On the dispersion management of fluorite whispering-gallery mode resonators for Kerr optical frequency comb generation in the telecom and mid-infrared range

0.98 MB·English

by  Guoping Lin

On the dispersion management
of fluorite whispering-gallery mode
resonators for Kerr optical frequency
comb generation in the telecom
and mid-infrared range
5
1
GuopingLin∗andYanneK.Chembo
0
2 FEMTO-STInstitute[CNRSUMR6174],OpticsDepartment,
15BAvenuedesMontboucons,25030Besanc¸oncedex,France.
n
∗[email protected]
a
J

0
1 Abstract: Opticalwhisperinggallerymode(WGM)resonatorshavebeen
very attracting platforms for versatile Kerr frequency comb generations.

] We report a systematic study on the material dispersion of various optical
s
c materialsthatarecapableofsupportingqualityfactorsabove109.Usingan
i analyticalapproximationofWGMresonantfrequenciesindiskresonators,
t
p
we investigate the effect of the geometry and transverse mode order on
o
the total group-velocity dispersion (GVD). We demonstrate that the major
.
s radii and the radial mode indices play an important role in tailoring the
c
i GVD of WGM resonators. In particular, our study shows that in WGM
s
disk-resonators,thepolarfamiliesofmodeshaveverysimilarGVD,while
y
h theradialfamiliesofmodesfeaturedispersionvaluesthatcandifferbyup
p to several orders of magnitude. The effect of these giant dispersion shifts
[ are experimentally evidenced in Kerr comb generation with magnesium

1 fluoride. From a more general perspective, this critical feature enables
v to push the zero-dispersion wavelength of fluorite crystals towards the
9 mid-infrared (mid-IR) range, thereby allowing for efficient Kerr comb
2
generationinthatspectralrange.Weshowthatbariumfluorideisthemost
3
interesting crystal in this regard, due to its zero dispersion wavelength
2
0 (ZDW)at1.93 µmandanoptimaldispersionprofileinthemid-IRregime.
. WeexpectourresultstofacilitatethedesignofdifferentplatformsforKerr
1
frequencycombgenerationsinbothtelecommunicationandmid-IRspectral
0
5 ranges.
1
© 2015 OpticalSocietyofAmerica
:
v
OCIScodes: (140.4780)Opticalresonators;(260.1180)Crystaloptics;(260.2030)Dispersion;
i
X (190.4380)Nonlinearoptics,four-wavemixing.
r
a
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1. Introduction
Optical frequency combs have been very useful scientific and technological tools in a wide
rangeofapplicationssuchasopticalclocks,precisionspectroscopy,lowphasenoisemicrowave
generationandopticalcommunications.Thegeneralapproachforproducingfrequencycombs
isbasedonmode-lockinginlasers.Analternativeapproachhasbeenrecentlydemonstratedina
silicamonolithictoroidalmicroresonator[1],wherecavityenhancedfour-wavemixing(FWM)
occurs.Asaresult,thistypeofcombusingthethird-ordersusceptibilityinresonatorsisusually
referredtoasKerropticalfrequencycomb.Theemergenceofsuchcombsprovidesacompact
andenergyefficientsolution.Therefore,extensiveamountofworksinboththeoriesandexper-
imentshavebeendevotedtothisfield.Forexample,theunderstandingofthecombformationis
nowveryelaborate,andinvolvesbothcoupled-modeandspatiotemporalmodels[2–7];various
applicationshavealsobeenexperimentallyexploredrangingfromlowphasenoisemicrowave
oscillations[8,9],ultra-shortpulsegeneration[10,11],andopticalcommunications[12].
