A silicon Brillouin laser

Nils T. Otterstrom,1 Ryan O. Behunin,1 Eric A. Kittlaus,1 Zheng Wang,2 and Peter T. Rakich1

1Department of Applied Physics, Yale University, New Haven, CT 06520 USA.2Department of Electrical and Computer Engineering,

University of Texas at Austin, Austin, TX 78758 USA.(Dated: December 12, 2017)

Brillouin laser oscillators offer powerful and flexible dynamics, which have been leveraged in a va-riety of physical systems to create mode-locked lasers, microwave oscillators, and optical gyroscopes.Until recently, however, the prospects for adapting Brillouin laser technologies to silicon photonics—one of the leading integrated photonics platforms—seemed bleak because Brillouin interactions werepuzzlingly absent from silicon waveguides. Only with the recent advent of new photonic-phononicwaveguides have Brillouin interactions become one of the strongest and most tailorable nonlinearitiesin silicon. In this paper, we harness these engineered nonlinearities to demonstrate Brillouin lasingin silicon for the first time. Moreover, we show that this laser produces an intriguing regime of three-wave dynamics, in which optical self-oscillation yields Schawlow-Townes-like linewidth narrowing ofa distributed, spatially heavily damped phonon field. These dynamics reveal that macroscopic Bril-louin lasers—which produce optical and not acoustic self-oscillation—can yield phonon linewidthnarrowing, as long as the temporal acoustic dissipation rate is smaller than that of the opticalfields. Beyond this study, these laser technologies open the door to a range of new applications formonolithic integration within silicon photonic circuits.

INTRODUCTION

Through a mastery of the optical and electronic proper-ties of silicon, the field of silicon photonics has produced avariety of chip-scale optical technologies [1–10] for appli-cations ranging from high-bandwidth communications [11]to biosensing on a chip [12]. The potential for these andother possible technologies has piqued interest in strategiesto reshape the coherence and spectral properties of lightfor a large array of on-chip functionalities. One promis-ing approach to customize on-chip light involves the use ofthe nonlinear optical properties of silicon to create opticallaser oscillators [6]. For example, Raman and Kerr non-linearities have been harnessed to create all-silicon Ramanlasers [13–15] and for demonstrations of long-wavelengthKerr frequency combs in silicon resonators [16, 17]. Bril-louin interactions, produced by the coupling between lightand sound, offer a complementary set of behaviors and ca-pabilities for laser technologies. By harnessing these non-linearities in a variety of physical systems, Brillouin lasershave been designed to yield everything from frequency-tunable laser emission [18] and mode-locked pulsed lasers[19] to low-noise oscillators and optical gyroscopes [20–26]. While Brillouin lasers have been widely studied inglass waveguides [20, 27] and resonators [28, 29], Brillouinlasing has yet to be demonstrated in silicon.

Brillouin laser oscillation occurs when round-trip opti-cal gain balances round-trip loss within an optical cavity.In Brillouin lasers, optical amplification is produced by anonlinear light-sound coupling called stimulated Brillouinscattering (SBS). These Brillouin interactions are typi-cally exceptionally strong, overtaking Kerr and Ramannonlinearities in most transparent media. Interestingly,however, the same silicon waveguides that radically en-hance Raman and Kerr nonlinearities produce exceedinglyweak Brillouin couplings. Only with the advent of a newclass of suspended waveguides—which tightly confine bothlight and sound—have appreciable nonlinearities becomepossible through forward stimulated Brillouin scattering

[30–33]. These suspended structures were used to demon-strate large optical Brillouin gain [31, 32] and recently netamplification [33, 34]. However, innovative strategies areneeded to translate these new Brillouin interactions intosilicon laser oscillators [35, 36].

In this letter, we demonstrate a Brillouin laser in sil-icon for the first time. This laser utilizes a novel formof on-chip Brillouin scattering, termed stimulated inter-modal Brillouin scattering (SIMS), that couples light fieldsguided in distinct optical spatial modes [35, 37–40]. Usinga Brillouin-active silicon waveguide, we leverage the single-sideband gain of the SIMS process to create a Brillouinlaser in a 4.6 cm-long racetrack resonator. This structureproduces large optical gain with low propagation loss, per-mitting Brillouin lasing at pump powers as low as 11 mW.This result represents an important step toward the re-alization of Brillouin laser technologies in complex siliconphotonic circuits.

