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Fast

and

slow

light

enhancement

using

cascaded

microring

resonators

with

the

Sagnac

reflector

Duy-Tien

Le

a

_,

_Manh-Cuong

_Nguyen

b

_,

_Trung-Thanh

_Le

c,∗

a_Posts_and_{Telecommunications}_Institute_of_Technology_(PTIT)_and_{Finance-Banking}_University,_Hanoi,_Viet_Nam
b_Le_Quy_Don_Technical_University,_Hanoi,_Viet_Nam

c_{International}_School_(IS-VNU),_Vietnam_National_University_(VNU),_Hanoi,_Viet_Nam

a

r

t

i

c

l

e

i

n

f

o

Articlehistory:

Received3September2015

Receivedinrevisedform31October2016
Accepted7November2016

Keywords:
Microringresonator
Fastlight
Slowlight
Siliconwaveguides
FDTD

Transfermatrixmethod
Multimodeinterference(MMI)
Microresonators

a

b

s

t

r

a

c

t

AcascadedmicroringresonatorbasedonsiliconwaveguideswithanMMI(Multimode
Interference)basedSagnacreectorisproposedinthisstudy.Bycontrollingthecoupling
coefcientswiththeusedoftheMMIbasedSagnacreector,thedoubleofbothpulse
delayandadvancementfortheslowandfastlightcanbeachieved.Thenewstructurecan
producethefastandslowlightphenomenonononechipwithadoubleofthetimedelay
andpulseadvancement.ByusingtheSagnacreector,thedeviceisverycompact.Transfer
matrixmethodandFDTD(FiniteDifferenceTimeDomain)simulationareusedtoobtainthe
characteristicsofthedevice.Thetransmission,phase,groupdelayandpulsepropagation
areanalyzedindetail.OurFDTDsimulationsshowagoodagreementwiththeanalytical
theory.

â2016ElsevierGmbH.Allrightsreserved.

<b>1.</b> <b>Introduction</b>

Inrecentyears,opticalmicroringresonatorshavebeenofgreatinterestforapplicationsinopticalcommunicationssuchas
opticaldelaylines,opticalswitches,modulators,lters,dispersioncompensatorsetc.[1,2].Micro-ringresonatorstructures
consistsofanumberofsinglemicro-ringresonatorscascadedinseriesorinparallelcanbeusedforhigherorderlterswith
extendedfreespectralratios[3]orswitching[4],modulatingapplications[5],fastandslowlight[6].

Analysisofthegroupdelayandtransmissioncharacteristicsofcascadedmicroringresonatorsusedforopticalltersand
dispersioncompensatorshavebeenstudied[79].However,thesestructureshavepositivegroupdelayandmainlydesigned
forpulsedelayapplications.Slowandfastlightgenerationareemergingasaveryattractiveresearchtopic.Varioustechniques
havebeendevelopedtorealizefastlightandslowlightinatomicvaporsandsolid-statematerials[10].Oneapplicationamong
thesetechniquesistocontrolthegroupvelocity

<i>v</i>

goflightpulsestomakethempropagateeitherveryslow(

<i>v</i>

g<c)orvery
fast(

<i>v</i>

g>cor

<i>v</i>

gisnegative),wherecisthevelocityoflight.

Inthisstudy,weproposeanewcascadedmicroringstructurebasedonsiliconwaveguideswithaSagnacloopreector.

TheSagnacloopreectorhasbeenappliedtomanyapplicationstructuressuchaslteringandfastlightstructures[11,12].
Bycontrollingthecouplingcoefcientsofthecouplerusedinmicroringresonatorsintheproposedstructure,negativeand
positivegroupdelaycanbeobtained.Thismeansthatthelightvelocitycanbecontrolledandthereforethefastandslow

Correspondingauthor.

E-mailaddresses:,(T.-T.Le).