Compared with Kerr comb generation platforms based on chip-scale waveguide racetrack
resonators[13],themechanicallypolisheddiskresonatorshaveamuchwiderrangeofmaterial
choicesandcanfeaturecombgenerationwithverylowpumppowerinthesub-mWlevel[9],
whichresultsfromtheultra-highquality(Q)factorsandsmallmodevolumesintheseWGM
resonators. Although Kerr frequency combs in fluoride WGM resonators have been demon-
strated in the telecommunication window and recently at 2.5 µm in the mid-IR regime [14],
there is still not a systematic study on the optimal choice of materials and the sizes of res-
onators.Moreover,anomalousGVDisgenerallydesirableforcombgenerationandthespan-
ningofKerrcombsisalsostronglydependentonthedispersionprofile.Recently,ananalytical
approximationforthedispersionofWGMsinthediskgeometryhasbeenderivedbytwodif-
ferent groups, showing a good agreement with finite-element simulations and high-precision
freespectralrange(FSR)measurements[15,16].Itenablesanaccuratestudyoftheimpactof
geometry on the GVD. On the other hand, barium fluoride (BaF ) with a good transmission
2
rangecoveringthewholemid-IRwindowhasbeenrecentlyshowntobecapableofsupporting
Qfactorsabove109 [17],andthisresultbroadensthechoiceofcrystallinecandidatesforKerr
combgeneration.
Inthisarticle,wereportthestudyofmaterialdispersionfordifferentIRandmid-IRoptical
materialsusingSellmeierequations.Wepointoutthatfluoridecrystallinematerialsgenerally
havesignificantlyflattermaterialdispersionprofilethanothermaterials.Asystematicandthe-
oretical investigation of the impact of resonator geometry and WGM indices on the GVD is
carried out, showing that the major radii and the radial WGM indices are key parameters to
considerforGVDtailoringinWGMresonators.TheimpactofgeometryonGVDisalsoex-
perimentallyevidencedinanexperimentshowinghowprimaryKerrcombswithsignificantly
different multiplicity can be generated with the same MgF resonator, but pumped in WGMs
2
(a)
R
r
(b)
v
G gain
v v
0 Fiber taper 0
Pump Primary comb
Fig.1. (a)Geometryofthediskresonator.(b)SchematicofKerrfrequencycombgenera-
tionsinafibertapercoupledresonatorsetup.WGMR:WGMresonator.v istheprimary
G
parametricgainpeakspacingfromthepumpfrequencyv .
0
with different radial orders. We also investigate theoretically MgF and BaF platforms for
2 2
potentialbroadbandcombgenerationsinthetelecomandmid-IRregimes.
2. InsearchoftheoptimalmaterialsforKerrcombgeneration
AsfarasKerrfrequencycombgenerationisconcerned,theprimarystephasbeenthesearch
of the optimal materials. Although the on-chip ring resonator as comb generation platforms
ismuchmorecompact,mechanicallypolishedhighQWGMcrystallineresonatorsarestillof
greatinterest,duetotheflexiblechoiceofmaterialsandultra-highQfactorsleadingtopower
efficientoperation.
In these axi-symmetric disk resonators, the rim part can be treated as a toroidal boundary
withthemainradiusRandthecurvatureradiusr,asillustratedinFig.1(a).Figure1(b)gives
a typical schematic drawing of Kerr comb generation. A single frequency continuous-wave
(cw)pumplasercanbecoupledintotheWGMRthroughafibertaperoraprism[18].Under
certainconditionsthathavebeenanalyzedinrefs.[2,3,7],thebuilt-upintracavitylightleadsto
themodulationinstability(MI)gainwithoffsetv fromthepumpfrequencyv .Theparametric
G 0
FWM then creates the first two sidebands where WGMs with single or multiple free spectral
range(FSR)relativetothepumpmodemeettheMIgain.Subsequentparametricprocesscre-
atesacombwithmultiple-FSRspacing,generallyreferredtoastheprimarycomb.Depending
on the dispersion and pump condition, broadband Kerr combs with single-FSR spacing and
thousands of frequency lines can be further produced. As Kerr comb arises from MI gain, it
thus prefers an anomalous dispersion profile similar to MI gain in optical fibers. In addition,
theMItheoryinKerrcombgenerationpredictsthattheoffsetv oftheparametricgainpeak
√ G
frequencyscaleswiththedispersionfollowingv ∝1/ GVD[2,3,7].Hence,weakdispersion
G
isgenerallyfavorableforbroadbandKerrcombgeneration.Itshouldbenotedthattheprimary
KerrcombshowninFig.1(b)alsoyieldsrolls(orTuringpatterns)inthespatialdomain[7,19].