From a scientific perspective, this Brillouin laser isintriguing because the exceedingly large Brillouin gaingrants access to a novel limit of three-wave laser dynam-ics. In particular, these dynamics reveal that macroscopicBrillouin lasers—which exhibit optical and not phononself-oscillation—can yield Schawlow-Townes-like phononlinewidth narrowing, as long as the temporal acoustic dis-sipation rate is smaller than that of the optical fields. Herewe show that this temporal dissipation hierarchy and re-sulting phonon linewidth narrowing occur despite largeacoustic spatial damping and a cavity geometry that pre-vents any form of phonon feedback. This unconventionalcombination of spatial and temporal dynamics is madepossible by the disparate velocities of the interacting lightand sound waves. In this regime of operation, the emittedStokes wave also experiences significant spectral compres-sion but eventually converges to the pump linewidth, be-coming highly coherent with the incident pump field. Inthis way, this Brillouin laser system behaves analogouslyto a self-driven acousto-optic frequency shifter, which mayenable new forms of demodulation and sensing in on-chip

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Building on this work, it may be possible to combinethese laser technologies with modulators, detectors, andfilters in the same active silicon layer, highlighting one ofthe major benefits of silicon photonics. Independent ofthe material system, this Brillouin laser topology offers anumber of properties and capabilities that alleviate sev-eral challenges facing the robust integration of Brillouinlasers in on-chip optical platforms. In particular, this laserconcept permits precise control of the Brillouin frequency,device size, and energy transfer dynamics, offering a flex-ible and robust design space for chip-scale Brillouin lasertechnologies. In addition, we alleviate the need for on-chipisolator and circulator technologies, necessary to integrateconventional backward Brillouin lasers, by engineering thespatial dynamics of this system to produce laser emissionin the forward direction.

SYSTEM DESCRIPTION AND OPERATION

We use a highly-engineerable form of stimulated inter-modal Brillouin scattering to create a Brillouin laser ofthe design diagrammed in Fig. 1a. A racetrack opticalresonator is formed from a multi-mode optical waveguidethat provides low-loss guidance of both symmetric (red)and antisymmetric (blue) TE-like spatial modes, yieldingtwo distinct sets of high quality-factor cavity modes. Lightfields propagating in these two spatial modes nonlinearlycouple as they traverse the Brillouin-active regions of theracetrack system (dark gray). In these Brillouin-active re-gions, the waveguide is designed to produce engineerableBrillouin coupling and tight confinement of both opticaland elastic waves. When phase matching and energy con-servation are satisfied, pump-light (of frequency ωp) prop-agating in the antisymmetric (blue) mode amplifies light inthe symmetric mode (red) through stimulated inter-modalBrillouin scattering (see Fig. 1b-e). When the round-tripoptical loss is balanced by the round-trip Brillouin gain,Brillouin oscillation occurs and yields coherent laser emis-sion at the Stokes frequency ωs = ωp − Ωb, where Ωb isthe Brillouin frequency. In what follows, we describe theunderlying principles of operation and the unique laserdynamics produced by this system.

We begin by outlining the linear optical properties ofour racetrack system. Since the symmetric and antisym-metric guided modes have different propagation constants,k1(ω) and k2(ω), respectively, the resonance conditions,k1(ω)L = 2πm and k2(ω)L = 2πn, where L is the race-track circumference, produce a distinct set of resonant fre-quencies ωm

1 and ωn2 (see Fig. 1f-g). Throughout this Let-

ter, we use subscripts 1 and 2 to denote the symmetric andantisymmetric modes, respectively, and use superscripts mand n for resonance indices for each corresponding modetype. A schematic illustrating the distinct resonance con-dition for each transverse spatial mode is shown in Fig. 1f-g. Evanescent coupling between a single-mode bus waveg-uide and the racetrack system permits light to couple intoand out of the symmetric and antisymmetric racetrackmodes with different coupling rates. The contribution to

the transmission spectrum from each spatial mode is iden-tified in Fig. 1f-g; these resonant contributions combineto produce the multimode transmission spectrum of Fig.1h.

Optical amplification is produced within the Brillouinactive segments (dark gray) of the racetrack resonator(Fig. 1a). Through inter-modal Brillouin scattering, apump wave traveling in the antisymmetric mode (ωp) pro-duces amplification of the Stokes wave propagating in thesymmetric mode (ωs) (see Figs. 1d-e). For this interac-tion to occur, both energy conservation (ωp = ωs + Ωb)and phase matching (k2(ωp) = k1(ωs) + q(Ωb)) must besatisfied. Here, q(Ω) is the acoustic dispersion relation.Together these conditions require k2(ωp) = k1(ωp−Ωb) +q(Ωb). This expression is the basis for the succinct dia-grammatic representation that captures the requirementsfor phase matching and energy conservation, shown inFigs. 1b-c. Phonons that satisfy this condition must lieon the acoustic dispersion curve, q(Ω), (Fig. 1c) and con-nect the initial (open circle) and final (solid circle) op-tical states identified on the optical dispersion curve ofFig. 1b. From this diagram, we see that a backward-propagating phonon mediates Stokes scattering betweenpump and Stokes photons propagating in the antisym-metric and symmetric waveguide modes, respectively. Bycontrast, the anti-Stokes process is mediated by a forwardpropagating phonon. Since distinct phonon wavevectorsmediate Stokes and anti-Stokes processes, these interac-tions are decoupled, permitting single-sideband amplifica-tion (for further discussion, see Ref. [35]).

