columbia university - optics dissipative kerr …...have given rise to a host of phenomena, such as...

REVIEW SUMMARY

OPTICS

Dissipative Kerr solitons inoptical microresonatorsTobias J Kippenberg Alexander L Gaeta Michal Lipson Michael L Gorodetsky

BACKGROUND Laser frequencycombswhichconsist of equidistant laser lines have revolu-tionized time-keepingmetrology and spectros-copy Conventional optical frequency combsbased on mode-locked lasers are still mostlyconfined to scientific laboratories In recentyears there has been progress in the devel-opment of optical frequency combs based oncompact chip-scale microresonators (ldquomicro-combsrdquo) with suchmicrocombs operating in thedissipative soliton regime Dissipative solitonsrely on a double balance of nonlinearity and dis-persion as well as dissipation and gain and arean example of self-organization in driven dissi-pative nonlinear systems Dissipative temporalsolitons are providing the long-sought-for re-

gimeof coherent broadbandmicrocomb spectraand in additionprovide an experimental settingin which to study dissipative soliton physicsMicrocombs are now capable of producing co-herent octave-spanning frequency combswithmicrowave to terahertz repetition rates at lowpumppower and in chip-scale devices andhavebeen used in a wide variety of applicationsowing to bandwidth and coherence providedby the dissipative temporal soliton states

ADVANCESUnderlying these recent advanceshas been the observationof temporal dissipativeKerr solitons (DKSs) in microresonatorswhich represent self-enforcing stationary lo-calized solutions of a damped driven and de-

tuned nonlinear Schroumldinger equation whichwas originally used to describe spatial self-organization phenomena They correspond tosolitons (localized patterns) in ldquoopenrdquo systemsmdashthat is systems that exhibit dissipation DKSsopposite to fiber solitons therefore rely on adouble balance of nonlinearity and dispersionas well as parametric gain and loss and dependon the laser-cavity detuning as a control pa-rameter Methods have been established thatenable the reliable generation of suchDKSs in awide rangeofmicroresonatorplatforms includingplatforms based on silicon nitride (Si3N4) that

are compatible with pho-tonic integration In addi-tion a variety of previouslyunknown and nonantici-pated soliton effects havebeenobserved such as sol-iton crystallization Raman-

Stokes solitons and previously unseen solitonbreather dynamicsMoreover dissipative solitonshave been shown to be suprisingly robust againstimperfections in the resonator mode structureDissipative soliton states inmicroresonators havetriggered a large number of applications includ-ing in terabit-coherent optical communicationsatomic clocks ultrafast distance measurementsdual-comb spectroscopy photonic integrated fre-quency synthesizers and the calibration of astro-physical spectrometers for exoplanet searches

OUTLOOK The reliable generation of DKSsin microresonators has established a nascentresearch field at the interface of soliton physicsfrequency metrology and integrated photonicsand provided impetus to microcomb sourcesEmerging frontiers include using advances innanofabrication and materials synthesis torealize ultralow-propagation-loss photonic nano-structures with unusual dispersion propertieswhich could explore dissipative solitons in newand unexplored parameter regimes and allowthe synthesis of even broader spectra that intime domain could correspond to single-cyclepulses andwhose spectral bandwidth could beextended to the mid-infrared or visible rangeBeyond the narrow class of materials used forDKSs so far (Si MgF2 SiO2 and Si3N4) manyother materials exist with distinct advantagessuch as III-V semiconductors which are alreadywidely used in light-emitting or laser diodesBeyond existing applications DKSs could beapplied to optical coherence tomographyRamanspectral imaging chip-scale atomic clocks basedonoptical transitions orultrafast photonic analog-to-digital conversion and have a potential tomake frequency metrology and spectroscopyubiquitous

RESEARCH

Kippenberg et al Science 361 567 (2018) 10 August 2018 1 of 1

This list of author affiliations is available in the full article onlineCorresponding author Email tobiaskippenbergepflch (TJK)mgrqcru (MLG)Cite this article as T J Kippenberg et al Science 361eaan8083 (2018) DOI 101126scienceaan8083

Dispersion NonlinearityLossGain

BA

C10

10

0

-20

-10

0

-20

-10

Pow

er (d

Bm)

Pow

er (d

Bm)

DKSs in optical microresonators (A) Principle of DKSs representing a double balance ofdispersion and nonlinearity as well as (parametric) gain and cavity loss (B) Optical fieldenvelope of a single DKS containing the localized soliton on top of a continuous-wave (CW)background field (C) Graphic image of dissipative soliton formation in a CW laserndashdrivenphotonic chipndashbased microresonator generating a continuously circulating DKS which infrequency space corresponds to a coherent optical frequency comb The optical frequencycomb is shown with two dispersive waves that arise from higher-order dispersionC

REDITSIM

ON

HOENLEPFL

ON OUR WEBSITE

Read the full articleat httpdxdoiorg101126scienceaan8083

on Septem

ber 10 2018

httpsciencesciencemagorg

Dow

nloaded from

REVIEW

OPTICS

Dissipative Kerr solitons inoptical microresonatorsTobias J Kippenberg1 Alexander L Gaeta2 Michal Lipson3 Michael L Gorodetsky45

The development of compact chip-scale optical frequency comb sources (microcombs)based on parametric frequency conversion in microresonators has seen applicationsin terabit optical coherent communications atomic clocks ultrafast distancemeasurements dual-comb spectroscopy and the calibration of astophysicalspectrometers and have enabled the creation of photonic-chip integrated frequencysynthesizers Underlying these recent advances has been the observation oftemporal dissipative Kerr solitons in microresonators which represent self-enforcingstationary and localized solutions of a damped driven and detuned nonlinearSchroumldinger equation which was first introduced to describe spatial self-organizationphenomena The generation of dissipative Kerr solitons provide a mechanism by whichcoherent optical combs with bandwidth exceeding one octave can be synthesized andhave given rise to a host of phenomena such as the Stokes soliton soliton crystalssoliton switching or dispersive waves Soliton microcombs are compact are compatiblewith wafer-scale processing operate at low power can operate with gigahertz toterahertz line spacing and can enable the implementation of frequency combsin remote and mobile environments outside the laboratory environment on Earthairborne or in outer space

Solitons are robust waveforms that preservetheir shape upon propagation in dispersivemedia and can be found in a variety of non-linear systems ranging fromhot plasma (1)ferrite films fiber optics (2) cold atoms (3)

hydrodynamics and biology to cloud and sanddune formation Although the initial promisesof optical solitons for increased bandwidth oftelecommunication were not implementeddissipative solitons circulating in cavities ofmode-locked lasers (4) are actively used Similar to thephysics of open quantum systems the ldquoopenrdquosoliton systemsmdashdissipative solitons (5)mdasharehighlyrelevant to actual experimental systems in whichdissipation cannot be neglected This is in con-trast to the mathematical treatment of solitonsthat have focused on conservative integrable sys-tems Dissipative solitons (5) balance loss andgain in an active media along with the balanceof nonlinearity and dispersion (Fig 1) Althoughdissipative optical solitons are known to occurinmode-locked lasers (4) dissipative optical soli-tons can also occur in systems that have para-metric gainmdashgain from four-wave mixing thatis the underlying process that enables Kerr fre-quency comb (6 7) formation inmicroresonators

Such dissipative solitons sustained by parametricgain were first studied in fiber cavities (8) In2014 it was observed that such dissipative Kerrsolitons (DKSs) can spontaneously form in pa-rametrically driven Kerr frequency combs inoptical microresonators (9) In these systemsoptical sidebands are generated bymeans of four-wave mixing processes (10 11) and undergo a self-organization process that leads to the emergenceof a soliton pulse train which canmathematicallyand rigorously be mapped (12) to the Lugiato-Lefever equation (LLE) (13ndash15) an equation ex-tensively studied in applied mathematics fordecadesThe equation was originally developed to de-

scribe spatially localized pattern formation indriven nonlinear systems in which it can leadto the formation of ldquodissipative structurerdquo Sincetheir observation in microresonators such DKSshave been generated in a wide variety of micro-resonators ranging from bulk crystalline (9 16)and silica microdisks (17) to photonic chip-scaledevices (17ndash20) under both continuous-wave (CW)and pulsed excitation (21) In the frequency do-main the pulse train constitutes an optical fre-quency comb (22ndash24) Such optical frequencycombs pioneered by using mode-locked lasersbased on pulse trains are key ingredients ofoptical atomic clocks and have proven invaluablefor a wide range of scientific and technologicalapplications [for example reviewed in (25)] Al-though optical comb technology has maturedsince its first introduction two decades ago basedon mode-locked lasers and is commerciallyavailable such devices are typically not amena-

ble to wafer-scale integration and have compar-atively narrow line spacing in the hundreds ofmegahertz range DKSs in microresonators haveprovided a route to compact low power in chip-frequency comb generation AlthoughKerr combshave been known for more than one decade (7)[and have been reviewed in (6 26ndash28)] the dis-covery of the DKS regime (9) has unlocked theirfull potential by providing a route to broadbandand fully coherent microresonator-based fre-quency combs overcoming earlier challengesof low coherence (29ndash31) Such soliton-basedmicrocombs in chip-integratedmicroresonatorshave achieved low-power octave-spanning fre-quency combs in various spectral windows thatnow encompass the near-infrared (32 33) tele-communication (34 35) andmid-infrared spectralwindow (18 36) with repetition rates from only afew gigahertz (37) to terahertz The observation ofDKS in microresonators yields a merging ofsoliton physics and high-precision frequency combapplications stimulating a renaissance in dissipa-tive soliton research and enabling many techno-logical applications Soliton microcombs havealready been applied succesfully to dual-combspectroscopy in thenear- (38ndash40) andmid-infrared(36) scanning comb spectroscopy (41) as well asthe demonstration of a self-referenced comb forthe counting of optical frequencies [both with(42) and without external broadening (43)]Likewise soliton microcombs have been used inmassively parallel communication (44) in pairsat both the transmitter and receiver side forlow-noise microwave generation (45) as well asfor chip-scale dual-combndashbased light detectionand ranging (LIDAR) (46 47) Moreover solitonmicrocombs have been used to realize an allintegrated photonics-based chip-frequency syn-thesizer (48) and been used as a microphotonicastrocomb for exoplanet searches (49 50) Soli-ton microcombs exhibit surprisingly rich non-linear dynamics and have led to the observationof a variety soliton dynamics effects such asbright soliton formation dark pulses flat-toppedldquoplaticonsrdquo formation (51) soliton Cherenkov ra-diation (20) Raman self-frequency shift (52ndash54)Stokes soliton (55) and breather solitons (56ndash60)Moreover the inherently multimode nature of mi-croresonators gives rise to new and unanticipateddynamics such as avoided-crossingndashinduced disper-sive wave formation (61 62) or soliton-intermodebreathing dynamics Likewise soliton crystalliza-tion (63) and switching (64) have been observedto occur Today a growing understanding of thenonlinear dynamics in soliton microcombs hasbeen established and new dynamics have beenobserved whose exploration continues to develop

Dissipative Kerr solitons

The topic of this Review is a soliton that has longbeen theoretically discussed (13ndash15) but only re-cently experimentally observed temporal DKSsin which losses in a passive nonlinear opticalcavity are compensated by a parametric inter-action via a CWexternal laser pump in a resonatorcontaining a Kerr nonlinearity (10) Such DKSstates first studied in fiber cavities (8) have