Atthisdate,KerropticalfrequencycombshavebeenreportedonWGMresonatorsmadeof
fusedsilica,magnesiumfluoride,calciumfluoride(CaF )andsapphire[11,14,20–24],which
2
areallcapableoffeaturingQfactorsabove109.Recently,suchhighQfactorshavealsobeen
demonstrated on barium fluoride [17]. We thereby take into account this material with others
for the investigation of the material dispersion (GVD ) using their Sellmeier equations [25].
M
TheGVD isusuallycharacterizedbygroupdelaydispersion(inps/km/nm):
M
λ ∂2n(λ)
GVD =− (1)
m c ∂λ2
200
m) SiO2
n150 Telecom Al2O3
m/ MgF
C-band 2
k CaF
ps/100 BaF22
(
M
D 50
V
G
al 0
ri Mid-IR
e
at -50 2.5 - 10 μm
M
-100
1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
Wavelength (μm)
Fig. 2. Comparison of material dispersion for different optical materials: fused silica
(SiO ),sapphire,MgF ,CaF andBaF .DashedlinesdesignatethecorrespondingZDWs:
2 2 2 2
1.27 µm, 1.31 µm 1.34 µm, 1.55 µm and 1.93 µm, respectively. The vertical blue and
brownzoneshighlightthetelecommunicationC-bandandmid-IRwindows.Notethatthe
GVDMprofileofMgF2investigatedhereisderivedfortheordinaryray.Theextraordinary
onehasverysimilarprofilewithaslightlydifferentZDW at1.57µm.
whereλ isthewavelength,cisthespeedoflightinvacuum,n(λ)istherefractiveindex.
Figure2displaysthecorrespondingmaterialGVD asafunctionofwavelengthfrom0.8to
M
6 µm. It should be noted that the dispersion is anomalous when GVD is greater than zero. A
broadbandcombgenerationwillrequireasmallGVDvalueinthisregime.Ontheotherhand,a
GVDthatistoolargewillalsomakeKerrcombgenerationdifficult,asthecorrespondingpara-
metricgaincouldbetooclosetothepumpfrequencyanddonotmeetanyWGMs(thephase
matching condition will therefore never be fulfilled). One can also see that fluoride materials
haveflatterdispersionprofilesthantheothers.ThecorrespondingZDW ofthesematerialsare
listedinTable1.Alsoshownaresomeopticalcharacteristicsofthesematerials[26].
One can note that fused silica, quartz and MgF have anomalous GVD around the tele-
2 M
comwavelength,whileallthefluoridecrystalsaretransparentwellintothemid-IRregimeand
also feature anomalous GVD . Among them, BaF is characterized by a flat high transmis-
M 2
Table1.Characteristicsofbulkfusedsilica,Sapphire,MgF ,CaF ,BaF atT≈300Kfor
2 2 2
Kerrcombgenerationa
Name Transmission λ Refractive (dL/L)/dT dn/dT (10−6/K)
0
(µm) (µm) indexatλ (10−6/K)
0
Fusedsilica 0.2-4.0 1.27 1.4473 0.55 9.6
Sapphire 0.19-5.2 1.313 1.7502 6.65(a)/7.15(c) 13.6(o)/14.7(e)
MgF 0.13-7.7 1.34 1.3717 9.4(a)/13.6(c) 0.9(o)/0.3(e)
2
CaF 0.12-10 1.55 1.4260 18.9 -11.5
2
BaF 0.14-13 1.93 1.4648 18.4 -16.2
2
a (dL/L)/dT along the a and c axes are denoted. o: ordinary ray; e: extroordinary ray.
dn/dT forfluoridecrystalsandsapphirearegivenat1.15µmand0.589µm,respectively.
dn/dT forfusedsilicaat1.13µmistakenfrommanufacturingspecifications.
sion covering the whole mid-IR regime (2.5−10 µm). Moreover, the reported 2 nm surface
roughness in BaF [17], also confirms its potential for reaching ultra-high Q factors that are
2
only limited by the material absorption across the whole mid-IR regime. Also shown in the
tablearetheirthermalexpansioncoefficients(dL/L)/dT andthermo-opticcoefficientsdn/dT.