Laser oscillation of the symmetric cavity mode (ωm1 )

occurs when Brillouin gain matches the round-trip loss,producing coherent laser emission at the Stokes frequency(ωs = ωm

1 ). These lasing requirements are met by inject-ing pump light (of power Pp) into the system through thebus waveguide. As the incident pump wave is tuned to ωn

2 ,pump light resonantly circulates in the nth antisymmetriccavity mode. When this pump power exceeds the thresh-old power (Pp > Pth), the Stokes field builds from ther-mal noise to yield appreciable line-narrowing and coherentStokes emission at frequency ωp − Ωb. This Stokes lightis coupled out of the resonator through the bus waveguide(Fig. 1a), which routes the light off-chip.

OBSERVATIONS OF LASER OSCILLATION

The silicon Brillouin laser system depicted in Fig. 2a isfabricated from a single-crystal silicon-on-insulator (SOI)wafer (for details see Methods). Throughout the device,light is guided by total internal reflection using a ridgewaveguide with the cross-section diagrammed in Fig. 2d.The Brillouin-active regions (Fig. 2b) were created by re-moving the oxide undercladding to create a continuously-suspended optomechanical waveguide that produces largeinter-modal Brillouin gain. These Brillouin-active sectionsof the waveguide are mechanically supported by a periodicarray of nanoscale tethers as depicted in Fig. 2a. Removalof the oxide under-cladding has negligible impact on theguidance of the optical modes. In addition to low-loss op-

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FIG. 1: Silicon Brillouin laser device concept (not to scale). (a) Schematic of laser cavity and basic operation. The Brillouinlaser consists of a multi-mode racetrack cavity with two Brillouin-active regions (dark gray). Pump light (blue) is coupled intothe antisymmetric spatial mode of the racetrack resonator via a single-mode bus waveguide. SIMS mediates energy transfer fromthe pump wave (antisymmetric waveguide mode) to the Stokes wave (symmetric waveguide mode). Panel (b) diagrams idealizeddispersion relations for the two optical modes and the phonon wavevector mediating SIMS. The acoustic dispersion relation q(Ω)for the Brillouin-active phonon mode is depicted in (c). Critical to SIMS phase matching, this phonon branch is cut off in frequencyfor vanishing wavevector. Pump light with frequency ωp and wavevector kp (plotted as an open circle to indicate initial state)can scatter to the Stokes wave with frequency ωs and wavevector ks (final state indicated by a closed circle) if a Brillouin-activephonon satisfies energy conservation and phase matching. In this representation, the Brillouin-active phonon vector (q(Ω),Ω)must connect the initial and final optical state coordinates. Once excited, this phonon does not mediate the anti-Stokes processas there is no optical dispersion relation that can satisfy phase matching (red no symbol). (d) and (e) illustrate SIMS energytransfer in the Brillouin-active segment of the laser cavity where an acoustic phonon mediates pump depletion and Stokes growth.Panel (f-h) depict the idealized transmission spectrum for the multimode ring cavity in the case when (f) µ2 = 0, (g) µ1 = 0, and(h) µ2 > µ1 > 0. (f) depicts the transmission spectrum produced by the symmetric mode (fi). Cavity resonances are indexed bythe letter m (fii). The resonance condition for this mode is given in (fiii). Panel (g) shows the transmission spectrum producedby the antisymmetric mode (gi), with cavity resonances indexed by the letter n (gii). The resonance condition for this modeis given in (giii). The optical properties of the multimode resonator is shown in (h), resulting in the transmission spectrum in(hii). Narrower (broader) resonant features correspond to the symmetric (antisymmetric) resonances of the racetrack system.(hii) Brillouin lasing occurs when the resonance condition for the pump and Stokes waves are simultaneously satisfied.

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FIG. 2: Diagram of the silicon Brillouin laser and Brillouin-active segments. (a) Schematic of the silicon Brillouin laser cavity(not to scale). The laser consists of a racetrack cavity with two long Brillouin-active regions. The Brillouin-active regionsare composed of continuously-suspended photonic-phononic waveguides supported by an array of nano-scale tethers (see slottedwaveguide sections). Panel (b) shows a cross-section of the Brillouin-active waveguide segment; this segment is suspended (seeMethods) in order to guide both sound and light. (c) plots the laser cross-section near the racetrack bends and the coupler,where the multimode waveguide is not suspended. The presence of the oxide undercladding has negligible impact on the guidanceof the optical modes. The silica substrate, however, does not permit phononic guidance around the bends. (d) Dimensions ofthe suspended Brillouin-active waveguide. (e) Strain profile εxx(x, y) of the 6 GHz Lamb-like acoustic mode that mediates theinter-modal Brillouin scattering process. (f) and (g) plot the x-directed electric field profiles (Ex) of the TE-like symmetric andantisymmetric optical modes, respectively.