RESEARCH

Kippenberg et al Science 361 eaan8083 (2018) 10 August 2018 1 of 11

1Eacutecole Polytechnique Feacutedeacuterale de Lausanne (EPFL) Instituteof Physics Lausanne CH-1015 Switzerland 2Department ofApplied Physics and Applied Mathematics Columbia UniversityNY 10027 USA 3Department of Electrical EngineeringColumbia University NY 10027 USA 4Faculty of PhysicsLomonosov Moscow State University Moscow 119991 Russia5Russian Quantum Center Moscow 143025 RussiaCorresponding author Email tobiaskippenbergepflch (TJK)mgrqcru (MLG)

on Septem

ber 10 2018


Dow

nloaded from

recently been discovered (9) as states in Kerrfrequency combs (7) Although DKS formationhas similarities to solitonmode-locking in femto-second lasers it does not require additionalsaturable absorbers to stabilize them and theydiffer fundamentally because the pump laserfrequency is a part of the soliton spectrum Theexternal coherent pump provides a central con-trol parameter of the soliton and in additionconstitutes one of the comb linesmdashwhich has nocounterpart in conventional mode-locked lasersEarly interest in optical cavities filled with anonlinear medium and pumped bymeans of CWsource was initially stimulated by optical bi-stability and its application to optical switching(65) demonstrated by using iterative discrete-time input-output relations (known today asIkeda map) so that such resonators exhibit mul-tistability However it was found afterward (66)that this system can also support very localizedoptical pulses If the intensity in the ring re-sonator changes only slightly per round tripthen the system can be described by a mean-field equationwith periodic boundary conditions(67) If the medium has Kerr nonlinearity andquadratic dispersion the mean field equationhas the form of a nonlinear Schroumldinger Equa-tion (NLS) (68) with dissipative anddriving termsDKSs were first studied and observed with ex-ternally injected pulses (8) Although these ex-periments conjectured (8) that soliton formationmay play a role only later experiments revealedfirst evidence of such a regime in terms of atransition from chaotic to low-phase-noise Kerrcomb states (30) as well as evidence of pulseformation (which is consistent with a parameterregime allowing for soliton formation) (69) Thefirst experiments that unambiguously identifiedDKSs in microresonators found DKS states byrecording the transmission during Kerr combformation Strikingly a series of steps on the red-detuned side of the resonance was observedThese steps as is now well understood are asign of soliton formation and correspond to thecontinued reduction of the number of solitonsin the cavity from N N ndash 1 Stably accessingsoliton states requires overcoming the thermalinstabilities associated with the drop in intra-cavity power and led to the development of awide range of techniques so that now stablyaccessing Kerr soliton states can occur in a widerange of microresonator platformsThe formation of DKSs in a crystalline micro-

resonator are shown in Fig 1 (9) A fundamentalanalytical property of DKSs is that they require ared-detuned pump laser from resonance as alsoshown experimentally by using a Pound-Drever-Hall error signal Although a red-detuned laserexcitation in microresonators is conventionallyunstable because of the thermal nonlinearitythe presence of a DKS can in turn self-stabilizethe system for red detuning This however nec-essitates overcoming transient thermal effects thatare associated with the discrete steps in cavitytransmission Understanding and overcomingthe thermal instability in the red-detuned soliton-formation regime as first accomplished with

optimized frequency tuning schemes (9) has beena pivotal experimental technique to enable stableDKS formation in microresonators Single DKSscan equally be accessed in fully planar architec-tures (70) notably in photonic-chipndashbased siliconnitride (Si3N4) resonators (71) despite the fact thatthis platform has a significantly lower quality (Q)factor and can exhibit dispersion imperfec-tions and strong thermal effects Photonic-chipndashbased microresonators based on Si3N4 areamenable to wafer-scale manufacturing and in-tegration and because of their higher non-linearity (compared with that of crystals or silica)enable broadbandDKS (72) with high repetitionrates (gt100 GHz) Although tuning into solitonstates in integrated Si3N4 microresonators hasalso been achieved with a slow laser-tuningmethod (73) the strong thermal effects havelead to the development of different methodssuch as ldquopower kickingrdquo (74) fast tuning by useof heaters (19) and most recently use of single-sideband modulator schemes (75) Using thesetechniques DKSs have been generated in a widevariety of microresonators ranging from bulk

crystalline (9 45) and silica microdisks (17) andmicrospheres (76) to photonic chip-scale devicesin Si and Si3N4 (20 77) and fiber cavities (21)under both CW and pulsed excitation (21) More-over schemes have been devised for soliton feed-back stabilization based on the soliton power(78) or by using the effective laser detuning ofthe soliton state (79)

Dissipative soliton regime in Kerrfrequency combs

We briefly review the physics of the parametricprocess inmicroresonators discussed in detail in(30 80) Kerr frequency combs were initially dis-covered in silica microtoroids and experimentsproved that the parametrically generated (11 81)sidebands were equidistant to at least one partin 10minus17 as compared with the optical carrierIn these early experiments the combs repeti-tion rate was in the terahertz range and afemtosecond-laser frequency comb was used tobridge and verify the equidistant nature of theteeth spacing It is today understood that suchhighly coherent combsonly exist in certain regimes


Dispersion NonlinearityLoss Gain

FROG interferometer delay (ps)

FROG

0 70

ωSH

G2

π 71 ps

4TH

z

0

-60

dB

Wavelength (nm)

Pow

er (2

0 dB

div

)

1520 1580

cw pumplaser

500 kHz

1 kHzGHzRBW1409

D

C

B

A

E

Fig 1 Soliton microcombs technology (A) Graphic image of dissipative soliton formation in a CWlaser-driven crystalline WGM microresonator enabling conversion of a CW laser to a train of DKSpulses (B) Principle of DKSs representing a double balance of dispersion and nonlinearity as well as(parametric) gain and cavity loss (C) Field envelope of a temporal soliton (D) Experimental opticalspectrum of a DKS in a crystalline MgF2 resonator (Inset) Detected microwave beatnote of thesoliton pulse train (E) Frequency-resolved optical gating spectrum showing localized optical pulsesseparated by the soliton-microcomb line spacing [Images are adapted from (9)]

RESEARCH | REVIEWon S

eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

Physics of Kerr comb formationThe resonance frequencies of a microresonatorcanbeTaylor-expanded around the pumpedmodew0 so that

wm frac14 w0 thornXj

Djmj=j eth1THORN

whereD12p is themean free spectral range (FSR)D2 is the group velocity dispersion (GVD) (positivefor anomalousGVD) and jisinZ If the FSR (becauseof D2 in Eq 1) increases with frequency the reso-nator has anomalous GVD as required for para-metric oscillations and soliton formation It isoften useful to introduce the integrated disper-sion which describes the deviation of a given res-onance from an equidistant frequency grid

Dint(m) = wm ndash (w0 ndash D1m)

For a microresonator with anomalous GVD thethird-order Kerr nonlinearity leads to a nonlinearcoupling between different modes The nonlinearcoupling coefficient g frac14 ℏw2

0cn2=n2V0 is definedas a Kerr frequency shift per photon n and n2 arethe refractive and nonlinear optical indices re-spectively V0 is the effective (nonlinear) volumeof the pumped mode and c is the speed of light

Parametric oscillation (11 81)mdashthe emergence ofsymmetrically spaced signal and idler sidebandsaround the pumpmdashoccurs when the parametricgain exceeds the cavity decay rate The thresholdcondition is equivalently understood as a Kerr-induced shift that is comparable with the cavitydecay rate and thus the onset of cavity bistabil-ity (g nc e k=2 where nc is the number of pho-tons in the mode) Unlike conventional lasers inwhich the threshold (11 81) scales as ordmV0Qthe threshold for parametric oscillation scaleswith ordmV0Q

2 which highlights the dramaticreduction of parametric threshold possible forultrahigh-Q microresonators When scanning thelaser into resonance from the blue-detuned side(Fig 2C) the first pair of sidebands that oscillateare those that are closest to the maximum of theparametric gain curve The sideband numberis approximately

mi asympffiffiffiffiffiffikD2

r

If this distance corresponds to one single FSRthe subsequent cascade leads to fully coherentfrequency combs Such combs are referred toas ldquonatively spaced combrdquo ldquoTuring rollsrdquo or(coherent) modulation instability (MI) combs

This coherent operating regime was initially ob-served in optical silica microtoroids (7) The co-herence in these systems is achieved becausethe comb formation leads to a native combwhose spacing corresponds to a single FSR Inthis scenario one can create fully coherent andrelatively broadband-frequency combs which hasbeen the case for toroid resonators in the tele-communication band or crystalline resonators inthe mid-infrared spectra range (82) and gener-ally integrated resonators with a large FSR (31)which lead to a large accumulated dispersionparameter D2 Not all platforms however yieldintrinsically low phase noise It was observed thatthe comb formation can give rise to low coher-ence statesmdashin particular in integrated platformsbased on for example Si3N4 Hydex glass SiAlN or AlGaAs Counterintuitively however evenfor ultrahigh-Q resonators such as crystalline reso-nators or silica disks low coherence was observedwhich is associated with subcomb formation

Subcomb formation and chaoticcomb states

In the case of resonators that have lowQ (such asSi3N4 and other integrated resonators) or lowdispersion (such as MgF2 crystalline resonators


2) Secondary subcombs

δδδ δ

ξ0 = 0 ξξminus 2ξminus2ξ

ωδ

1) Primary combs

ω

Pump

3) Subcomb merging

4) Dissipative Kerr soliton formationsech2 envelope

ω

ω

δδ minus ξ

δ + ξ

Δ

Kerr cavityInput

ω

tCW

P

P

CW

OutputSolitons

t

P

ω

P

Frequencycomb

1

2

3

4

5

1 2 3 4 5

1 2 3 4 5

C

B

A

D

-1 20 30100

Intr

acav

ity p

ower

ζ0 (normalized laser detuning)

Optical power spectra

Temporal intracavity power

Δ Δ Δ

101 modes

1 round trip

nmiddotδ ne Δ n = 12

40

au

Fig 2 Numerical simulations of DKSs in optical microresonators(A) Temporal and spectral input and output from a CW laserndashdrivenresonator supporting DKS (B) The mode proliferation in the case of aresonator that exhibits subcomb formation with eventual transition toDKS (C) Intracavity field as a function of the laser detuning Shown are theregions of modulation instability (marked ldquo1rdquo) breather soliton (yellow

marked ldquo2rdquo) and stable soliton formation (green marked ldquo3rdquo to ldquo5rdquo)Different trajectories corresponding to multiple simulations are shown inyellow (bold line) The different steps designate transitions betweendifferent soliton states (D) intracavity waveform corresponding to thedifferent chaotic MI breather solitons and stable DKS states [Images areadapted from (9)]


eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

in the telecommunication band) the first pairof sidebands (generated when scanning a pumplaser into resonance from the blue-detuned side)are separated by many FSRs which leads to the

formation of subcombs (because mi frac14ffiffiffiffikD2

q≫1)