These parameters are particularly important for WGM applications. The self-thermal locking
method can be used for WGM pumping for microlasers [27] and combs [1]. However, it is
onlyapplicableforthematerialthatpossessesthesamesignsofthesetwoparameters,suchas
MgF ,fusedsilicaandSapphire.Ontheotherhand,theotherscouldbenefitfromself-thermal
2
compensationlikeathermalglasses.Hence,theyarelesssensitivetotheenvironmenttempera-
ture fluctuation, due to the fact that the frequency variation dv of a WGM is determined by
(dv/v)/dT ≈(dn/n)/dT −(dL/L)/dT [28]. Considering another useful locking technique,
theself-injectionlockingmethodusingRayleighscatteringinducedfeedbackisapplicablein
bothmaterials[9,29].
3. GeometrydispersioneffectsofWGMsindiskresonators
Itisknownthatthegeometrydispersioncanhaveastrongimpactinthetotaldispersion,such
asthatofthesiliconchannelwaveguide[30].Similarly,italsoplaysanimportantroleinres-
onatorstructures.AslightinWGMresonatorsisconfinedundertheequatorialboundaryofthe
disk, the size and the curvature of the rim are expected to affect the total dispersion value. A
recentstudyalsoshowsthatthedispersionofasphericalresonatorcanbetailoredbyacoating
layer[31].Here,wefocusonmonolithicresonatorswithoutanycoatingtopreservetheirhigh
Qfactorsthatareonlylimitedbytheabsorptionofthehostmaterial.
A simple analytical approximation of the eigenfrequencies of WGMs derived by two
groups [15,16] will be used in this work. The WGM resonant wavelength in a toroid can be
expressedasfollows:
2πn(λ)R (cid:16)m(cid:17)1/3 (cid:18) 1(cid:19)(cid:18)R(cid:19)1/2
=m−α + p+ (2)
q
λ 2 2 r
wheren(λ)isdeterminedbytheSellmeierequationofthematerial,m,q,parethemodenum-
bersintheazimuthal,radialandpolardirections,α isthenegativeqthrootoftheAiryfunction.
q
Theindex pissimilartothatofWGMsinspheresandisrelatedtotwomomentumnumbersl
andmby p=l−m,indicatingthep+1antinodesofthefielddistributioninthepolardirection
which can be identified through excitation mapping [32]. Other methods from spectroscopy
andemissionpatternsforidentifyingWGMshavealsobeendemonstrated[33,34].Inatypical
mm-sizediskresonator,musuallyrangesfrom103to105foropticalresonances.
ItshouldbenotedthatKerrcombshavebeengeneratedusingtwodifferentsetsofWGMs.
ThefirstoneisthefamilyofWGMswithspacingofoneFSR,v =v −v andtheother
FSR m m−1
oneisagroupofpolarWGMswithv −v spacing[21],wherev=c/λ istheresonantfre-
p p−1
quency.Here,wefocusontheKerrcombsgeneratedusingthelongitudinalFSR,andwhichcan
feature hundreds to thousands of comb lines, interesting for many spectroscopy applications.
WethereforeanalyzethetotaldispersionoftheselongitudinalFSRspacedWGMs.Thedisper-
sionofWGMsunderlyingKerrcombgenerationischaracterizedbythevariationintheFSR,
∆v =(v −v )−(v −v ). This value can be derived by numerically solving Eq. 2
FSR m+1 m m m−1
withtheunderlyingSellmeierequations.Toconvert∆v intothewidelyuseddispersionpa-
FSR
rameterGVDinunitsofps/nm/km,weappliedthefollowingapproximatedexpression[35]:
n(λ)3R2
GVD≈4π2 ∆v . (3)
c2λ2 FSR
1.55
14 (a) (b)
) 1
)12 m R=1.0mm
km/nm180 TCe-lbecaonmd h λ (μ0 1.5 r=0.1mm 0.5
(ps/ 6 decreasing engt1.45 10μm
D 4 R el 0
v
V R=5,4,3,2,1,0.5mm a
G 2 W 1.4
r=0.1mm
0
-2 1.35
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1 2 3 4 5
Wavelength (μm) Radius R (mm)
Fig.3.(a)CalculatedgeometrydispersioninfluenceonthetotalGVDforaMgF diskwith
2
differentRandr=0.1mm.Dashedline:materialGVDM curve.(b)Calculatedzerototal
dispersionwavelengthsλ0 forRfrom0.5mmto5mm.Inset:distributionoftheelectric
fieldamplitudeforafundamentalWGMswithR=1mmandr=0.1mm.Note:λ0isthe
resonantwavelengththathasitstotaldispersionclosestto0.