tical modes (Figs. 2f-g), these Brillouin-active waveguidesections also support guidance of a 6 GHz phonon mode(Fig. 2e) which mediates efficient Brillouin coupling be-tween symmetric and antisymmetric optical modes. Thedesign of the Brillouin-active waveguide structure is identi-cal to that described in Ref. [35], which yields a peak inter-modal Brillouin gain coefficient of Gb

∼= 470 W−1m−1 at acenter frequency of 6.03 GHz with a resonance bandwidthof 13 MHz (full-width at half-maximum).

To ensure that our Brillouin laser has numerous opti-cal modes that satisfy the resonance frequency and phase-matching requirements, we use the racetrack cavity ge-ometry of Fig. 2a. This racetrack system has a circum-ference of 4.576 cm, yielding cavity free spectral ranges(FSRs) of 1.614 GHz and 1.570 GHz for the symmetricand antisymmetric modes, respectively. To maximize theround-trip Brillouin gain, each Brillouin-active region is2.223 cm long, making up more than 97% of the racetracklength. The bus waveguide evanescently couples to theracetrack system, providing access to both cavity modes;it behaves as a directional coupler which is designed to

couple strongly to the antisymmetric and weakly to thesymmetric mode.

By analyzing the multimode transmission spectra, wedetermine both the linear losses and the bus-ring couplingefficiencies; these parameters allow us to estimate intra-cavity powers and predict the laser characteristics. Trans-mission measurements of our racetrack resonator revealtotal Q-factors of 2.4×106 (4×105) for the symmetric (an-tisymmetric) mode. Through further analysis (see Supple-mentary Section 3.7), we identify an average intra-cavitypropagation loss of 0.26 dB cm−1 (0.69 dB cm−1) and thebus-ring coupling of µ2

1 = 0.06 (µ22 = 0.71) for the symmet-

ric (antisymmetric) mode. Here, µ2 is the fractional powercoupled between the bus and the racetrack waveguide (seeSupplementary Section 3.7 for more information). Usingthese linear loss measurements combined with an inde-pendently measured gain coefficient of 470 W−1m−1 [35],we estimate that input pump powers (intracavity pumppowers) as low as 9 mW (16 mW) should be adequateto reach threshold for laser oscillation of the symmetriccavity modes.

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FIG. 3: Experimental apparatus and laser threshold behavior. (a) Apparatus used for heterodyne spectroscopy. Continuous-wave pump light (Agilent 81600B, linewidth = 13 kHz) used to initiate Brillouin lasing is amplified using an erbium-doped fiberamplifier (EDFA) and subsequently coupled on-chip using grating couplers. A frequency-shifted (+44 MHz) local oscillator issynthesized on a separate reference arm using an acousto-optic modulator (AOM). Tuned to a cavity resonance, pump lightstrongly couples into the racetrack resonator and can initiate lasing if the Stokes frequency is also resonant with the laser cavity.A small fraction of this Stokes light is coupled to the bus waveguide and thereafter coupled off chip. The optical beat note betweenthe Stokes and local oscillator waves is detected by a fast photodiode, and its spectrum is recorded by a radio frequency spectrumanalyzer. (b) Heterodyne spectra of spontaneously scattered Stokes light from a linear waveguide (multiplied by 107) and thelinewidth-narrowed intracavity laser spectrum above threshold. The spontaneous Brillouin scattering spectrum from the linearwaveguide with the same cross section as the laser waveguide shown in Fig. 2d determines the intrinsic Brillouin gain bandwidth(13.1 MHz). Above threshold, as is the case here for the laser spectrum of panel (b), all measurements of the Stokes-pump beatnote power spectrum are resolution-bandwidth limited. Panel (c) plots theory and experiment for the output laser power vs.input pump power. These data clearly demonstrate laser threshold behavior. Intracavity pump powers are estimated using thetransmitted pump power and the detuning from resonance (ωn

2 ), while intracavity Stokes power is determined from the measuredbus Stokes power and comparison with the theoretical model (see Supplementary Section 4.1).

As discussed, Brillouin coupling can only be producedbetween symmetric (ωm

1 ) and antisymmetric (ωn2 ) cavity

modes if light traveling in these modes satisfies the energyconservation and phase-matching conditions required bythe inter-modal Brillouin process. In other words, efficientlaser operation requires coupling between cavity modesthat meet the Brillouin resonance condition ωn

2−ωm1 = Ωb,

within the uncertainty of the resonator linewidths (see Fig.1h). This condition was a crucial consideration of our race-track resonator design; because the FSRs of the two cavitymodes differ by 3.1%, this resonance frequency conditionis satisfied by symmetric and antisymmetric cavity modepairs that occur frequently (every 0.40 nm) across the C-band (from 1530-1565nm).