In this process depicted in Fig 2 first a primarycomb is generated with a line spacing D Uponincreasing pump power secondary sidebandsaround the primary sidebands are generatedleading to the formation of subcombs The initialsubcombs all share the same repetition rate dImportantly the primary and secondary spacingdo not need to be commensurate Therefore oncethe pumping power is further increased the sub-combs merge and lead to the counterintuitivesituation in which more than one single combtooth can occupy a cavity resonance (30) Thisscenario leads the comb to exhibit beatnotes thatcanexhibitmultipletswhich for sufficiently strongpump power merge into broad beatnotes Thiscomb state therefore exhibits low coherencemak-ing it unsuitable for metrology Although thecomb spectral envelope of these combs is recordedwith an optical spectrum analyzer to be spectrallysmooth the underlying intracavity waveform inthe chaotic MI regime is vastly varying Manyearly reported comb spectra (including photonicintegrated resonator platforms or crystalline reso-nators in the telecommunicationband) are chaoticin origin and exhibit low coherence It was firstdemonstrated in Si3N4microresonators (30) thattransitions to low noise also exist Such a transi-tion to low noise state was shortly thereafter alsoobserved in silica resonators (83) and evidencewas found that the coherent regime is con-comitant with pulse trains (69) These resultsindicated that the major challenge and road-block of high noise could be overcome unlockingthe full potential of chip-scale frequency combsToday it is understood that these transitions are

likely associated with DKSs (9 13 15) which giverise to coherent comb states Indeed althoughthe LLE provides an accurate description of sol-iton statesmany phenomena require correctionsbeyond the established models and are to beexplored in Kerr microresonators

Transition to the DKS regime

An unusual and unexpected observation was ex-perimentally made in crystalline resonators (9)when analyzing the scan across the cavity res-onance during the comb formation process anunusual set of discrete stepndashlike behavior in thetransmitted power occurred which exhibited aremarkably regular and repeatable step heightTransientmeasurements of the combrsquos beatnotein this regime showed low phase noise indicatinga coherent comb-formationregimeThisobservationconfirmed an earlier theoretical prediction (84)that used numerical simulation of the laser-pumpscan based on the coupled-mode equations (CMEs)(85) These simulations (9) equally revealed thatduring the laser scan across the resonance un-expectedly sharp and discrete stepndashlike transi-tions to low-noise states appear (Fig 2) Thenumerical simulation predicted that the regionswith discrete steps are associated with solitonsinside the cavity and the discrete steps in thetransmission trace are associated with the an-nihilation of individual solitons one by one Thenumerical simulation uses the bare cavity detun-ing as a parameter which does not correspondto the actual (effective) detuning from the res-onance because of the Kerr frequency shiftcaused by the pump Actual soliton formation canonly occur when the laser transitions to the red-detuned side of the cavity resonance where theintracavity field is bistable (Fig 3A) The sim-ulation reveals the primary sidebands subcombformation and chaotic MI followed by the for-mation of stable DKS inside the cavity The

numerical simulation was repeated multipletimes and the yellow curves in Fig 3A showthe evolution of all possible trajectories whereasblue denotes the numerical simulation of a singlelaser scan trajectory Although numerical sim-ulations of dissipative solitons using amean fieldmodel equation (the LLE) have been carried outextensively simulations of the actual laser scan-ning process relevant to the microresonator caseare a new development and pivotal in the under-standing of DKS The simulations also predictedthe existence of a single soliton state with a sech2

spectral envelope (Fig 2D) These predictions arein agreement with experiments Although MIoccurs for the effectively blue-detuned regionthe regime of soliton formation occurs in the bi-stable regime where two solutions existmdashwherethe laser is effectively red-detuned (Fig 3A) Ex-perimentally this canbe verified byusing aPound-Drever-Hall error signal which can differentiatebetween effective red- and blue-detuning (Fig 3A)

Stably accessing DKS in the presence ofthermal nonlinearity

Accessing the DKS is in practice compoundedby thermal effects A fundamental consequenceof the existence of bright solitons is the opera-tion in the bistable regime which causes theoptical pump to be effectively red-detuned fromthe (Kerr and thermally shifted) cavity resonanceAlthough blue-detuned excitation is thermallystablemdashthe resonator and laser form an auton-omous feedback loop that stabilizes the lasercavity detuningmdashthe opposite is true for a red-detuned operation (required for soliton forma-tion) (86) Moreover the series of discrete stepsin resonator transmission laser scandue to solitonformation compound the stable access of solitonsbecause concomitant with a discrete step is achange in intracavity power and as a conse-quence temperature change This in turn via


Pump laser detuningBlue detuned Red detuned

~ 100 nm

Rela

tive

norm

aliz

ed in

trac

avity

pow

er

0

Laser scan

λ

Wavelength

Spec

tral

pow

er

A CB

0 10

100

2004

1

Tran

smis

sion

(au

)

Scan time (ms)

Red detunedBlue detuned

Relative mode number μ0minus100002minus

0

300

MH

z

25 nm

(ωμ-ω0-μD12π)

Pow

er (a

u)

Pump laser wavelength (au)

Laser scanLower branch

Upper bra

nch

Fig 3 Transition to the dissipative soliton states and solitoncrystals (A) (Top)The cavity bistability in the intra-cavity power dueto the Kerr nonlinearity (Bottom) The integrated dispersion profile[Dint(m)] of a measured progression of resonances exhibiting anomalousdispersion (quadratic variation of the integrated dispersion) (Middle)A series of steps on the red-detuned side of the resonance indicative

of dissipative soliton formation (B) The evolution of the intracavitypower as a function of laser detuning revealing in particular aseries of discrete steps associated with soliton formation (C) Morecomplex arrangements of a large number of solitons that areordered in crystals (63 134) but that contain defects such asvacancies or defects [Images are adapted from (9 63)]


eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

the thermal effect (temperature-dependent re-fractive index) changes the relative laser cavitydetuning In early work this challenge was over-come by using an optimized laser scan (9) Sincethen several techniques have been developedfrom ldquopower kickingrdquo (74) to very fast laser mod-ulation using single sideband modulators (75)as well as fast thermal on-chip tuning (19) andcarrier injection (77) All developed techniqueshave in common that once the DKS state isreached the thermal response of the cavity is dom-inated by the DKS causing the system to beself-stabilized in the presence of a red-detuned(typically several linewidth) strong pump fieldThis is due to the fortuitous circumstance thatthermal effects make accessing the soliton statemore difficult once the solitons are generatedthe thermal nonlinearity induced by the solitonself-stabilizes the laser-cavity detuning con-stituting an autonomous feedback loop Onceaccessed DKS in microresonators can be pas-sively stable for hours Another useful featureof the DKS regime is that if probed with a mod-ulated laser the soliton state (64) exhibits tworesonances a C and S resonance which cor-respond to themodulation response of the cavityand solitonic solution and reveal the effectivelaser detuning Experimentally the detuningof the laser determines the soliton duration inwhich detuning further from resonance decreases

the soliton duration until the point at which thesoliton existence range limit is reached (Fig 2Cgreen region) In addition to continous wavepumping schemes pulsed pumping can be usedAlthough the generation of the Kerr soliton mi-crocombs can occur at record low power [tensof milliwatts have been reported for octave span-ning spectra (48)] the efficiency of the processis low because of the detuned nature of the pumpduring theDKS formation Periodic pulsed pump-ing at a frequency similar to the cavitiesrsquo inverseround-trip time can enhance this efficiency byapproximately the ratio of the round-trip timeto the pulse duration This scheme was recentlyshown for DKSs in fiber cavities (21) exhibitinga locking behavior to the drive pulse and leadingto a substantial increase in the conversion ef-ficiency This scheme also has been demonstratedfor Si3N4 microresonators for efficient and broad-band comb generation (49) Because of the higherefficiency of pulsed pumping thermal effectsare suppressed making tuning to the DKS statepossible by using slow laser scans

Theoretical modeling and numericalsimulations of DKS in microresonators

Kerr comb formation can in addition to CMEsdescribed above also be described by a meanfield The two descriptions are equivalent butonly the latter enables a connection to the soliton

to be established The internal optical fieldenvelope Aethf tTHORN in a microresonator with a self-focusing Kerr nonlinearity and only second-order GVD may be described with the LLE(12 13 16 72) where f is the angular coordinatein a ring resonator in a frame copropagating theenvelope with the group velocity and t is theslow time (slower than the round trip)

A

t i

D2

2

2A

f2 igjAj2Athorn ethk=2thorn iDTHORN A frac14 ffiffiffiffiffiffiffi

hksp

eth2THORN

Here the input laser power is given by P frac14ℏw0jsj2 h = kexk is the coupling efficiencydetermined as a ratio of the output coupling ratekex to the total loss rate k (h = 12 corresponds tocritical coupling and h asymp 1 corresponds to strongovercoupling) ℏ is the reduced Planckrsquos constantand w0 is the pumped optical frequency D1 fallsout from the equation in transforming to arotating frame f frac14 ϕ D1t consideration ofterms with j gt 2 leads to appearance of higher-order derivatives in Eq 2 and D = w0 ndash w is thepump detuning It is convenient to switch to adimensionless equation

iYt

thorn 1

2

2Y

q2thorn jYj2Y frac14 z0Y iYthorn if eth3THORN


minus100

0

50

100

Disp

ersi

on (p

s (n

mmiddotk

m)- 1)

2200200018001600140012001000

Wavelength (nm)

TE 730 nm times 1200 nm

TE 730 nm times 1650 nmTE 730 nm times 2000 nm

minus50

Bulk Si3N4

Frequency (THz)188 190 192 194 196 198

-05

0

05

1

15

2

25

D int2

π (G

Hz)

Mod

e in

dex

n eff(λ

)

Wavelength λ

Mat

eria

l ind

ex n

(λ)

Normal AnomalousCore index

Cladding index

Normal GVDAnomalous GVD

Wavelength λ

High confinement

Low confinement

A B D

C

E

500 nm1 mm10 microm10 microm 10 microm

Fig 4 Dispersion engineering in photonic chipndashbased integratedwaveguides and resonators (A and B) Effective refractive index of a photonicintegrated waveguide as a function of wavelength revealing that tightconfinement waveguides give rise to anomalous group velocity dispersion(positive curvature of the effective index curve) By contrast weak confinementleads to normal GVD (C) Shown is the measured integrated dispersion

parameter Dint(m) of a Si3N4 microresonator by use of frequency comb-assistedspectroscopy revealing purely anomalous GVD (D) Engineering of the GVDof a Si3N4 waveguide cladded with silica opening an anomalous GVD window inthe telecommunication band (98) (E) Integrated microresonator platformsshowing Kerr comb formation to date including diamond AlN AlGaAs silica onsilicon and Si3N4 DKSs have been generated in silica Si3N4 and Si


eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

(q frac14 fffiffiffiffiffiffik

2D2

q f frac14

ffiffiffiffiffiffiffiffiffiffi8ghPink2ℏw0

q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk

t = kt2) Without the right part Eq 3mdashknownas the nonlinear Schroumldinger (NLS) equationmdashisintegrable with a sech-shaped soliton solution(87) An exact stationary solution of Eq 3 in theform of solitonic pulses on CW background isalso known (14 88 89) when only a driving termwithout losses is considered Although this solu-tion provides a good approximation in the limit oflarge detuning z0 it gives little insight in the un-derstanding of the DKS dynamics without num-

erical simulations and asymptotic approximationsThe damped driven NLS (Eq 3) is frequentlyreferred to as the LLE (13) an important equationfirst introduced to describe two-dimensionalspatial transversal solitonic field patterns in non-linear Fabry-Perot etalons and later reformulatedfor longitudinal temporal solitons (15) The samelongitudinal equation was however earlier ana-lyzed in application to plasma physics (90 91)where solitons on background and existenceboundaries were found

In (85) an alternative approach for Kerr combsimulation was proposed on the basis of discreteanalysis of each comb line nonlinearly coupledwith all others The system of equations in thisway may consist of hundreds of CMEs with mil-lions of nonlinear terms that nevertheless can beefficiently numerically integrated by using thefast Fourier transform (92) The CME descriptionis equivalent to LLE (11) and may be consideredas its discrete Fourier transform The CME ap-proach is useful to analyze dynamics for each


Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)

