelegth. Additioal logic ca be added to reset ad re-attempt the wavelegth lockig should it fail o its iitial attempt. The optical spectrum aalyzer traces i Fig. 9 demostrate the system lockig a microrig resoator to a laser (a ASE source is used to backgroud image the microrig s resoace). Iitially, the microrig resoace is at 1559.2 m, ad the laser is offset at 1560 m. Over the course of 50 s, the microrig is tued higher i wavelegth util the cotrol system detects the error sigal ad establishes the lock. To record the optical traces of Fig. 9, the ramp speed (rate at which the tuig occurs) of the system was drastically reduced, such that the wavelegth lockig would occur over the course of secods. I subsequet trials, the ramp speed was icreased to achieve wavelegth lockig i the ms time frame. I future implemetatios, the speed of the ditherig sigal ca easily be icreased to >1 MHzto allow the wavelegth lockig to occur i the μs time frame. At that poit, the fudametal limits o the speed of the wavelegth lockig will be determied by the iitial offset betwee the microrig resoace ad the laser wavelegth, ad the rate at which the itegrated heater ca tue the temperature of the resoator. Fig. 10 shows a oscilloscope measuremet of the heater voltage durig the wavelegth lockig process. Here, the ramp speed has bee decreased to allow lockig i the ms regime. The graph of Fig. 10 has bee aotated with the stages of the state machie, with the period (I) desigatig the heater voltage while the system is i the reset state, (II) desigatig the period i which the voltage is ramped, ad (III) desigatig the wavelegth-locked hold state, which occurs oce the microrig resoator ad wavelegth become aliged.
PADMARAJU et al.: WAVELENGTH LOCKING AND THERMALLY STABILIZING MICRORING RESONATORS USING DITHERING SIGNALS 509 Fig. 7. Simulatios of the eye patter of a 10 Gb/s optical sigal propagatig through a microrig with o ditherig (left), ad ditherig of magitudes 0.1K (middle) ad 0.2 K (right). Fig. 8. A state diagram describig the fuctioality of the wavelegth lockig circuitry (red box, Fig. 3). Fig. 10. Oscilloscope trace of the heater voltage as the microrig is wavelegth locked to the laser source. Aotated are the reset state (I), ramp state (II), ad hold state (III). Fig. 9. Optical traces show the microrig resoace beig tued ad wavelegth locked to a laser source. IV. THERMAL STABILIZATION The wavelegth lockig method we have demostrated serves as a effective meas to iitialize the microrig-based photoic lik. Oce the lik has bee iitialized, it is ecessary to guard it agaist thermal fluctuatios. Covetioally, to maitai the local temperature of the microrig, the heat geerated by the itegrated heater is icreased or decreased i respose to decreases or icreases i the ambiet temperature [10]. To implemet this, the thermal ditherig system was cascaded with a feedback system (as schematized i Fig. 3) to thermally stabilize the microrig resoator. To test the system, 10-Hz siusoidal thermal fluctuatios of magitude 5 K were geerated usig a exteral visible laser [9]. I order to verify the thermal stabilizatio, wavelegth scas were performed of a resoace adjacet i wavelegth to the resoace that the thermal ditherig & feedback system was locked to (see Fig. 11). As Fig. 11 shows, with the thermal ditherig & feedback system implemeted, the microrig resoace stays locked to the laser wavelegth, with the dyamic tuig of the heater couteractig the thermal fluctuatios iflicted o the microrig (see Fig. 12). I cotrast, without thermal stabilizatio, the resoace fluctuates severely, beig washed out i the wavelegth sca (see Fig. 11). This thermal stabilizatio method is robust eough that wavelegth lockig ca occur eve i thermally volatile eviromets. To demostrate this the microrig resoator was wavelegth locked while the microrig resoator was subjected to siusoidal thermal fluctuatios. Fig. 13 shows the heater voltage durig this process. At the momet i time that the microrig resoator is wavelegth locked, the system immediately begis the thermal stabilizatio mechaism (as evideced by the siusoidal couter-tuig of the heater voltage i Fig. 13). V. SCALABILITY AND POWER EFFICIENCY I order for the demostrated cotrol system to be adapted for commercial microrig-based liks it must be scalable. Scalability requires meetig two criteria, the first beig that the cotrol system is sufficietly low-power such that the aggregate power cosumptio of iitializig ad stabilizig all the microrig