We investigate Brillouin lasing by injecting continuous-wave (cw) pump light (ωp) into an antisymmetric cavitymode (ωn

2 ) while analyzing the emission of Stokes lightfrom a symmetric cavity mode (ωm

1 ). These cavity modes

are chosen such that they satisfy the Brillouin resonancecondition (ωn

2 − ωm1 = Ωb), as illustrated in Fig. 1hii.

Experiments are conducted at room temperature us-ing the apparatus in Fig. 3a. The cw pump wave (ωp)is generated by a tunable external cavity laser (Agilent81600B) corresponding to a vacuum wavelength (λp) of1535 nm. This pump wave is amplified using an erbium-doped fiber amplifier (EDFA) before it is coupled on-chipusing a grating coupler. The pump wave resonantly buildswithin the antisymmetric racetrack mode when it is tunednear the cavity resonance (ωn

2 ). Under these conditions,Stokes light with frequency centered about ωs is emittedfrom the symmetric cavity mode into the bus waveguide.This Stokes light exits the chip through a second gratingcoupler for spectral analysis.

High-resolution spectral analysis of the emitted Stokeslight is performed using heterodyne detection (Fig. 3a).Using this technique, the spectrum of the light exiting

6

the bus waveguide is analyzed by combining these emittedlight fields with a reference laser-field (or optical local os-cillator); as these light-fields interfere on a photo-receiver,they produce unique microwave tones (or beat notes) thatpermit analysis of the emitted spectrum with kHz spectralresolution. As seen in Fig. 3a, the optical local oscillatoris synthesized by passing pump light through an acousto-optic modulator (AOM), which frequency shifts the pumplight from ωp → ωp + ∆, where (∆/2π) = 44 MHz. Theoptical beat frequencies produced by interference of theStokes and local oscillator waves is detected by a fast pho-todiode. The resulting electrical spectrum is recorded us-ing a radio frequency spectrum analyzer. Through carefulcalibration of the detector response and the optical localoscillator power, these microwave spectra provide an ac-curate measurement of the Stokes power and coherenceproperties of the optical spectrum.

Using this heterodyne detection method, we performpower and linewidth measurements of the output Stokeswave as a function of injected pump power. Figure 3cshows the emitted Stokes-wave power as the pump-wavepower is increased. These data reveal a threshold on-chippump power of 10.6 mW, corresponding with an intra-cavity power of 19 mW. This laser threshold shows goodagreement with our prior estimate and with the power re-quired for net amplification in a linear waveguide [35]. Fur-ther analysis of this efficiency curve reveals that this race-track Brillouin laser exhibits an on-chip slope efficiency of3% (see Supplementary Section 1.3.b).

As the pump power is increased, the linewidth of theemitted Stokes light also exhibits dramatic spectral com-pression that is characteristic of laser oscillation. Figure3b compares the Stokes spectrum emitted by the laser(red) with the spontaneous Stokes spectrum emitted froman identical Brillouin-active waveguide segment (gray) inthe absence of optical feedback. We see that optical feed-back produces a ∼ 103-fold spectral compression; the rel-atively broad spontaneous Stokes spectrum (FWHM ∼= 13MHz) is compressed to a resolution-limited value of 20kHz.

LASER DYNAMICS

From the observed threshold and the line-narrowing be-haviors, it is evident that this system produces opticalself-oscillation from stimulated Brillouin scattering. How-ever, we will see that this silicon Brillouin laser yields anunusual combination of spatial and temporal dynamics.In what follows, we first outline the salient features of thespatial dynamics of this and other Brillouin lasers. Wenext describe the most common form of linewidth narrow-ing in Brillouin lasers and then proceed to analyze thetemporal dynamics of our silicon Brillouin laser system.