Here,wewillstudytheimpactofgeometrydispersionbyassumingWGMresonatorsmade
of MgF , which has a small GVD over a large range when compared with fused silica and
2
quartz.WefirstconsiderthecaseofthefundamentalWGMfamilywithq=1and p=0and
discussthesizeimpactofRandr.AsonecanseeinEq.(2),thesecondtermontherightsideis
muchlargerthanthethirdtermduetothelargevalueofminmm-sizeresonators.Wethusstudy
onlytheimpactofthemainradiusRonthetotaldispersion,becausetheeffectofrissimilarto
theeffectofthepolarnumber pthatwillbeintroducedlater.Figure3(a)showsthecalculated
totalGVDfordifferentradiiRvaluesfrom0.5mmto5mmaroundthezerodispersionregime.
ThedashedlinepresentsthematerialGVD .
M
OnecannotethatsmallerRshiftsthezeroGVDfurtherintothelongerwavelengths,which
resultsfromanormalgeometryGVDaddingontothematerialGVD .ThetotalGVDprofile
M
comparedtothematerialoneiseventuallypulleddown.Similareffectshavealsobeenreported
[23,36].Thewavelengths(λ )oftheWGMsclosesttozeroGVDasafunctionoftheradiiRis
0
showninFig.3(b).Theinsetprovidesacalculatedelectricfielddistributionofafundamental
WGMresonatingatλ inaMgF diskwithR=1.0mmandr=0.1mmusingananalytical
0 2
expression given in [15]. From this figure, one can conclude that small anomalous GVD can
be engineered by carefully choosing the major radius of the resonator. This is a key step for
designingaKerrfrequencycombsetup.Forexample,theGVDforthefundamentalmodesina
R=0.5mmMgF resonatorat1.55µmisreducedto0.72ps/km/nm,whichisaboutonetenth
2
oftheoriginalmaterialGVD(7.0ps/km/nm).Carefulengineeringofthisparameter,theGVD
reductioncanbeevenlarger.
The ability of the GVD tailoring in a WGM resonator after fabrication is also critical.
Thereby, we have also investigated the effect of different transverse mode orders on the to-
talGVD.Figure4showsthecorrespondingcalculationforaMgF resonatorwithR=1.0mm
2
andr=0.1mm.Theblackdashedlinesmarksthematerialdispersionitself.Theupperimages
arethecorrespondingcalculatedfielddistributionpatterns.TheimpactofWGMswithdifferent
qindicesonthetotalGVDispresentedinFig.4(a).Ontheotherhand,themodeindex pplays
alessinterestingroleintailoringthetotalGVDasshowninFig.4(b).Theinsetisazoomin
oftheGVDprofilearoundtheZDW.Thereddashedlinehighlightsthetelecomwavelengthof
1.55µm.Thefundamentalradialmodewithq=1possessesaGVDof3.0ps/km/nm,whichis
muchlargerthanthepreviousvalueforthefundamentalmodeinaresonatorwithR=0.5mm.
(a) q= 1 q= 2 q= 3 (b) p= 0 p= 1 p= 2
10μm 10μm
14 Telecom 14
)12 C-band )12
m q= 1,p= 0 m p= 0,q= 1
/n10 q= 2,p= 0 /n10 p= 1,q= 1
m 8 q= 3,p= 0 m 8 p= 2,q= 1
k k
/ /
s 6 s 6
p p 0.04
( (
D 4 D 4 0.02
V R=1mm V 0
G 2 G 2
r=0.1mm -0.02
0 0 -0.04
1.4511.4521.453
-2 -2
1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
Wavelength (μm) Wavelength (μm)
Fig.4.CalculatedtotalGVDfordifferenttransverseWGMsinaMgF diskwithR=1mm
2
andr=0.1mm.Top:ElectricfieldamplitudedistributionofWGMs;Bottom:GVDasa
functionoftheresonantwavelength.ForWGMswithtransverseindicesof(a)q=1,2,3,
p=0and(b)p=0,1,2,q=1.Dashedline:materialGVDM curve.