510 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 3, FEBRUARY 1, 2014 Fig. 11. The microrig resoace whe subjected to thermal fluctuatios (T.F.), with ad without the ditherig & feedback system implemeted. Also show for compariso is the resoace without the stabilizatio system uder ormal, thermally stable coditios. Fig. 12. Heater voltage whe the microrig resoator is beig exposed to thermal fluctuatios ad, with ad without, the ditherig system actively couteractig said thermal fluctuatios. Fig. 13. Heater voltage as the microrig resoator is wavelegth locked while beig subjected to siusoidal thermal fluctuatios. resoators i the optical lik does ot exceed the powerefficiecy improvemet gaied by the use of microrig resoators. Secodly, the mechaisms i the cotrol system must be compatible with the typical WDM cofiguratio of microrig resoators, i which they are cascaded alog a commo waveguide bus. To get a estimate of the power cosumptio we replicate the approach of [10], tabulatig the active compoets of the cotrol circuitry (see Fig. 3) ad referecig established power cosumptio figures whe these compoets are implemeted i coservative CMOS techology. We ote that the majority of the circuitry devoted to wavelegth lockig (see Fig. 3, red box) is composed of digital logic elemets that oly draw power whe switchig logic states. Uder the assumptio that the optical lik s operatig time will be much loger tha the iitializatio period, the power cosumptio of these elemets are egligible. The feedback cotrol ca be decomposed ito 4 op-amps [10], with the TIA, uity buffer, ad summers cotributig aother 4 op-amps, for a total of 8 op-amps. Op-amps with the required MHz badwidth characteristics have routiely bee implemeted i CMOS techology with power cosumptios as low as 40 μw [15]. The ditherig sigal ca be implemeted usig a oscillator, with a example oscillator coverig the required sub-mhz to few-mhz rage while exhibitig power cosumptios as low as 20 μw [16]. Lastly, aalog multipliers are also routiely implemeted i CMOS techology, with a coservative example havig a power cosumptio of 45 μw [17]. The aggregate power cosumptio of the cotrol circuitry ca the be estimated to be 385 μw. To express this i the popular fj/bit metric we assume that the microrig resoator is operatig o a 10 Gb/s data sigal, yieldig a power cosumptio of 38.5 fj/bit for the cotrol circuitry. This estimate falls well withi the strictest pj/bit power budgets required of some evisioed applicatios [1]. Whe evaluatig the power cosumptio of this solutio, the power cosumptio of the itegrated heater should also be take ito cosideratio. However, this power cosumptio ca be treated idepedetly of the cotrol circuitry, ad will be a fuctio of the heater efficiecy, required tuig distace, ad variatio i temperature [10]. Advaces i itegrated heater desig cotiue to improve o their efficiecy [18] [20], idicatig that they will evetually adhere to the power cosumptio budget of microrig-based optical liks. Whe cosiderig the other importat criteria, that the cotrol method is compatible with WDM implemetatio, we ote that there are o iheret features of the ditherig techique that precludes the use of WDM. The ditherig is a process local to the microrig resoator ad does ot affect adjacet cascaded microrig resoators. Additioally, by usig differet ditherig frequecies for differet microrig resoators, the orthogoality priciple ca be leveraged (2) cos(f t)cos(f m t)dt =0,f f m. (2) This feature elimiates crosstalk i error sigals eve whe microrig resoators overlap i the spectral regime. Hece, WDM implemetatio ca be readily achieved, with each microrig resoator i the cascaded array iitializig ad stabilizig idepedetly. VI. CONCLUSION The demostrated system has bee show to be able to effectively iitialize ad thermally stabilize idividual microrig resoators. Our experimetal testbed limited us to testig the system agaist thermal fluctuatios of magitude 5 K. However, the temperature rage a implemeted solutio could cover is much larger. There are o iheret limitatios o the voltages supplied by the cotrol circuitry, ad itegrated heaters typically have very large temperature tuig rages (tested to be >80 K o our device). This should be sufficiet to cover the
PADMARAJU et al.: WAVELENGTH LOCKING AND THERMALLY STABILIZING MICRORING RESONATORS USING DITHERING SIGNALS 511 temperature variatios i evisioed applicatios. For istace, withi a data ceter rack, the projected temperature variatio is at most 20 K [21]. As discussed, the ovel use of thermal ditherig to geerate the error sigal has the advatage of givig the feedback system relative immuity to power fluctuatios. This reders the system robust agaist fluctuatios i the received power, ad resiliet to Fabry-Perot artifacts i the optical path. Additioally, the use of low-speed aalog ad digital ICs i the experimetal implemetatio leds credibility to the system s ability to scale i a eergy-efficiet maer to the multiple microrig resoators that comprise a WDM photoic itercoect. Thus, the fabricatio offset ad thermal issues that curretly plague microrig resoators ca be resolved i future commercial implemetatios, allowig them to be maifested i a variety of applicatios for the purpose of deliverig magitudes of order greater itercoect badwidth tha available with traditioal electroic itercoects. APPENDIX The Fourier represetatio of a ormalized 2π-periodic ideal square wave is give i (3) f(t, φ) = 4 π si(t + φ). (3) As per the costructio give i Sectio I, whe the geerated optical modulatio is i-phase with the drivig ditherig sigal, the product is give as (4) f(t, 0)f(t, 0) = 16 ( )( si(t) ) si(t). (4) The DC compoet is give as the itegral of (4). Usig the orthogoality priciple (2), the cross-terms ca be elimiated, leavig the o-zero terms give i (5) f(t, 0)f(t, 0)dt = 16 = 16 ( 1 2 ) si 2 (t) 2 dt 1 2 = 16 ( )( ) 1 π 2 =1 (5) 2 8 where the ifiite summatio has bee solved as a modified Basel series. Similarly, whe the geerated optical modulatio is out-of-phase with the drivig ditherig sigal, the product ad dc compoet are give as (6) ad (7), respectively, f(t, 0)f(t, π) = 16 ( ( si(t) ) si(t + π) ) (6) f(t, 0)f(t, π)dt = 16 si 2 (t) 2 dt = 1. (7) Hece, the dc compoet takes o a value of {1, 1}, i compariso to usig ormalized siusoidal waves (sectio I), i which the dc compoets have values of {1/2, 1/2}. ACKNOWLEDGMENT The authors gratefully ackowledge R. Grote for his group idex calculatio. Additioally, the authors are also grateful to CMC Microsystems ad IME Sigapore for eablig work i the desig ad fabricatio of the silico photoic chips. REFERENCES [1] A. V. Krishamoorthy, R. Ho, X. Zheg, H. Schwetma, J. Lexau, P. Koka, G. Li, I. Shubi, ad J. E. Cuigham, Computer systems based o silico photoic itercoects, Proc. IEEE, vol. 97, o. 7, pp. 1337 1361, Jul. 2009. [2] R. Urata, H. Liu, C. Lam, P. Dashti, ad C. Johso, Silico photoics for optical access etworks, i Proc. IEEE 9th It. Cof. 