We begin by highlighting the conditions for and basicspatial dynamics of Brillouin lasing. In Brillouin lasers,optical self-oscillation requires (1) optical feedback and(2) sufficient optical gain to compensate for round-tripoptical loss. In such systems, optical gain is supplied bystimulated Brillouin scattering—a phase-matched nonlin-

ear process that produces stimulated optical spatial gainfor the Stokes wave [41, 42]. Thus, with an optical cav-ity for the Stokes field, optical self-oscillation ensues whenthe Stokes spatial mode reaches a lossless state, a con-dition that occurs when round-trip optical gain balancesround-trip optical loss. While the phonon field plays animportant role in producing gain (i.e., mediating this op-tomechanical three-wave coupling), by comparison it isspatially heavily damped [41, 42] and does not experiencefeedback within a Brillouin laser; rather, its emission canbe viewed as a byproduct of laser oscillation. In this way,a Brillouin laser operates in direct analogy to a singly-resonant optical parametric oscillator (OPO) [41], wherethe Stokes field plays the role of the resonant signal waveand the acoustic field performs the function of the ballis-tic (i.e., cavity-less) idler wave. Thus, in general, Brillouinlasers require only an optical cavity; no phonon feedbackis necessary for laser oscillation. (for further discussion,see Supplementary Section 1.6.b-c)

Beyond the characteristic spatial dynamics, Brillouinlasers are often designed to produce optical line-narrowingof the Stokes wave. This behavior arises in systems wherethe optical temporal dissipation rates for the pump andStokes fields, γp and γs respectively, are much smallerthan the acoustic dissipation rate Γ [29], which we re-fer to as the standard dissipation hierarchy. Thus, thismost frequently-studied regime of dynamics results whenthe phonon field is more rapidly damped than the opti-cal fields in both space and time. Given this standarddissipation hierarchy, the laser dynamics produce Stokesemission that is spectrally much narrower than the inci-dent pump wave (Fig. 4aii). As a result, the emittedStokes wave can be viewed as a purified (and frequency-shifted) version of the incident pump field—even when theincident pump wave has a large amount of phase noise.When operating this system well above laser threshold, aheterodyne measurement between these pump and Stokesfields in a photo-receiver produces a noisy beat-signal (Fig.4aiii) with a microwave spectral width that mirrors thelinewidth of the incident pump field (Figure 4aiv). It isimportant to note that in the standard dissipation hier-archy, both the Stokes and phonon fields experience spec-tral compression through gain filtering. However, only theStokes field experiences Schawlow-Townes-like narrowing,and at high powers, the phonon linewidth saturates to avalue determined by the noise of the source laser (Fig. 4a).

This silicon Brillouin laser produces an unconventionalcombination of spatial and temporal regimes of dynamics.Consistent with the well-known spatial dynamics of stimu-lated Brillouin scattering [41], the phonon field is spatiallyheavily damped compared to the optical fields. In addi-tion to large acoustic propagation losses, the geometry ofthe laser cavity prevents any form of phonon feedback (seeSupplementary Section 1.6.d). Thus, as with other Bril-louin lasers, optical self-oscillation occurs as a result ofoptical Stokes feedback; the optical cavity in this systempermits optical self-oscillation of the Stokes wave to occurwhen round-trip optical gain balances round-trip opticalloss (see Supplementary Section 1.1). However, in contrastto the most widely-studied regime of temporal dynamics

7

(standard dissipation hierarchy), optical self-oscillation inthis system yields a distinct set of behaviors and spectralcharacteristics. These dynamics arise from an inverteddissipation hierarchy where the phonon temporal dissipa-tion rate is much smaller than the optical dissipation ratesfor the pump and Stokes cavity modes (i.e. γp, γs Γas illustrated in Fig. 4bi). This is possible because theBrillouin gain in this silicon system is exceptionally large(Gb

∼= 400W−1m−1). Thus, this Brillouin laser simulta-neously operates where the spatial acoustic decay length(63 µm) is far smaller than the optical decay length (∼0.1-1 m), while the intrinsic phonon lifetime (77 ns) exceedsthat of the optical fields (∼ 2−12 ns) — a condition madepossible by the disparate velocities of light and sound.

In this regime of operation, optical self-oscillation ofthe Stokes field produces an unconventional set of tem-poral dynamics and resulting spectral characteristics. Aslaser oscillation ensues, the spontaneously scattered Stokeswave is resonantly-enhanced and begins to self-oscillate,resulting in spectral compression of both Stokes andphonon fields. Thus, in this system, both Stokes andphonon fields experience a large degree of linewidth nar-rowing from the spontaneous Brillouin linewidth (com-pression factors of ∼ 103 and > 104, respectively). How-ever, due to the inverted dissipation hierarchy, only thephonon field experiences linewidth narrowing below thatof the pump field, and at high powers, the Stokes linewidthconverges to the the pump linewidth, becoming highly co-herent with the pump field. In this limit of operation, theemitted Stokes wave becomes a frequency-shifted copy ofthe pump wave—aside from a small degree of excess noiseimparted by the phonon field (Fig. 4bii). In this way,the system yields a combination of spatial and temporaldynamics similar to that produced in superradiant lasers[43], where the output laser light becomes a spectral copyof the incident pump field.