However,onecannotethatforthesecondradialorderWGMswithq=2,theGVDhasbeen
furtherreducedto 0.086ps/km/nm,thatisonly 1/80ofthematerialGVD and1/35ofthat
M
forq=1.Inthiscase,Kerrcombgenerationscanbesignificantlydifferent,dependentonwhich
radialordermodeisexcited.ComparingFig.4withFig.3,onecanseethatthemanagement
ofGVDbychoosingWGMswithdifferentqindicesismoreremarkable.Besides,wealsofind
that the change of GVD due to the q index variation is much larger in the resonator with a
smallerR.
4. Experimental observation of significant different primary combs in a single MgF
2
resonator
To experimental verify that the geometry dispersion plays an important role in WGM res-
onators, we have carried out experiments on a disk resonator made of MgF . The resonator
2
was fabricated by carefully polishing the rim of a disk preform with a radius of 6 mm and
a thickness of 1 mm. It features an intrinsic Q factor of 109. For a well polished resonator,
WGMs with lower order mode indices can possess the same Q factors and cannot be distin-
guishedthroughQfactormeasurements.Ithasbeenexperimentallyshownthatonecanchoose
thefibertapertoexciteWGMswiththeradialindicesfromq=1toq=11simultaneously[36].
Hence,afibertaperwasusedtocouplethelightintoandoutfromtheresonatorasillustrated
in Fig. 1 (b). The throughput of the fiber taper was then monitored with a photodetector and
ahigh-resolutionspectrumanalyzer(APEX2440B),respectively.Thetunablecwpumplaser
with a sub-KHz spectral linewidth at 1552 nm was used. It was then thermally locked to ex-
citedifferentWGMs[1,27].Dependingonthepumpcondition,bothprimaryKerrcombsand
fulldevelopedKerrcombswereobserved.ThevalueofsingleFSRspacing(v )canthusbe
FSR
measured.
The giant shift in dispersion predicted by our calculations should therefore allow for the
possibility to generate primary combs with significantly different multiplicities, by pumping
0 0 0
(a) (b) (c)
-10 -10 -10
m)-20 4-vFSR -20 164-vFSR -20 229-v
B FSR
d-30 -30 -30
(
r
e-40 -40 -40
w
o
P-50 -50 -50
-60 -60 -60
-70 -70 -70
-100 -50 0 50 100 -2 -1 0 1 2 -3 -2 -1 0 1 2 3
Frequency offset (GHz) Frequency offset (THz) Frequency offset (THz)
Fig.5.Experimentalobservationofthreeprimarycombsshowingverydifferentspacing
of(a)4-v ,(b)164-v and(c)229-v .ThreedifferentWGMswerepumpedinthe
FSR FSR FSR
sameresonator,andthelargedifferenceofGVDexplainsthelargedifferenceofprimary
combmultiplicity.
differentradialmodesinthesameresonator.Inparticular,oneshouldbeabletoobservevery
different primary comb spacing in the same resonator with a fixed incident pump power, es-
pecially in the case of fiber taper coupling where many different radial order WGMs can be
excitedatthesametime.Figure5presentthreeexperimentallyobtainedprimarycombsinthe
same MgF resonator when three different WGMs were pumped. The spacing of 23.7 GHz ,
2
971.9GHzand1351.1GHzcorrespondsto4-v ,164-v and229-v ,respectively.The
FSR FSR FSR
corresponding ratios between the largest spacing and the smallest one is about 50. Although
thedetuningbetweenthepumpandtheresonancecanalsochangethespacingoftheprimary
comb,thedifferenceisusuallylimitedduetotherequirementofon-resonancepumping.Inour
case,weusuallyseearatiooflessthan2whenchangingthedetuningwithinonemodewith
a fixed incident pump power. Therefore, we believe that the geometry dispersion differences
causethisobservation,whichisexpectedaspreviouslydiscussed.