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<b>Fig.2.</b>Transmission,phaseandgroupdelaycharacteristicsofthesinglemicroringresonator.

lightcanbeinducedbythestructure[1315].Here,weuseaSagnacloopreectorbasedonan1ì2MMI(Multimode
Interferencecoupler)attheendofthestructuretoenhancethefastandslowlight.TheuseofanMMIbasedreectorforthe
reectiontodoublethepulsedelayandpulseadvancement.Itisshownthatthegroupdelay,timedelayandadvancement
aredoubledcomparedtothecasewithoutusingtheMMISagnacloopreector.Weusesiliconmicroringresonatorsbecause
ofhighqualityoffabricationbyusingCMOScompatibleprocessanddevicecompactnesswithahighindexcontrastsystem.

<b>2.</b> <b>Design</b>

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<b>Fig.3.</b> Inputandoutputpulsesatthesinglemicroringresonator.

2.1. Singlemicroringresonator

ForasinglemicroringresonatorasshowninFig.1(b),theoutputfieldcanberelatedtotheinputfieldbytheexpression
[16]

H1=
E2
E1 =

1−˛1exp



j1



1−˛11exp



j1



(1)

whereE1,E2arethefieldamplitudeattheinputandoutput;1 and1=



1−|1|2arethetransmissionandcoupling
coefficientsofthecoupler;˛1 isthelossfactorintheringwaveguideand1=2_NeffLR1istheaccumulatedphaseshift
overtheringwaveguide.Neff istheeffectiverefractiveindexofthewaveguide,isthewavelengthandLR1=2R1isthe
circumferenceoftheringwaveguide.

Theeffectivephaseshiftofthemicroringresonatorcanbedefinedby

single=arg



_E
2

E1



=artan



˛12sin (ω)



1₊˛12



₋(1₊2_)˛
1cos (ω)



(2)

Thenormalizedgroupdelayisgivenbyn=−
dsingle

dω .Theabsolutegroupdelayisd=Tn,whereTistheunitdelayofthe
signalpropagatingoverthemicroringwaveguide.Theresonanceisoccurredatthephase1=2m,wheremisaninteger.
Atresonance,1>˛1theringresonatorandwaveguideisunder-coupledandleadingtopulseadvancementorfastlight;
when1<˛1,theyareover-coupledandleadingtopulsedelayorslowlight;thecriticalcouplingoccurswhen1=˛1.

The transmission, phase and group delay of the single microring resonator at the transmission coefficients 1=
0.9975,0.9966and0.99respectivelyareshowninFig.2.Theparametersaresetasfollows:thelossfactorofthewaveguide

˛1=1dB/cm,thelengthofthemicroringwaveguideLR1=300␮m.Thesimulationshowsthatthepositiveandnegative
groupdelaycanbeachievedbyadjustingthecouplingcoefficientofthecoupler.Itisassumedthatasiliconwaveguidewith
aheightof220nmandwidthof400nmandrefractiveindexNeff =2.25.

Wenowinvestigatethepulsepropagationoverthesingleringresonator.ItisassumedthattheinputpulseisGaussian
andcanbeexpressedas[17]

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<b>Fig.4.</b>Transmissioncharacteristicsofthecascadedmicroringresonators(a)=1=0.99and(b)=1=0.9975.

where0istheresonancewavelengthofthesinglemicroringresonator,THW=Tb/2isthebithalfwidthat1/e2intensityand
Tbisthebitperiod.FromthesimulationsofFig.2,theresonancewavelengthis0=1.54817␮m.Theinputandcorresponding
outputpulseswiththetransmissioncoefficients1=0.9975, 0.9966and0.99areshowninFig.3,wheretheinputpulse
widthTp=50ps[18].Thesimulationsshowthatpulsedelayof20pscanbeobtainedwhen1=0.99andwhen1=0.9975
thepulseadvancementof12psisobtained.

2.2. Cascadedmicroringresonators

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<b>Fig.5.</b>Inputandoutputpulsesatthecascadedmicroringresonatorstructure.

lightcanbeobtained.Here,weconsideraSCISSORasshowninFig.1withaSagnacloopreflector.Forsimplicity,weassume
thatNringresonatorsareidentical.Asaresult,thetransferfunctionoftheSCISSORcanbewrittenby

HSCISSOR=H1H2...HN=(
E2
E1
)
N
=



−˛exp



j



1₋˛exp



j





N

(4)

Here=1and˛=˛1isthelossfactorintheringwaveguideand=2_NeffLR.