To elucidate our experimental observations and to pin-point the underlying dynamics, we derive simple analyt-ical models that describe the basic spatial and temporalbehavior of laser oscillation in this system (see Supple-mentary Section 1). Building on established treatments ofBrillouin laser physics [29, 44, 45], we develop a mean-fieldtreatment of our system to explore the temporal dynamics— a reduction that is adequate to predict the salient noiseand linewidth narrowing characteristics of this silicon Bril-louin laser. All of the parameters of our model are corrob-orated through independent measurements, such as trans-mission spectra and Brillouin scattering in linear waveg-uides, aside from a conversion factor relating the intracav-ity laser power to output laser power (see SupplementarySection 3.5,3.8).

Well above threshold, this model predicts that opticalself-oscillation produces Stokes emission that is highly co-herent with the incident pump field, with an excess phasenoise linewidth (∆νb) given by

∆νb =Γ

4πβ2(nthb + nths + 1). (1)

Here, Γ is the intrinsic acoustic dissipation rate, β2 is thecoherently-driven phonon occupation number, nthb is the

thermal occupation number of the phonon field, and nthsis average thermal occupation number of the symmetricmode of the optical resonator (with acoustic and opticalfrequencies of Ωb = 2π×6.02 GHz and ωs = 2π×195 THz,these correspond to room temperature values of nthb ≈ 103

and nths ≈ 0, respectively). As a result of the three-wavedynamics of this system, the pump-Stokes coherence pro-vides a direct window into the spectrum of the distributedacoustic field (see Supplementary Section 1.3), revealingthat Brillouin lasing in this regime produces linewidth nar-rowing of the coherent acoustic field.

These phonon linewidth narrowing dynamics are rem-iniscent of those produced in optomechanical self-oscillation (sometimes called phonon lasing) [46–49]. How-ever, in contrast to phonon lasers, this Brillouin laser doesnot possess a phonon cavity that permits acoustic feed-back for phonon self-oscillation —an essential characteris-tic of phonon lasing (see Supplementary Section 1.6.b-d).Rather, this system is analogous to an extreme limit ofsingly-resonant OPO physics, with a slow, ballistic, long-lived idler wave (see Supplementary Section 1.6.e).

Related regimes of temporal dynamics have previouslybeen considered and studied in the context of cavity-optomechanical systems, which possess both acoustic andoptical feedback [47, 50]. This work extends these con-cepts to macroscopic Brillouin laser systems, in which theacoustic field is spatially heavily damped and does not ex-perience feedback within the system. Here we show thatdespite large acoustic spatial damping and lack of phononfeedback, optical self-oscillation of the Stokes wave (Bril-louin lasing) can still produce linewidth narrowing of theacoustic field, as long as the temporal acoustic dissipationrate is smaller than that of the optical fields. Interest-ingly, it is likely that other Brillouin laser systems havealso operated near or in this temporal dissipation hierar-chy [27, 51, 52]. However, these studies did not report acharacterization or analysis of the phonon linewidth nar-rowing dynamics. Thanks to the stability of this mono-lithic silicon system, we are able to study this combinationof temporal and spatial dynamics for the first time.

While closed-form analytical expressions for phase noiseare tractable well below and above threshold, stochasticnumerical simulations are necessary to model the noisecharacteristics in the vicinity of laser threshold (verticalgray bar of Fig. 4e). The comparison between simulationsand experiment (Fig. 4e) show good qualitative agreementbetween the predicted and measured linewidths.

The experimental data in Fig. 4e were obtained usingtwo complementary measurement methods that allow usto determine the phonon coherence at a range of laserpowers (see Fig. 4c-d). Above threshold, small variationsin pump frequency translate to frequency instabilities thatare not intrinsic to this system. Using the apparatus ofFig. 3a, the heterodyne spectrum must be acquired on a30 millisecond timescale to avoid distortions produced byslow frequency hops generated by the pump laser. Theseshorter acquisition times effectively limit the resolutionbandwidth of this measurement approach to 20 kHz.

At higher emitted powers, we are able to overcome thisresolution limitation by using a separate sub-coherence de-

8

c

a

b

e

i ii iii iv Γ >> γ

Γ

γ

Frequency

Brillouin resonanceOptical cavity resonance

Resp

onse

ωs ωp

Δνp

Δνs<< Δνp

Δνs

Frequency

1st order Stokes spectrum

Pump spectrum

Spec

tral

den

sity

Ωb

Δνb ~ Δνp

Frequency

Δνb

Beat note spectrum

Spec

tral

den

sity

Photodiode

Beat note

Ωb = ωp- ωs

1st Stokes

Pump

Time Time

i ii iii iv Γ << γ

Γ

γ

Brillouin resonanceOptical cavity resonance

Frequency

Resp

onse

ωs ωp

Δνp

Δνs ~ Δνp

1st order Stokes spectrum

Pump spectrum

Δνs

FrequencySp

ectr

al d

ensi

tyΩb

Δνb<< Δνp

Beat note spectrum

Frequency

Spec

tral

den

sity

Δνb

Beat note

Ωb = ωp- ωs

1st Stokes

Pump

Time Time

Photodiode

Self-heterodyneStochastic simulation

Laser Threshold

Resolution bandwidth

Beat note linewidth

Simulation trend

Standard Dissipation Hierarchy

Inverted Dissipation Hierarchy

Pump

AOM

Silicon BrillouinLaser

d

Pump

AOM

Silicon BrillouinLaser

Heterodyne Spectrum

τd

Δνb

Heterodyne Spectrum

Δνb

Delay line

10- 15 10- 13 10- 11 10- 9 10- 7

1

10

100

1000

Peak Spectral Density (W/Hz)