Fig 5 The influence of dispersion and Raman effect on DKS(A) A nearly purely quadratic dispersion (anomalous GVD)ndashdominatedDKS generated in a crystalline microresonators (B) Soliton-inducedCherenkov radiationmdasha soliton whose spectral envelope encompassesthe normal GVD regime A dispersive wave is emitted where the phase

matching condition [Dint(m) = 0] is satisfied (C) A spectrum with dualdispersive waves covering a full octave in a Si3N4 microresonatorwith terahertz mode spacing (D) A platicon state (dark pulses)generated in the normal dispersion regime (E) A Raman solitongenerated from the single soliton state in a silica microdisk


eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

comb line and also in cases in which local dis-persion for particular lines is perturbed andnot easily described by the Taylor series (61)

Microresonator platforms forsoliton formation

Soliton Kerr frequency combs inmicroresonatorswere first observed (9) in crystalline MgF2 res-onators (45 93) with ultrahigh-Q factor lead-ing to a low threshold for nonlinear effects (94)The platforms inwhich solitons can be generatedhave been extended significantly and includeplatforms that are amenable to wafer-scaleprocessing such as silica microdisk resonators(17) and photonic integrated platforms notablySi3N4 (19 20 95 96) Such photonic chipndashbasedresonators based on Si3N4 enable integration ofwaveguides and resonators enable further func-tionality such as heaters and are space compatible(70) Moreover solitons have been generated inSimicroresonators in themid-infrared (77) wheretwo-photon absorption is negligible A chal-lenge in this case is attaining sufficiently high

Q for efficient nonlinear parametric frequencyconversionOne platform that is particularly well suited is

based on Si3N4 a material that is already a partof the complementarymetal-oxide semiconductor(CMOS) process for the strain engineering oftransistors and as a capping layer Its high re-fractive index of n asymp 2 enables tight optical con-finement waveguides and microring resonatorsand the high bandgap (~3 eV) mitigates multi-photon absorption in the telecommunicationband Recent measurements have revealed thatSi3N4 has intrinsic material limited Q-factorsthat can exceed 107 (97) Although the Q-factorsattained in Si3N4 are still substantially belowthose of crystalline resonators the significantlytighter optical confinement and higher Kerr-nonlinearity have allowed DKS generationMoreover DKSs have been observed in fiberFabry-Perot microcavities (21) Although moreplatforms exist in which Kerr comb generationhas been shownmdashincluding in particular com-pound semiconductors such as AlGaAs or AlNmdash

soliton formation has to date not been demon-strated in these materials Irrespective of theplatform or material used a requirement forsoliton formation is attaining anomalous groupvelocity dispersion As one approaches the band-gap of the material the normal GVD will startto dominate This limitation has been overcomewith waveguide geometryndashbased dispersion en-gineering (98)

Dispersion engineering

Optical microresonators exhibit variation of theFSR because of the effect of material and geo-metric (resonator) dispersion Material dispersioncommonly expressed via the Sellmeier equationsposes stringent limits on the wavelength range inwhich anomalous GVD is possible By contrast theresonator dispersion can be engineered via theresonator geometry and can be engineered tobe anomalous GVD overcoming the inherentmaterial dispersion (98) For a nanophotonicwaveguide the dependence of the effective re-fractive index on thewavelengths leads inherently


Fig 6 Applications of soliton microcombs Applications of areas ofmicrocombs (from top clockwise) ultrafast dual soliton microcomb distancemeasurements (LIDAR) (46) chip-scale atomic clocks photonic Radar (135)

dual-microcomb spectroscopy (38) optical coherence tomography low-noise microwave generation (45) integrated optical frequency synthesizer(136) and astrophysical spectrometer calibration for exoplanet detection (49)


eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

to normal GVD when the mode is weakly con-fined and to anomalous GVD for tight confine-ment This is illustrated in Fig 4 and has beenapplied to achieve Kerr comb generation inSi3N4 despite the material having normal ma-terial dispersion (71) Moreover coatings canbe used to further engineer dispersion properties(99) A limitation arises from the whispering-gallery mode (WGM) potential induced normalGVD for small waveguide radii (99) The combbandwidth can be significantly extended whenmaking use of higher-order soliton broadeningeffects If higher-order dispersion terms are sig-nificant and the comb extends into regionswherethe dispersion changes sign the comb bandwidthwill be significantly extended (20 35 72 100ndash103)Comb lines near zeros in the integrated dispersionprofile (Fig 3A) form dispersive waves coupled tosolitons also known as soliton Cherenkov radia-tion (103ndash106) An analogous effect may occurbecause of dispersion perturbation induced bymode coupling (62 106) Proper design of theresonators (73) also allows for suppressing thenonfundamental modes that because of modecouplings with multiple avoided mode crossings(61) may prevent generation of solitons Becauseof the very short duration of the DKS in micro-resonators the LLE needs to be generalized toinclude higher-order dispersion (72 105) Ramanscattering and self-steepening terms (107ndash109)to describe the soliton microcombs

Dispersive waves in DKS

For solitons whose spectral bandwidth extendsinto the normal GVD regime the LLE has to beextended with higher-order dispersion termsThis gives rise in particular to the effect ofsoliton-induced Cherenkov radiation also knownas dispersive wave formation This process wasfirst described for supercontinuum generation inoptical fiber (104) and applies also to DKSs inoptical microresonators (72) In this case thesoliton develops an oscillatory tail on the trailingor leading edge of the pulse The spectral locationof this dispersive wave corresponds approximatelyto the condition where Dint(m) = 0 where theresonator FSRmatches that of the dispersionlesssoliton Dispersive waves offer the practical ad-vantage that they enable extension of the spec-tral coverage of the soliton Kerr comb to thenormal dispersion regime and increase the pow-er in the spectral ends of the comb Suitablequartic dispersion enables also the creation ofDKSs with two dispersive waves that can covera full octave (Fig 5C) The soliton Cherenkovradiation also entails a spectral recoil (105 110)

Raman effect in DKS and theStokes soliton

The Raman effect is particularly relevant inamorphous media such as silica or Si3N4 buthas been observed to be negligible in crystal-line material of MgF2 because of the narrowRaman gain bandwidth The Raman effect leadsto a Raman self-frequency shift a soliton thatis shifted with respect to the pump [also calledfrequency-locked Raman soliton (53)] The Raman

effect can also lead to a new type of soliton theStokes soliton which is a soliton generated bythe Raman gain and locked to the pump soliton(Fig 5E) (55) Recently it was also shown thatDKSs experience a limitation on their temporalduration and bandwidth because of the stim-ulated Raman scattering (111) once the combspacing is less than the Raman gain bandwidthIn addition it was shown that in the presenceof a large Raman gain (such as in diamond)comb generation can only occur if the FSR issufficiently large to suppress Raman lasing

Counterpropagating solitons andspatial multiplexing

WGM microresonators support both counter-propagating traveling modes which allows sim-ultaneous excitation of DKS in both clockwiseand counterpropagating directions (112 113)The two combs however have been observedto lock to each other an effect likely associatedwith Rayleigh backscattering Such locking wasrecently demonstrated and enables the creationof two phase-locked pulse streams of solitonKerrcombs which can be used for dual-comb distancemeasurements (47) Such dual-comb soliton pulsestreams can also be generated by using spatialmultiplexing of solitonsmdashthe generation of DKSsin several spatial mode families of one and thesame resonator (114)

Normal dispersion solitonic pulses

The LLE also exhibits solutions for normal GVDa solution known as dark soliton (short dip inCW background) However because of periodicboundary conditions purely dark solitons withopposite sign of phase on both sides are im-possible inmicroresonators whereas low-contrastgray solitons have a quite narrow hardly ac-hievable range of existence (115 116) Neverthelessmode-locked Kerr frequency combs in the normaldispersion regime were experimentally demon-strated in different materials (117ndash120) It wasrevealed by using numerical simulations thatthese experimental results may be interpretedby use of a type of solitonic pulses calledldquoplaticonsrdquo flat-topped bright or dark (depend-ing on the duration) pulses that can be softlyexcited and stably exist in microresonators withnormal dispersion in case of bichromatic oramplitude-modulated pump (120) as well as witha local dispersion perturbation (51) In real mi-croresonators such a perturbation may occurbecause of the normal mode coupling betweendifferent mode families (119 122) Platicons rep-resent bound states of opposing switching waves(123) that connect upper and lower branches ofbistable resonance to satisfy periodic boundaryconditions Theymay be identified in experimentsbecause of their characteristic optical spectrumwith pronounced wings (Fig 5) In (124) it wasdemonstrated that the dynamics of platiconsin the presence of the third-order dispersion isquite peculiar and drastically different frombright solitons In (125) a possibility of stablecoexistence of dark and bright solitons in caseof nonzero third-order dispersion was revealed

Applications of DKSsAccessing DKSs in microresonators has unlockedin a short timeframe several promising applica-tions The ability of DKSs to generate broadband-coherent combs with gigahertz repetition ratesand short pulse streams has a number of applica-tions in which either compactness is required orthe high repetition rates are advantageous (Fig 6)

Microwave-to-optical link

Soliton Kerr microcombs have enabled the count-ing of the cycles of light by creating a directmicrowave-to-optical link byusing self-referencingassisted with external fiber-based broadening(42 126) and by directly exploiting the broad-band nature of DKS (43)

Optical frequency synthesizers

DKSs based on a dual-comb approach have beenused to create an integrated optical frequencysynthesizer that allows synthesizing an opticalfrequency from a radio-frequency reference (48)

Massively parallel coherentcommunications

Kerr combs with hundreds of equally spacedlines can serve as a reference for wavelength di-vision multiplexing in massively parallel coher-ent telecommunications with demonstrated datarates exceeding 50 terabits per second at boththe receiver and transmitter ends (44)

Dual-comb spectroscopy

Two slightly different frequency combs can formon a photodetector a radio-frequency comb (25)with the repetition rate equal to both combsrsquooptical repetition rates difference DKSs havebeen used for such dual-comb measurements inboth near- and mid-infrared (37ndash40) and alloweven full integration on chip

LIDAR

Dual-microcomb may be implemented for ul-trafast distance measurements with submi-crometer resolution (46) using counterclockwisesolitons (47)

Low-noise microwave generation

Soliton Kerr microcombs can also be used forcompact ultrastable microwave generation es-pecially if augmented with compact atomic cellreferences (126 128)

Astrophysical spectrometer calibration

The large line spacing makes DKSs ideal forcalibrating astrophysical spectrometers for ex-oplanet searches In contrast to conventionallaser frequecy combs that require filtering thisis not necessary for DKSs Recent work demon-strated that DKSs provide calibration sufficientfor search of Earth-like planets (49 50)Beyond these already demonstrated applica-

tions there is application potential in severaldomains in particular for the visible range Op-tical coherence tomography dual-comb coherentanti-Stokes Raman spectroscopy (129) or photonicmicrowave signal processing (130) are obvious



eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

new directions but qualitatively and quantita-tively new nonlinear physics and devices areexpected when advancing the ability to engineerdispersion over broad bandwidth in photonicchips and when investigated more complexphotonic devices and geometries such as photonicband-gap waveguides and cavities

Further developments and outlook

The fast progress in recent yearsmdashin the under-standing of soliton formation the broad range ofplatforms in which solitons in driven dissipativenonlinear microresonators have been generatedand the rapid rise of applications and proof ofconceptsmdashindicate a fruitful playground for fu-ture progress Yet challenges exist in particulartowardmaking the technology viable increasingbandwidth of solitons increasing the efficiencyand increasing the photonic integration More-over despite decades of research integrated pho-tonic resonators presently still exhibit propagationlosses that are more than three orders of mag-nitude higher than those of standard opticalfibers Overcoming this challenge would allowdramatic improvements in power efficiency andlikely requires further research in photonic ma-terials and fabrication processes Several areascan be identified that may offer advances andnew avenues both technological in nature as wellas fundamental such as new and improved ma-terials and new platforms Although the advancescannot be predicted several developments canbe anticipated