Thetransmission,phaseandgroupdelayofthecascadedmicroringresonatorforN=1,2,3areshowninFigs.4and5.It
isassumedthatthetransmissioncoefficientofthecoupleris1=0.99and0.9975.Thesimulationresultsshowthatslow
andfastlightareinducedbyadjustingthecouplingcoefficients.Inaddition,thepulsedelayandpulseadvancementare
increasedbyNtimescomparedwiththesinglemicroringresonator.

2.3. CascadedmicroringresonatorswiththeSagnacreflector

Fig.1showsthecascadedmicroringresonatorwiththeSagnacreflector.Inthisstudy,weusean1×2MMIcouplerin
theSagnacreflector.Asaresult,thetransferfunctionoftheproposedstructureinFig.1canbeexpressedby

H=(2j˛sss)



−˛exp



j



1−˛exp



j





2N

(5)

wheresands=



1−|s|2arethetransmissionandcouplingcoefficientsofthecoupleroftheSagnacreflectorand˛sis
thelossfactorintheringwaveguideoftheSagnacreflector.

Fig.6(a)and(b)showsthetransmission,phase,groupdelayandoutputpulsespropagatingoverthestructurewithand
withoutSagnacreflector.ItisassumedthatthestructureconsistingofNidenticalmicroringresonators(N=1and2)with
thetransmissioncoefficientof1=0.99.ByusingtheSagnacreflector,weobtainthepulsedelaysof43psand83psfor
N=1and2respectively,comparedwith20psand40pswithoutusingtheSagnacreflector.

When1=0.9975,theundercoupledconditionoccurs.Therefore,thefastlightcanbeinducedbyusingtheproposed
structure.Fig.7(a)and(b)showsthetransmissioncharacteristicsandoutputpulsespropagatingoverthestructurewithand
withoutSagnacreflector.Itisshownthatpulseadvancementsof25psand50psareachievedwhentheSagnacreflectoris
used(comparedwith12psand24pswithouttheSagnacreflector).

Bycontrollingthecouplingcoefficientsofringresonators,thefastandslowlightcanbeachieved.Thepulsedelayand
advancementcanbeincreasedbyNtimesifNidenticalringresonatorsareused.Fig.8showsthetimedelayandadvancement
ofthepulsepropagatingthroughourprosedstructure.WecanseethatbyusingtheSagnacreflector,thepulsedelayand
advancementcanbedoubledcomparedwiththeconventionalSCISSORstructure.

Toverifytheaccuracyofthetransfermatrixanalysis,wecomparetheresultsobtainedwiththeFDTD.ForourFDTD
simulations,theradiusofthemicroringresonatoristobeR=5␮m,thewaveguidewidthisWa=400nm,thegapbetween
themicroringwaveguideandthestraightwaveguideischosentobeg=160nminorderforthepowertransmissioncoupling
(||2₎_to_be_||2₌_0.9_as_shown_in_Fig.₁₀_(a)._Here_we_take_into_account_the_wavelength_dispersion_of_the_silicon_waveguide
usingtheexpressionNeff(␭)=4.7020−1.6667for=1.5−1.6␮m(Fig.10(b)).

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<b>Fig.8.</b>TimedelayandadvancementwithandwithouttheSagnacreflector.

<b>Fig.9.</b>Directionalcouplerusedformicroringresonator.

<b>Fig.10.</b> FDTDsimulations(a)transmissioncoefficientatdifferentgapand(b)wavelengthdispersionofthesiliconwaveguidewithawidthof400nm(the
insetshowsthefieldat=1.55␮m).

tothetransmissioncalculatedbytheanalyticaltheory.Figs.11(b)and12(b)showtheFDTDfielddistributionsatonand
off-resonances.

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<b>Fig.11.</b>FDTDsimulationoftheproposedstructurewithoneringresonatorandSagnacreflector.

<b>Fig.12.</b>FDTDsimulationoftheproposedstructurewithtworingresonatorsandSagnacreflector.

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grantnumber“103.02-2013.72”andVietnamNationalUniversity,Hanoi(VNU)underprojectnumberQG.15.30.
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