5. Towardsthemid-IRregime
AsshowninTable1,BaF hasalargerzeromaterialdispersionwavelength(λ )of1.93 µm,
2 0
comparedwithothermaterialsstudiedhere.Inaddition,theGVD ofBaF intheanomalous
M 2
regimealsogrowsmuchslower.Thisisanimportantsignatureindicatingthatitcouldbevery
favorableforthebroadbandcombgenerationinthemid-IRregime.AsshowninFig.2,2.5µm
marksthefirstKerrfrequencycombgeneratedinthemid-IRregime[14],whereaMgF WGM
2
resonatorwasutilized.ItalsoindicatesthatcombgenerationisfeasibleforamaterialGVD of
M
25ps/km/nm.ItshouldbenotedthatBaF hasanonlinearrefractiveindexcoefficient(n ,in
2 2
10−16cm2/W)of2.85,whichislargerthan1.90ofCaF and0.92ofMgF [25].Ifweassume
2 2
aBaF resonatorwiththesameQfactorandthesamegeometrydispersion,onecanthusexpect
2
aKerrcombgenerationcapabilityatλ =5.5µm.ConsideringthatthedemonstratedKerrcomb
at2.5µminMgF hasaveryhighphasestability[14],thecapabilityofgeneratingsuchcombs
2
furtherintothewavelengthabove5µminBaF isthushighlydesirable.
2
Here,wepresentthecorrespondingGVDcalculationonWGMsmadeofBaF forKerrcomb
2
generations towards mid-IR. Figure 6 (a) shows the GVD for the fundamental WGMs in res-
onatorswithdifferentRandr=0.1mm.AsBaF hasamaterialGVD at1.93µm,itisthereby
2 M
feasibletopushthezerodispersionWGMresonancewavelengthλ closeto2.5µm,whichcan
0
potentiallyfacilitatebroadbandcombgenerationsintothisregime.Thecorrespondingλ asa
0
2.4
14 (a) Mid-IR (b)
m/nm)1120 Rr==05.,14m,3m,2,1,0.5mm λ (μm)02.3 Rr==01..10mmmm 1
k 8 h 0.5
(ps/ 6 engt 2.2 10μm
D 4 decreasing el 0
GV 2 R Wav 2.1
0
2
-2
2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Wavelength (μm) Radius R (mm)
14 (c) Mid-IR q= 1 (d) q= 1 q= 2 q= 3
)12
m 2
n10 R=1mm
m/ r=0.1mm 3 10μm
k 8 4
/
ps 6 5
D ( 4 6 q= 4 q= 5 q= 6
V
G 2
0
-2
2 2.5 3 3.5 4
Wavelength (μm)
Fig. 6. (a) Calculated geometry dispersion influence on the total GVD for a disk BaF
2
withdifferentRandr=0.1mm.Dashedline:materialGVDM curve..(b)Calculatedzero
total dispersion wavelengths λ0 for R from 0.5 mm to 5 mm. Inset: distribution of the
electric field amplitude for a fundamental WGM with R=1 mm and r=0.1 mm. (c)
CalculatedradialmodeorderinfluenceonthetotalGVDforadiskBaF withR=1mm
2
andr=0.1mm.WGMindices:q=1,2,3,4,5,6andp=0.(d)Thecorrespondingelectric
fieldamplitudedistributionofWGMs.
functionofthemainradiiRareshowninFig.6(b).Theinsetgivesthefielddistributionofa
fundamentalmodeatabout2.2µminaresonatorwithR=1mmandr=0.1mm.
Figure 6 (c) presents the GVD as a function of WGM resonance wavelengths for different
radialordermodesinaBaF resonatorwithR=1mmandr=0.1mm.Thecorrespondingfield
2
distributionisshowninFig.6(d).Asonecansee,GVDvaluesat3.5 µmfordifferentradial
WGMsareanomalousandreducedwiththeincreaseofqindices.Inthecaseofprismcoupling
setup,onecanchangethecouplingangletoselectivelyexciteWGMswithhighqvalues.The
corresponding wavelengths and indices of those modes closest to the zero total dispersion in
Fig. 6 (c) are listed in Table 2. For q=1 and q=2, the change in ZDW of about 240 nm is
observed.ZDW beyond3µmisfeasibleinhigherqordermodes.
Table2.IndicesofWGMsandzerototaldispersionwavelengthforMgF withR=1mm
2
andr=0.1mm
Wavelength m q p Wavelength m q p
(λ (WGM)) (λ (WGM))
0 0
2.1863 4176 1 0 2.8877 3100 4 0
2.4290 3733 2 0 3.1100 2860 5 0
2.6608 3386 3 0 3.3272 2656 6 0