Advanced dispersion engineering andnew nonlinear materials

The coherent synthesis of an even broader spec-trum possibly corresponding to single-cyclepulses depends critically on dispersion engineer-ing Dispersion engineering may open single oreven subcycle waveforms inside dielectric resona-tors as well as new coupled soliton complexesSuch advanced dispersion engineering is in itsinfancy at present and could use multilayer co-atingsmore complexwaveguides that use coupledresonators or photonic crystal waveguide geo-metries or combinations of existing materialsTo date solitons have been generated in a smallclass of materials (MgF2 Si3N4 Si and SiO2) andevident potential exists when using III-V semi-conductors in particular which offer atomic-scale deposition processes and tailored band gap(131) Likewise solitons are surprisingly stableattractorsmdashand are robust to dispersion disorderThis could enable machine-learning algorithmsto design complex highly multimode systemswhose composite mode coupling induced abroadband desired dispersion landscape thatsupports solitons If achieved it may becomepossible to engineer arbitrary and much morecomplex and flat dispersion landscapes that couldhost multiple a-GVDndashhosted soliton complexes

Quantum technology

A further area under exploration is the use ofparametric interactions a process that is ca-pable of generating correlated pairs of photons

for the generation of quantum states One ap-plication lies in generating quantum randomnumber generation using optical parametric os-cillations (132) Moreover the parametric fluo-rescence in Kerr nonlinear microresonators hasalready been successfully used for correlatedphoton states and higher-order correlated photonstates (133) Beyond technological advances out-lined above there are also fundamental openquestions An emerging question is the behaviorof coupled parametric soliton arrays The behav-ior of such driven dissipative nonlinear systemsis an emergent question that is also experimen-tally accessible and may give rise to unanticipatedeffects These possible advances reveal that DKS-based microcombs may offer advances to a widerange of areas from timekeeping and spectros-copy to sensing and telecommunications andare emerging as candidate building blocks intomorrowrsquos photonic technologies

REFERENCES AND NOTES

1 N J Zabusky M D Kruskal Interaction of ldquosolitonsrdquo in acollisionless plasma and the recurrence of initial statesPhys Rev Lett 15 240ndash243 (1965) doi 101103PhysRevLett15240

2 L F Mollenauer R H Stolen J P Gordon Experimentalobservation of picosecond pulse narrowing and solitons inoptical fibers Phys Rev Lett 45 1095ndash1098 (1980)doi 101103PhysRevLett451095

3 K E Strecker G B Partridge A G Truscott R G HuletFormation and propagation of matter-wave soliton trains Nature417 150ndash153 (2002) doi 101038nature747 pmid 11986621

4 P Grelu N Akhmediev Dissipative solitons for mode-lockedlasers Nat Photonics 6 84ndash92 (2012) doi 101038nphoton2011345

5 N Akhmediev A Ankiewicz Dissipative Solitons LectureNotes in Physics (Springer-Verlag 2005)

6 T J Kippenberg R Holzwarth S A DiddamsMicroresonator-based optical frequency combs Science332 555ndash559 (2011) doi 101126science1193968pmid 21527707

7 P DelrsquoHaye et al Optical frequency comb generation from amonolithic microresonator Nature 450 1214ndash1217 (2007)doi 101038nature06401 pmid 18097405

8 F Leo et al Temporal cavity solitons in one-dimensional Kerrmedia as bits in an all-optical buffer Nat Photonics 4471ndash476 (2010) doi 101038nphoton2010120

9 T Herr et al Temporal solitons in optical microresonatorsNat Photonics 8 145ndash152 (2014) doi 101038nphoton2013343

10 A B Matsko A A Savchenkov D Strekalov V S IlchenkoL Maleki Optical hyperparametric oscillations in awhispering-gallery-mode resonator Threshold and phasediffusion Phys Rev A 71 033804 (2005) doi 101103PhysRevA71033804

11 T J Kippenberg S M Spillane K J Vahala Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Qtoroid microcavity Phys Rev Lett 93 083904 (2004)doi 101103PhysRevLett93083904 pmid 15447188

12 Y K Chembo C R Menyuk Spatiotemporal Lugiato-Lefeverformalism for Kerr-comb generation in whispering-gallery-mode resonators Phys Rev A 87 053852 (2013)doi 101103PhysRevA87053852

13 L A Lugiato R Lefever Spatial dissipative structures inpassive optical systems Phys Rev Lett 58 2209ndash2211(1987) doi 101103PhysRevLett582209 pmid 10034681

14 I V Barashenkov IV Y S Smirnov Existence and stabilitychart for the ac-driven damped nonlinear Schroumldingersolitons Phys Rev E Stat Phys Plasmas Fluids RelatInterdiscip Topics 54 5707ndash5725 (1996) doi 101103PhysRevE545707 pmid 9965759

15 M Haelterman S Trillo S Wabnitz Dissipative modulationinstability in a nonlinear dispersive ring cavity Opt Commun91 401ndash407 (1992) doi 1010160030-4018(92)90367-Z

16 A B Matsko et al Mode-locked Kerr frequency combsOpt Lett 36 2845ndash2847 (2011) doi 101364OL36002845 pmid 21808332

17 X Yi Q-F Yang K Y Yang M-G Suh K Vahala Solitonfrequency comb at microwave rates in a high-Q silicamicroresonator Optica 2 1078 (2015) doi 101364OPTICA2001078

18 A G Griffith et al Silicon-chip mid-infrared frequency combgeneration Nat Commun 6 6299 (2015) doi 101038ncomms7299 pmid 25708922

19 C Joshi et al Thermally controlled comb generationand soliton modelocking in microresonators Opt Lett41 2565ndash2568 (2016) doi 101364OL41002565pmid 27244415

20 V Brasch et al Photonic chip-based optical frequency combusing soliton Cherenkov radiation Science 351 357ndash360(2016) doi 101126scienceaad4811 pmid 26721682

21 E Obrzud S Lecomte T Herr Temporal solitons inmicroresonators driven by optical pulses Nat Photonics 11600ndash607 (2017) doi 101038nphoton2017140

22 T Udem R Holzwarth T W Haumlnsch Optical frequencymetrology Nature 416 233ndash237 (2002) doi 101038416233a pmid 11894107

23 J L Hall Defining and measuring optical frequenciesThe optical clock opportunitymdashand more (Nobel lecture)Rev Mod Phys 78 1279ndash1295 (2006) doi 101103RevModPhys781279 pmid 17086589

24 S T Cundiff J Ye Colloquium Femtosecond opticalfrequency combs Rev Mod Phys 75 325ndash342 (2003)doi 101103RevModPhys75325

25 I Coddington N Newbury W Swann Dual-combspectroscopy Optica 3 414 (2016) doi 101364OPTICA3000414

26 Y K Chembo Kerr optical frequency combs Theoryapplications and perspectives Nanophotonics 5 214 (2016)doi 101515nanoph-2016-0013

27 A A Savchenkov A B Matsko L Maleki On FrequencyCombs in Monolithic Resonators Nanophotonics 5 363ndash391(2016) doi 101515nanoph-2016-0031

28 A Pasquazi et al Micro-combs A novel generation of opticalsources Phys Rep 729 1ndash81 (2018) doi 101016jphysrep201708004

29 P DelrsquoHaye et al Octave spanning tunable frequencycomb from amicroresonator Phys Rev Lett 107 063901 (2011)doi 101103PhysRevLett107063901 pmid 21902324

30 T Herr et al Universal formation dynamics and noise ofKerr-frequency combs in microresonators Nat Photonics 6480ndash487 (2012) doi 101038nphoton2012127

31 F Ferdous et al Spectral line-by-line pulse shaping ofon-chip microresonator frequency combs Nat Photonics 5770ndash776 (2011) doi 101038nphoton2011255

32 S H Lee et al Towards visible soliton microcombgeneration Nat Commun 8 1295 (2017) doi 101038s41467-017-01473-9 pmid 29101367

33 M Karpov M H Pfeiffer T J Kippenberg Photonic chip-basedsoliton frequency combs covering the biological imagingwindow arXiv170606445 [physicsoptics] (2017)

34 Q Li et al Stably accessing octave-spanning microresonatorfrequency combs in the soliton regime Optica 4 193ndash203(2017) doi 101364OPTICA4000193 pmid 28603754

35 M H P Pfeiffer et al Octave-spanning dissipative Kerrsoliton frequency combs in Si_3N_4 microresonators Optica4 684 (2017) doi 101364OPTICA4000684

36 M Yu et al Silicon-chip-based mid-infrared dual-combspectroscopy arXiv161001121 [physicsoptics] (2016)

37 M G Suh K Vahala Gigahertz-repetition-rate solitonmicrocombs Optica 5 65ndash66 (2018) doi 101364OPTICA5000065

38 M-G Suh Q-F Yang K Y Yang X Yi K J VahalaMicroresonator soliton dual-comb spectroscopy Science354 600ndash603 (2016) doi 101126scienceaah6516pmid 27738017

39 N G Pavlov et al Soliton dual frequency combs incrystalline microresonators Opt Lett 42 514ndash517 (2017)doi 101364OL42000514 pmid 28146515

40 A Dutt et al On-chip dual comb source for spectroscopyarXiv161107673 [physicsoptics] (2016)

41 M Yu Y Okawachi A G Griffith M Lipson A L GaetaMicroresonator-based high-resolution gas spectroscopyOpt Lett 42 4442ndash4445 (2017) doi 101364OL42004442 pmid 29088183

42 J D Jost et al Counting the cycles of light using a self-referenced optical microresonator Optica 2 706 (2015)doi 101364OPTICA2000706

43 V Brasch E Lucas J D Jost M GeiselmannT J Kippenberg Self-referenced photonic chip soliton Kerr



eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

frequency comb Light Sci Appl 6 e16202 (2016)doi 101038lsa2016202

44 P Marin-Palomo et al Microresonator-based solitonsfor massively parallel coherent optical communicationsNature 546 274ndash279 (2017) doi 101038nature22387pmid 28593968

45 W Liang et al High spectral purity Kerr frequency combradio frequency photonic oscillator Nat Commun 6 7957(2015) doi 101038ncomms8957 pmid 26260955

46 P Trocha et al Ultrafast optical ranging usingmicroresonator soliton frequency combs Science 359887ndash891 (2018) doi 101126scienceaao3924pmid 29472477

47 M-G Suh K Vahala Soliton microcomb rangemeasurement arXiv170506697 [physicsoptics] (2017)

48 D T Spencer et al An integrated-photonics optical-frequencysynthesizer arXiv170805228 [physicsapp-ph] (2017)

49 E Obrzud et al A microphotonic astrocombarXiv171209526 [physicsoptics] (2017)

50 M-G Suh et al Searching for exoplanets using amicroresonator astrocomb arXiv180105174 [physicsoptics](2018)

51 V E Lobanov G Lihachev T J KippenbergM L Gorodetsky Frequency combs and platicons inoptical microresonators with normal GVD Opt Express23 7713ndash7721 (2015) doi 101364OE23007713pmid 25837109

52 M Karpov et al Raman self-frequency shift of dissipativeKerr solitons in an optical microresonator Phys Rev Lett116 103902 (2016) doi 101103PhysRevLett116103902pmid 27015482