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512 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 32, NO. 3, FEBRUARY 1, 2014 [18] P. Dog, W. Qia, H. Liag, R. Shafiiha, D. Feg, G. Li, J. E. Cuigham, A. V. Krishamoorthy, ad M. Asghari, Thermally tuable silico racetrack resoators with ultralow tuig power, Opt. Exp., vol. 18, o. 19, pp. 20298 20304, 2010. [19] Q. Fag, J. Sog, X. Luo, L. Jia, M. Yu, G. Lo, ad Y. Liu, High efficiecy rig-resoator filter with isi heater, IEEE Photo. Techol. Lett., vol.24, o. 5, pp. 350 352, Mar. 2012. [20] M. R. Watts, W. A. Zortma, D. C. Trotter, G. N. Nielso, D. L. Luck, ad R. W. Youg, Adiabatic resoat microrigs (ARMs) with directly itegrated thermal microphotoics, i Proc. Cof. Lasers Electro-Opt., 2009, pp. 1 2. [21] R. R. Schmidt, E. E. Cruz, ad M. K. Iyegar, Challeges of data ceter thermal maagemet, IBM J. Res. Develop., vol. 49, o.4/5, pp.709 723, 2005. Kishore Padmaraju (S 07) received the B.S. degree (highest distictio) i electrical ad computer egieerig from the Uiversity of Rochester, Rochester, NY, USA, i 2009, ad the M.S. degree i electrical egieerig from Columbia Uiversity, New York, NY, USA, i 2011, where he is curretly workig toward the Ph.D. degree i the Departmet of Electrical Egieerig. His curret research iterests iclude cotrol systems for the iitializatio ad stabilizatio of silico photoics devices, as well as the use of silico photoic devices for advaced modulatio formats i optical commuicatio systems. Jaso J. Ackert received the B.Sc. degree i physics from the Uiversity of Waterloo, Waterloo, ON, Caada, i 2009. He is curretly workig toward the Ph.D. degree i the Departmet of Egieerig Physics, McMaster Uiversity, Hamilto, ON, Caada. His curret research focuses o moolithically itegrated silico photoic devices ad their applicatio to optical itercoects. Adrew P. Kights received the Ph.D. degree from the Uiversity of East Aglia, Norwich, U.K., i 1994 i the area of surface ad subsurface material characterizatio. His subsequet work took him first to the Uiversity of Wester Otario, Lodo, ON, Caada, where he performed research o the geeratio ad evolutio of implat iduced defects i silico, ad the to the Uiversity of Surrey, Surrey, U.K., as part of the U.K. Natioal Io Beam Cetre, researchig ovel fabricatio processes for micro- ad opto-electroic materials. I 2000, he joied Bookham Techology ad worked o a rage of silico-based, highly itegrated photoic devices. I 2003, he moved to McMaster Uiversity, Hamilto, ON, Caada, where he holds a faculty positio with the Departmet of Egieerig Physics. He curretly leads a research group workig o the iteractio of optical ad electrical fuctioality i silico-based structures. Dyla F. Loga received the B.Eg. ad Ph.D. degrees i egieerig physics from McMaster Uiversity, Hamilto, ON, Caada, i 2007 ad 2011, respectively. He recetly completed a postdoctoral fellowship at the Uiversity of Toroto, where he worked i itegrated waveguide oliear optics. He is curretly a Desig Egieer with Raovus Ic., Ottawa, ON, Caada. His research iterests iclude silico photoic device desig ad trasceiver architectures. Takashi Shiraishi (M 07) received the B.S. ad the M.S. degrees from the Tsukuba Uiversity, Ibaraki, Japa, i 2000 ad 2002, respectively, all i sciece ad egieerig. He is curretly workig as a Visitig Researcher with the Departmet of Electrical Egieerig, Columbia Uiversity, New York, NY, USA, ad also with the Photoics Laboratory, Fujitsu Laboratories Ltd., Kaagawa, Japa. His curret research focuses o the applicatio of silico photoic optical etworks to memory systems. Kere Bergma (S 87 M 93 SM 07 F 09) received the B.S. degree from Buckell Uiversity, Lewisburg, PA, USA, i 1988, ad the M.S. ad Ph.D. degrees from the Massachusetts Istitute of Techology, Cambridge, MA, USA, i 1991 ad 1994, respectively, all i electrical egieerig. She is curretly the Charles Batchelor Professor ad Chair of Electrical Egieerig at Columbia Uiversity, New York, NY, USA, where she also directs the Lightwave Research Laboratory. She leads multiple research programs o optical itercoectio etworks for advaced computig systems, data ceters, optical packet switched routers, ad chip multiprocessor aophotoic etworks-o-chip. Dr. Bergma is a Fellow of the Optical Society of America. She curretly serves as the CoEditor-i-Chief of the IEEE/OSA Joural of Optical Commuicatios ad Networkig.