53 C Miliaacuten A V Gorbach M Taki A V Yulin D V SkryabinSolitons and frequency combs in silica microringresonators Interplay of the Raman and higher-orderdispersion effects Phys Rev A 92 033851 (2015)doi 101103PhysRevA92033851

54 X Yi Q-F Yang K Y Yang K Vahala Theory andmeasurement of the soliton self-frequency shift andefficiency in optical microcavities Opt Lett 41 3419ndash3422(2016) doi 101364OL41003419 pmid 27472583

55 Q F Yang X Yi K Y Yang K Vahala Stokes solitons inoptical microcavities Nat Phys 13 53ndash57 (2017)doi 101038nphys3875

56 E Fermi J Pasta S Ulam Los Alamos Report LA-1940(1955) reproduced in A C Newell Nonlinear Wave Motion(AMS 1974)

57 A B Matsko A A Savchenkov L Maleki On excitationof breather solitons in an optical microresonator Opt Lett37 4856ndash4858 (2012) doi 101364OL37004856pmid 23202069

58 C Y Bao et al Phys Rev Lett 117 5 (2016)59 M Yu et al Breather soliton dynamics in microresonators

Nat Commun 8 14569 (2017) doi 101038ncomms14569pmid 28232720

60 E Lucas M Karpov H Guo M L GorodetskyT J Kippenberg Breathing dissipative solitons in opticalmicroresonators Nat Commun 8 736 (2017) doi 101038s41467-017-00719-w pmid 28963496

61 T Herr et al Mode spectrum and temporal solitonformation in optical microresonators Phys Rev Lett 113123901 (2014) doi 101103PhysRevLett113123901pmid 25279630

62 Q-F Yang X Yi K Y Yang K Vahala Spatial-mode-interaction-induced dispersive waves and their active tuningin microresonators Optica 3 1132 (2016) doi 101364OPTICA3001132

63 D C Cole E S Lamb P DelHaye S A Diddams S B PappSoliton crystals in Kerr resonators Nat Photonics 11671ndash676 (2017) doi 101038s41566-017-0009-z

64 H Guo et al Universal dynamics and deterministic switchingof dissipative Kerr solitons in optical microresonatorsNat Phys 13 94ndash102 (2017) doi 101038nphys3893

65 K Ikeda H Daido O Akimoto Optical turbulenceChaotic behavior of transmitted light from a ring cavityPhys Rev Lett 45 709ndash712 (1980) doi 101103PhysRevLett45709

66 D W M Laughlin J V Moloney A C Newell Solitary wavesas fixed points of infinite-dimensional maps in an opticalbistable ring cavity Phys Rev Lett 51 75ndash78 (1983) PRLdoi 101103PhysRevLett5175

67 T Hansson S Wabnitz Dynamics of microresonatorfrequency comb generation Models and stabilityNanophotonics 5 231 (2016) doi 101515nanoph-2016-0012

68 A Hasegawa F Tappert Transmission of stationarynonlinear optical pulses in dispersive dielectric fibersI Anomalous dispersion Appl Phys Lett 23 142ndash144(1973) doi 10106311654836

69 K Saha et al Modelocking and femtosecond pulsegeneration in chip-based frequency combs Opt Express21 1335ndash1343 (2013) doi 101364OE21001335pmid 23389027

70 V Brasch Q-F Chen S Schiller T J Kippenberg Radiationhardness of high-Q silicon nitride microresonators for spacecompatible integrated optics Opt Express 22 30786ndash30794(2014) doi 101364OE22030786 pmid 25607027

71 J S Levy et al CMOS-compatible multiple-wavelengthoscillator for on-chip optical interconnects Nat Photonics 437ndash40 (2010) doi 101038nphoton2009259

72 S Coen H G Randle T Sylvestre M Erkintalo Modeling ofoctave-spanning Kerr frequency combs using a generalizedmean-field Lugiato-Lefever model Opt Lett 38 37ndash39(2013) doi 101364OL38000037 pmid 23282830

73 A Kordts M H P Pfeiffer H Guo V BraschT J Kippenberg Higher order mode suppression in high-Qanomalous dispersion SiN microresonators for temporaldissipative Kerr soliton formation Opt Lett 41 452ndash455(2016) doi 101364OL41000452 pmid 26907395

74 V Brasch M Geiselmann M H P Pfeiffer T J KippenbergBringing short-lived dissipative Kerr soliton states inmicroresonators into a steady state Opt Express 2429312ndash29320 (2016) doi 101364OE24029312pmid 27958591

75 J Stone et al Thermal and nonlinear dissipative-solitondynamics in Kerr microresonator frequency combsarXiv170808405 [physicsoptics] (2017)

76 K E Webb M Erkintalo S Coen S G MurdochExperimental observation of coherent cavity solitonfrequency combs in silica microspheres Opt Lett 414613ndash4616 (2016) doi 101364OL41004613pmid 28005849

77 M Yu Y Okawachi A G Griffith M Lipson A L GaetaMode-locked mid-infrared frequency combs in a siliconmicroresonator Optica 3 854 (2016) doi 101364OPTICA3000854

78 X Yi Q-F Yang K Youl Yang K Vahala Active capture andstabilization of temporal solitons in microresonatorsOpt Lett 41 2037ndash2040 (2016) doi 101364OL41002037 pmid 27128068

79 E Lucas H Guo J D Jost M Karpov T J KippenbergDetuning-dependent properties and dispersion-inducedinstabilities of temporal dissipative Kerr solitons in opticalmicroresonators Phys Rev A 95 043822 (2017)doi 101103PhysRevA95043822

80 T Herr M L Gorodetsky T J Kippenberg in Dissipative KerrSolitons in Optical Microresonators (Wiley 2015)pp 129ndash162

81 A A Savchenkov et al Low threshold optical oscillations in awhispering gallery mode CaF2 resonator Phys Rev Lett 93243905 (2004) doi 101103PhysRevLett93243905 pmid15697815

82 C Y Wang et al Mid-infrared optical frequency combsat 25 mm based on crystalline microresonatorsNat Commun 4 1345 (2013) doi 101038ncomms2335pmid 23299895

83 J Li H Lee T Chen K J Vahala Low-pump-powerlow-phase-noise and microwave to millimeter-waverepetition rate operation in microcombs Phys Rev Lett109 233901 (2012) doi 101103PhysRevLett109233901pmid 23368202

84 M L Gorodetsky WE-Heraeus Seminar on Micro andMacro-cavities in Classical and Non-classical Light (2011)httpgoogltmniGm

85 Y K Chembo D V Strekalov N Yu Spectrum and dynamicsof optical frequency combs generated with monolithicwhispering gallery mode resonators Phys Rev Lett104 103902 (2010) doi 101103PhysRevLett104103902pmid 20366426

86 T Carmon L Yang K Vahala Dynamical thermalbehavior and thermal self-stability of microcavitiesOpt Express 12 4742ndash4750 (2004) doi 101364OPEX12004742 pmid 19484026

87 V E Zakharov A B Shabat Exact theory of two-dimensionalself-focusing and one-dimensional self-modulation of wavesin nonlinear media Sov Phys JETP 34 62 (1972)

88 T S Raju C N Kumar P K Panigrahi On exact solitarywave solutions of the nonlinear Schroumldinger equation with a

source J Phys Math Gen 38 L271ndashL276 (2005)doi 1010880305-44703816L02

89 W H Renninger P T Rakich Closed-form solutions andscaling laws for Kerr frequency combs Sci Rep 6 24742(2016) doi 101038srep24742 pmid 27108810

90 D J Kaup A C Newell Solitons as Particles Oscillators andin slowly changing media A singular perturbation theoryProc R Soc London Ser A 361 413ndash446 (1978)doi 101098rspa19780110

91 K Nozaki N Bekki Solitons as attractors of a forceddissipative nonlinear Schroumldinger equation Phys Lett A 102383ndash386 (1984) doi 1010160375-9601(84)91060-0

92 T Hansson D Modotto S Wabnitz On the numericalsimulation of Kerr frequency combs using coupled modeequations Opt Commun 312 134ndash136 (2014) doi 101016joptcom201309017

93 I S Grudinin et al High-contrast Kerr frequency combsOptica 4 434 (2017) doi 101364OPTICA4000434

94 V B Braginsky M L Gorodetsky V S Ilchenko Quality-factor and nonlinear properties of optical whispering-gallerymodes Phys Lett A 137 393ndash397 (1989) doi 1010160375-9601(89)90912-2

95 P-H Wang et al Intracavity characterization of micro-combgeneration in the single-soliton regime Opt Express24 10890ndash10897 (2016) doi 101364OE24010890pmid 27409909

96 M H P Pfeiffer et al Photonic Damascene process forintegrated high-Q microresonator based nonlinear photonicsOptica 3 20 (2016) doi 101364OPTICA3000020

97 X Ji et al Ultra-low-loss on-chip resonators withsub-milliwatt parametric oscillation threshold Optica 4619 (2017) doi 101364OPTICA4000619

98 M A Foster et al Broad-band optical parametric gain on asilicon photonic chip Nature 441 960ndash963 (2006)doi 101038nature04932 pmid 16791190

99 J Riemensberger et al Dispersion engineering of thick high-Q silicon nitride ring-resonators via atomic layer depositionOpt Express 20 27661ndash27669 (2012) doi 101364OE20027661 pmid 23262714

100 L Zhang et al Generation of two-cycle pulses andoctave-spanning frequency combs in a dispersion-flattenedmicro-resonator Opt Lett 38 5122ndash5125 (2013)doi 101364OL38005122 pmid 24281525

101 Y Okawachi et al Bandwidth shaping of microresonator-basedfrequency combs via dispersion engineering Opt Lett 393535ndash3538 (2014) doi 101364OL39003535 pmid 24978530

102 I S Grudinin N Yu Dispersion engineering of crystallineresonators via microstructuring Optica 2 221 (2015)doi 101364OPTICA2000221

103 C Bao et al High-order dispersion in Kerr comb oscillatorsJ Opt Soc Am B 34 715 (2017) doi 101364JOSAB34000715

104 N Akhmediev M Karlsson Cherenkov radiation emitted bysolitons in optical fibers Phys Rev A 51 2602ndash2607 (1995)doi 101103PhysRevA512602 pmid 9911876

105 A V Cherenkov V E Lobanov M L Gorodetsky DissipativeKerr solitons and Cherenkov radiation in opticalmicroresonators with third-order dispersion Phys Rev A 95033810 (2017) doi 101103PhysRevA95033810

106 X Yi et al Nat Commun 8 9 (2017) doi 101038s41467-017-00020-w pmid 28377584

107 M R E Lamont Y Okawachi A L Gaeta Route to stabilizedultrabroadband microresonator-based frequency combsOpt Lett 38 3478ndash3481 (2013) doi 101364OL38003478 pmid 24104792

108 H Kumar F Chand Dark and bright solitary wave solutionsof the higher order nonlinear Schroumldinger equation withself-steepening and self-frequency shift effectsJ Nonlinear Opt Phys Mater 22 135001 (2013)doi 101142S021886351350001X

109 C Bao et al Nonlinear conversion efficiency in Kerrfrequency comb generation Opt Lett 39 6126ndash6129 (2014)doi 101364OL39006126 pmid 25361295

110 M Erkintalo Y Q Xu S G Murdoch J M DudleyG Genty Cascaded phase matching and nonlinear symmetrybreaking in fiber frequency combs Phys Rev Lett109 223904 (2012) doi 101103PhysRevLett109223904pmid 23368122

111 Y Wang M Anderson S Coen S G Murdoch M ErkintaloStimulated Raman scattering imposes fundamentallimits to the duration and bandwidth of temporal cavitysolitons Phys Rev Lett 120 053902 (2018) doi 101103PhysRevLett120053902 pmid 29481150



eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

112 Q F Yang X Yi K Y Yang K Vahala Counter-propagatingsolitons in microresonators Nat Photonics 11 560ndash564(2017) doi 101038nphoton2017117

113 C Joshi et al Counter-rotating cavity solitons in a siliconnitride microresonator Opt Lett 43 547ndash550 (2018)doi 101364OL43000547 pmid 29400837

114 E Lucas et al Spatial multiplexing of soliton microcombsarXiv180403706v2 [physicsoptics] (2018) doi arxivorgabs180403706v2

115 A B Matsko A A Savchenkov L Maleki Normal group-velocity dispersion Kerr frequency comb Opt Lett 3743ndash45 (2012) doi 101364OL37000043 pmid 22212785

116 C Godey I V Balakireva A Coillet Y K Chembo Stabilityanalysis of the spatiotemporal Lugiato-Lefever model for Kerroptical frequency combs in the anomalous and normaldispersion regimes Phys Rev A 89 063814 (2014)doi 101103PhysRevA89063814

117 W Liang et al Generation of a coherent near-infrared Kerrfrequency comb in a monolithic microresonator with normalGVD Opt Lett 39 2920ndash2923 (2014) doi 101364OL39002920 pmid 24978237

118 S W Huang et al Mode-locked ultrashort pulse generationfrom on-chip normal dispersion microresonatorsPhys Rev Lett 114 053901 (2015) doi 101103PhysRevLett114053901 pmid 25699441

119 X Xue et al Mode-locked dark pulse Kerr combs innormal-dispersion microresonators Nat Photonics 9594ndash600 (2015) doi 101038nphoton2015137

120 J K Jang et al Dynamics of mode-coupling-inducedmicroresonator frequency combs in normal dispersionOpt Express 24 28794ndash28803 (2016) doi 101364OE24028794 pmid 27958523

121 V E Lobanov G Lihachev M L Gorodetsky Generation ofplaticons and frequency combs in optical microresonatorswith normal GVD by modulated pump Europhys Lett 11254008 (2015) doi 1012090295-507511254008

122 X X Xue et al Normal-dispersion microcombs enabled bycontrollable mode interactions Laser Photonics Rev 9L23ndashL28 (2015) doi 101002lpor201500107

123 P Parra-Rivas E Knobloch D Gomila L Gelens Darksolitons in the Lugiato-Lefever equation with normaldispersion Phys Rev A 93 063839 (2016) doi 101103PhysRevA93063839

124 V E Lobanov A V Cherenkov A E Shitikov I A BilenkoM L Gorodetsky Dynamics of platicons due to third-orderdispersion Eur Phys J D 71 185 (2017) doi 101140epjde2017-80148-0

125 P Parra-Rivas D Gomila L Gelens Coexistence of stabledark- and bright-soliton Kerr combs in normal-dispersionresonators Phys Rev A 95 053863 (2017) doi 101103PhysRevA95053863

126 S B Papp et al Microresonator frequency comboptical clock Optica 1 10 (2014) doi 101364OPTICA1000010

127 P DelrsquoHaye et al Nat Photonics 10 516ndash520 (2016)128 W Liang et al IEEE Photonics J 9 11 (2017)129 T Ideguchi et al Coherent Raman spectro-imaging with laser

frequency combs Nature 502 355ndash358 (2013)doi 101038nature12607 pmid 24132293

130 V Ataie D Esman B P-P Kuo N Alic S RadicSubnoise detection of a fast random event Science350 1343ndash1346 (2015) doi 101126scienceaac8446pmid 26659052

131 M Pu L Ottaviano E Semenova K Yvind Efficientfrequency comb generation in AlGaAs-on-insulator Optica 3823 (2016) doi 101364OPTICA3000823

132 Y Okawachi et al Quantum random number generatorusing a microresonator-based Kerr oscillator Opt Lett41 4194ndash4197 (2016) doi 101364OL41004194pmid 27628355

133 M Kues et al On-chip generation of high-dimensionalentangled quantum states and their coherent control

Nature 546 622ndash626 (2017) doi 101038nature22986pmid 28658228

134 H Taheri A B Matsko L Maleki Optical lattice trap for Kerrsolitons Eur Phys J D 71 153 (2017) doi 101140epjde2017-80150-6

135 P Ghelfi et al A fully photonics-based coherent radarsystem Nature 507 341ndash345 (2014) doi 101038nature13078 pmid 24646997

136 D T Spencer et al An optical-frequency synthesizer usingintegrated photonics Nature 557 81ndash85 (2018)doi 101038s41586-018-0065-7 pmid 29695870

ACKNOWLEDGMENTS

The authors thank M Karpov and A Lukashchuk for assistance inpreparing the figures as well as S Houmlhnle from IBM for renderingFig 1 Author contributions This Review was written by TJKand MLG with critical input and support of ALG and ML Allauthors contributed and discussed the content of the ReviewFunding This material is based on work supported by the DefenseAdvanced Research Projects Agency (DARPA) Defense SciencesOffice (DSO) and (MTO) grants HR0011-15-C-0055 (DODOS)W31P4Q-16-1-0002 (SCOUT) and W31P4Q-14-C-0050 (PULSE)The European Space Technology Centre with ESA (400011614516NLMHGM) the Swiss National Science Foundation via BRIDGE(176563) a Swiss National Science Foundation Grant and Ministry ofEducation and Science of the Russian Federation (projectRFMEFI58516X0005) and the Air Force Office of ScientificResearch Air Force Material Command USAF under awardsFA9550-15-1-0099 and FA9550-15-1-0303 Competing interestsTJK is a cofounder and shareholder of LiGenTec SA a start-upcompany offering Si3N4 photonic integrated circuits as a foundryservice TJK ALG MLG and ML are co-inventors of patents inthe field of publication Patents are owned by EPFL Cornell andthe Max Planck Society

101126scienceaan8083



eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from

Dissipative Kerr solitons in optical microresonatorsTobias J Kippenberg Alexander L Gaeta Michal Lipson and Michael L Gorodetsky

DOI 101126scienceaan8083 (6402) eaan8083361Science

this issue p eaan8083Scienceproviding a new basis for precision technologyof microresonator-generated frequency combs and map out how understanding and control of their generation is

review the developmentet alapplications from the realm of national laboratories to that of everyday devices Kippenberg combs being generated in optical microresonator circuits offering the prospect of shifting precision metrology

has revolutionized metrology and precision spectroscopy The past decade has seen frequencyminusminusrange of wavelengths light sources comprising equidistant laser lines spanning a largeminusminusThe ability to generate laser frequency combs

Shrinking optical metrology

ARTICLE TOOLS httpsciencesciencemagorgcontent3616402eaan8083

REFERENCES

httpsciencesciencemagorgcontent3616402eaan8083BIBLThis article cites 123 articles 6 of which you can access for free

PERMISSIONS httpwwwsciencemagorghelpreprints-and-permissions

Terms of ServiceUse of this article is subject to the

is a registered trademark of AAASSciencelicensee American Association for the Advancement of Science No claim to original US Government Works The title Science 1200 New York Avenue NW Washington DC 20005 2017 copy The Authors some rights reserved exclusive

(print ISSN 0036-8075 online ISSN 1095-9203) is published by the American Association for the Advancement ofScience

on Septem

ber 10 2018


Dow

nloaded from

REVIEW

OPTICS

Dissipative Kerr solitons inoptical microresonatorsTobias J Kippenberg1 Alexander L Gaeta2 Michal Lipson3 Michael L Gorodetsky45

The development of compact chip-scale optical frequency comb sources (microcombs)based on parametric frequency conversion in microresonators has seen applicationsin terabit optical coherent communications atomic clocks ultrafast distancemeasurements dual-comb spectroscopy and the calibration of astophysicalspectrometers and have enabled the creation of photonic-chip integrated frequencysynthesizers Underlying these recent advances has been the observation oftemporal dissipative Kerr solitons in microresonators which represent self-enforcingstationary and localized solutions of a damped driven and detuned nonlinearSchroumldinger equation which was first introduced to describe spatial self-organizationphenomena The generation of dissipative Kerr solitons provide a mechanism by whichcoherent optical combs with bandwidth exceeding one octave can be synthesized andhave given rise to a host of phenomena such as the Stokes soliton soliton crystalssoliton switching or dispersive waves Soliton microcombs are compact are compatiblewith wafer-scale processing operate at low power can operate with gigahertz toterahertz line spacing and can enable the implementation of frequency combsin remote and mobile environments outside the laboratory environment on Earthairborne or in outer space

Solitons are robust waveforms that preservetheir shape upon propagation in dispersivemedia and can be found in a variety of non-linear systems ranging fromhot plasma (1)ferrite films fiber optics (2) cold atoms (3)

hydrodynamics and biology to cloud and sanddune formation Although the initial promisesof optical solitons for increased bandwidth oftelecommunication were not implementeddissipative solitons circulating in cavities ofmode-locked lasers (4) are actively used Similar to thephysics of open quantum systems the ldquoopenrdquosoliton systemsmdashdissipative solitons (5)mdasharehighlyrelevant to actual experimental systems in whichdissipation cannot be neglected This is in con-trast to the mathematical treatment of solitonsthat have focused on conservative integrable sys-tems Dissipative solitons (5) balance loss andgain in an active media along with the balanceof nonlinearity and dispersion (Fig 1) Althoughdissipative optical solitons are known to occurinmode-locked lasers (4) dissipative optical soli-tons can also occur in systems that have para-metric gainmdashgain from four-wave mixing thatis the underlying process that enables Kerr fre-quency comb (6 7) formation inmicroresonators

Such dissipative solitons sustained by parametricgain were first studied in fiber cavities (8) In2014 it was observed that such dissipative Kerrsolitons (DKSs) can spontaneously form in pa-rametrically driven Kerr frequency combs inoptical microresonators (9) In these systemsoptical sidebands are generated bymeans of four-wave mixing processes (10 11) and undergo a self-organization process that leads to the emergenceof a soliton pulse train which canmathematicallyand rigorously be mapped (12) to the Lugiato-Lefever equation (LLE) (13ndash15) an equation ex-tensively studied in applied mathematics fordecadesThe equation was originally developed to de-

scribe spatially localized pattern formation indriven nonlinear systems in which it can leadto the formation of ldquodissipative structurerdquo Sincetheir observation in microresonators such DKSshave been generated in a wide variety of micro-resonators ranging from bulk crystalline (9 16)and silica microdisks (17) to photonic chip-scaledevices (17ndash20) under both continuous-wave (CW)and pulsed excitation (21) In the frequency do-main the pulse train constitutes an optical fre-quency comb (22ndash24) Such optical frequencycombs pioneered by using mode-locked lasersbased on pulse trains are key ingredients ofoptical atomic clocks and have proven invaluablefor a wide range of scientific and technologicalapplications [for example reviewed in (25)] Al-though optical comb technology has maturedsince its first introduction two decades ago basedon mode-locked lasers and is commerciallyavailable such devices are typically not amena-

ble to wafer-scale integration and have compar-atively narrow line spacing in the hundreds ofmegahertz range DKSs in microresonators haveprovided a route to compact low power in chip-frequency comb generation AlthoughKerr combshave been known for more than one decade (7)[and have been reviewed in (6 26ndash28)] the dis-covery of the DKS regime (9) has unlocked theirfull potential by providing a route to broadbandand fully coherent microresonator-based fre-quency combs overcoming earlier challengesof low coherence (29ndash31) Such soliton-basedmicrocombs in chip-integratedmicroresonatorshave achieved low-power octave-spanning fre-quency combs in various spectral windows thatnow encompass the near-infrared (32 33) tele-communication (34 35) andmid-infrared spectralwindow (18 36) with repetition rates from only afew gigahertz (37) to terahertz The observation ofDKS in microresonators yields a merging ofsoliton physics and high-precision frequency combapplications stimulating a renaissance in dissipa-tive soliton research and enabling many techno-logical applications Soliton microcombs havealready been applied succesfully to dual-combspectroscopy in thenear- (38ndash40) andmid-infrared(36) scanning comb spectroscopy (41) as well asthe demonstration of a self-referenced comb forthe counting of optical frequencies [both with(42) and without external broadening (43)]Likewise soliton microcombs have been used inmassively parallel communication (44) in pairsat both the transmitter and receiver side forlow-noise microwave generation (45) as well asfor chip-scale dual-combndashbased light detectionand ranging (LIDAR) (46 47) Moreover solitonmicrocombs have been used to realize an allintegrated photonics-based chip-frequency syn-thesizer (48) and been used as a microphotonicastrocomb for exoplanet searches (49 50) Soli-ton microcombs exhibit surprisingly rich non-linear dynamics and have led to the observationof a variety soliton dynamics effects such asbright soliton formation dark pulses flat-toppedldquoplaticonsrdquo formation (51) soliton Cherenkov ra-diation (20) Raman self-frequency shift (52ndash54)Stokes soliton (55) and breather solitons (56ndash60)Moreover the inherently multimode nature of mi-croresonators gives rise to new and unanticipateddynamics such as avoided-crossingndashinduced disper-sive wave formation (61 62) or soliton-intermodebreathing dynamics Likewise soliton crystalliza-tion (63) and switching (64) have been observedto occur Today a growing understanding of thenonlinear dynamics in soliton microcombs hasbeen established and new dynamics have beenobserved whose exploration continues to develop

Dissipative Kerr solitons

The topic of this Review is a soliton that has longbeen theoretically discussed (13ndash15) but only re-cently experimentally observed temporal DKSsin which losses in a passive nonlinear opticalcavity are compensated by a parametric inter-action via a CWexternal laser pump in a resonatorcontaining a Kerr nonlinearity (10) Such DKSstates first studied in fiber cavities (8) have

RESEARCH


1Eacutecole Polytechnique Feacutedeacuterale de Lausanne (EPFL) Instituteof Physics Lausanne CH-1015 Switzerland 2Department ofApplied Physics and Applied Mathematics Columbia UniversityNY 10027 USA 3Department of Electrical EngineeringColumbia University NY 10027 USA 4Faculty of PhysicsLomonosov Moscow State University Moscow 119991 Russia5Russian Quantum Center Moscow 143025 RussiaCorresponding author Email tobiaskippenbergepflch (TJK)mgrqcru (MLG)

on Septem

ber 10 2018


Dow

nloaded from










FROG

0 70

ωSH

G2

π 71 ps

4TH

z

0

-60

dB

Wavelength (nm)

Pow

er (2

0 dB

div

)

1520 1580

cw pumplaser

500 kHz

1 kHzGHzRBW1409

D

C

B

A

E



eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from



Djmj=j eth1THORN








r







δδδ δ


ωδ

1) Primary combs

ω

Pump

3) Subcomb merging


ω

ω

δδ minus ξ

δ + ξ

Δ

Kerr cavityInput

ω

tCW

P

P

CW

OutputSolitons

t

P

ω

P

Frequencycomb

1

2

3

4

5

1 2 3 4 5

1 2 3 4 5

C

B

A

D

-1 20 30100

Intr

acav

ity p

ower




Δ Δ Δ

101 modes

1 round trip


40

au




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from



q≫1)











~ 100 nm

Rela

tive

norm

aliz

ed in

trac

avity

pow

er

0

Laser scan

λ

Wavelength

Spec

tral

pow

er

A CB

0 10

100

2004

1

Tran

smis

sion

(au

)

Scan time (ms)



0

300

MH

z

25 nm

(ωμ-ω0-μD12π)

Pow

er (a

u)



Upper bra

nch




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






A

t i

D2

2

2A


hksp

eth2THORN


iYt

thorn 1

2

2Y



minus100

0

50

100

Disp

ersi

on (p

s (n

mmiddotk

m)- 1)

2200200018001600140012001000

Wavelength (nm)



minus50

Bulk Si3N4

Frequency (THz)188 190 192 194 196 198

-05

0

05

1

15

2

25

D int2

π (G

Hz)

Mod

e in

dex

n eff(λ

)

Wavelength λ

Mat

eria

l ind

ex n

(λ)


Cladding index


Wavelength λ

High confinement

Low confinement

A B D

C

E





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


2D2

q f frac14


q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk





Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from










FROG

0 70

ωSH

G2

π 71 ps

4TH

z

0

-60

dB

Wavelength (nm)

Pow

er (2

0 dB

div

)

1520 1580

cw pumplaser

500 kHz

1 kHzGHzRBW1409

D

C

B

A

E



eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from



Djmj=j eth1THORN








r







δδδ δ


ωδ

1) Primary combs

ω

Pump

3) Subcomb merging


ω

ω

δδ minus ξ

δ + ξ

Δ

Kerr cavityInput

ω

tCW

P

P

CW

OutputSolitons

t

P

ω

P

Frequencycomb

1

2

3

4

5

1 2 3 4 5

1 2 3 4 5

C

B

A

D

-1 20 30100

Intr

acav

ity p

ower




Δ Δ Δ

101 modes

1 round trip


40

au




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from



q≫1)











~ 100 nm

Rela

tive

norm

aliz

ed in

trac

avity

pow

er

0

Laser scan

λ

Wavelength

Spec

tral

pow

er

A CB

0 10

100

2004

1

Tran

smis

sion

(au

)

Scan time (ms)



0

300

MH

z

25 nm

(ωμ-ω0-μD12π)

Pow

er (a

u)



Upper bra

nch




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






A

t i

D2

2

2A


hksp

eth2THORN


iYt

thorn 1

2

2Y



minus100

0

50

100

Disp

ersi

on (p

s (n

mmiddotk

m)- 1)

2200200018001600140012001000

Wavelength (nm)



minus50

Bulk Si3N4

Frequency (THz)188 190 192 194 196 198

-05

0

05

1

15

2

25

D int2

π (G

Hz)

Mod

e in

dex

n eff(λ

)

Wavelength λ

Mat

eria

l ind

ex n

(λ)


Cladding index


Wavelength λ

High confinement

Low confinement

A B D

C

E





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


2D2

q f frac14


q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk





Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from



Djmj=j eth1THORN








r







δδδ δ


ωδ

1) Primary combs

ω

Pump

3) Subcomb merging


ω

ω

δδ minus ξ

δ + ξ

Δ

Kerr cavityInput

ω

tCW

P

P

CW

OutputSolitons

t

P

ω

P

Frequencycomb

1

2

3

4

5

1 2 3 4 5

1 2 3 4 5

C

B

A

D

-1 20 30100

Intr

acav

ity p

ower




Δ Δ Δ

101 modes

1 round trip


40

au




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from



q≫1)











~ 100 nm

Rela

tive

norm

aliz

ed in

trac

avity

pow

er

0

Laser scan

λ

Wavelength

Spec

tral

pow

er

A CB

0 10

100

2004

1

Tran

smis

sion

(au

)

Scan time (ms)



0

300

MH

z

25 nm

(ωμ-ω0-μD12π)

Pow

er (a

u)



Upper bra

nch




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






A

t i

D2

2

2A


hksp

eth2THORN


iYt

thorn 1

2

2Y



minus100

0

50

100

Disp

ersi

on (p

s (n

mmiddotk

m)- 1)

2200200018001600140012001000

Wavelength (nm)



minus50

Bulk Si3N4

Frequency (THz)188 190 192 194 196 198

-05

0

05

1

15

2

25

D int2

π (G

Hz)

Mod

e in

dex

n eff(λ

)

Wavelength λ

Mat

eria

l ind

ex n

(λ)


Cladding index


Wavelength λ

High confinement

Low confinement

A B D

C

E





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


2D2

q f frac14


q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk





Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from



q≫1)











~ 100 nm

Rela

tive

norm

aliz

ed in

trac

avity

pow

er

0

Laser scan

λ

Wavelength

Spec

tral

pow

er

A CB

0 10

100

2004

1

Tran

smis

sion

(au

)

Scan time (ms)



0

300

MH

z

25 nm

(ωμ-ω0-μD12π)

Pow

er (a

u)



Upper bra

nch




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






A

t i

D2

2

2A


hksp

eth2THORN


iYt

thorn 1

2

2Y



minus100

0

50

100

Disp

ersi

on (p

s (n

mmiddotk

m)- 1)

2200200018001600140012001000

Wavelength (nm)



minus50

Bulk Si3N4

Frequency (THz)188 190 192 194 196 198

-05

0

05

1

15

2

25

D int2

π (G

Hz)

Mod

e in

dex

n eff(λ

)

Wavelength λ

Mat

eria

l ind

ex n

(λ)


Cladding index


Wavelength λ

High confinement

Low confinement

A B D

C

E





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


2D2

q f frac14


q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk





Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from






A

t i

D2

2

2A


hksp

eth2THORN


iYt

thorn 1

2

2Y



minus100

0

50

100

Disp

ersi

on (p

s (n

mmiddotk

m)- 1)

2200200018001600140012001000

Wavelength (nm)



minus50

Bulk Si3N4

Frequency (THz)188 190 192 194 196 198

-05

0

05

1

15

2

25

D int2

π (G

Hz)

Mod

e in

dex

n eff(λ

)

Wavelength λ

Mat

eria

l ind

ex n

(λ)


Cladding index


Wavelength λ

High confinement

Low confinement

A B D

C

E





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


2D2

q f frac14


q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk





Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from


2D2

q f frac14


q Y frac14

ffiffiffiffi2gk

qA z0 = 2Dk





Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)Po

wer

(20

dBd

iv)

Pow

er (2

0 dB

div

)

Din

t2π

(MH

z)D

int2

π (G

Hz)

Din

t2π

(GH

z)

2001000

0 50 100-50-100

-100 1300 1400 1500 1600 1700 1800 1900 2000

0 200-100-200 100 1520 16001540 1560 1580Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Wavelength (nm)

Mode number

Mode number

Mode number

~ sech2 Pow

er (a

u)

Din

t2π

(GH

z)

0-40-80 40 80 1550 1600 1650 1700

1560

1500

1520 1600 1640

14501400

Mode number

Raman gain

Din

t2π

(MH

z)

0 200-100-200 100

0

0

10

0

-25

25

20

30

100

200

300

-10

-20

-30

0

0

100

200

300

Mode number

20 μm

Bus waveguide

100 microm

Microring

Microheater

B

A

C

D

E

1600 1800 2000 2200140012001000

10 μm

Time (au)

Pow

er (a

u)

Time (au)

Pow

er (a

u)

Time (au)




eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from













eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from




















LIDAR









eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from






Quantum technology

















































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from









































































eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from


























ACKNOWLEDGMENTS





eptember 10 2018

httpsciencesciencem

agorgD

ownloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from








REFERENCES






on Septem

ber 10 2018


Dow

nloaded from

columbia university - optics dissipative kerr …...have given rise to a host of phenomena, such as...

Documents