Δν b

(kH

z)

FIG. 4: Noise characteristics of the silicon Brillouin laser. (a) Didactic diagram illustrating the noise properties of a Brillouin laseroperating above threshold in the standard dissipation hierarchy where (i) the Brillouin gain bandwidth (or phononic decay rate,Γ) is much larger than the cavity linewidth (γ) of the optical resonator. In this limit, Brillouin lasing produces optical linewidthnarrowing of the Stokes wave, resulting in the optical spectrum plotted in (ii), where the Stokes wave linewidth is narrow comparedto that of the pump wave. (iii) diagrams a heterodyne measurement of the optical beat frequency between these two waves, whichinherits the noise of the pump laser. This results in a noisy beat note with a linewidth approximately equal to the pump wavelinewidth (iv). (b) Diagram showing the noise properties of this silicon Brillouin laser operating above threshold. (i) In this regimethe dissipation hierarchy is inverted (Γ << γ). As a result, Brillouin lasing produces phonon linewidth narrowing, and the Stokesbecomes a frequency-shifted copy of the pump, giving the optical spectrum plotted in (ii). As a result, heterodyne detectionproduces a clean beat note (iii) that directly reflects the phonon power spectrum (iv). (c) Standard heterodyne spectroscopyapparatus to measure the phonon linewidth, the linewidth given by the FWHM of the heterodyne spectrum. (d) Subcoherenceself-heterodyne apparatus used to probe the phonon dynamics at higher output Stokes powers (see Supplementary Section 2).The phonon linewidth is determined by measuring the fringe contrast or coherence between output Stokes and pump waves,yielding the blue data points in panel (e). (e) Experimental and theoretical comparison of pump-Stokes beat note (or phonon)linewidth as a function of peak Stokes spectral density. Below threshold we use the standard heterodyne spectroscopy (dark reddata points). At higher powers, this measurement becomes resolution bandwidth limited (see Supplementary Section 4.2). Forthis reason, we use the sub-coherence self-heterodyne technique which yields the blue data points (error bars represent the 95%confidence interval). Using this method, we measure a minimum phonon linewidth of ∼ 800 Hz (See Supplementary Section 4.2).

layed self-heterodyne measurement technique (Fig. 4d).This method allows us to determine the intrinsic laserlinewidth (and resulting phonon dynamics) despite theslow center-frequency hops induced by the pump laser[53, 54]. This method uses a delayed interferometer tomeasure the coherence between the pump and Stokeswaves (see Supplementary Section 2), yielding the bluedata points seen in Fig. 4e. These measurements demon-strate (1) the pump-Stokes phase coherence unique to thislimit of Brillouin dynamics and (2) phonon linewidth nar-rowing far below the linewidth of the pump laser (∼ 13kHz), in good agreement with the theoretical trend (seeSupplementary Section 2.2.a). Using this technique, we

observe a minimum phonon linewidth of ∼ 800 Hz (com-pression factor of ∼ 104 relative to the spontaneous Bril-louin linewidth).

DISCUSSION

In this paper, we have harnessed highly engineerableBrillouin nonlinearities to create a silicon Brillouin laser.These results open the door to a new class of silicon lasers,whose distinct and tailorable dynamics can be leveragedwithin silicon photonic circuits for a range of sensing andcommunications applications. Independent of its poten-

9

tial in silicon photonics, this versatile Brillouin laser con-cept could be adapted to any number of material systemsfor robust and scalable monolithic integration of Brillouinlasers.

As we contemplate these possibilities, it is importantto note that this Brillouin laser design —made possibleby the forward inter-modal scattering process— is quitedifferent from the geometries of previous Brillouin lasersystems. These fundamental distinctions in device con-cept alleviate a number of challenges facing on-chip inte-gration of Brillouin lasers. Since conventional Brillouinlasers are based on backward SBS, they invariably pro-duce Stokes laser emission in the backward direction. Inmany cases, this poses a significant challenge for integra-tion; such Brillouin lasers would likely require on-chip cir-culators and isolators to protect the pump laser from un-wanted feedback. However, the absence of mature isolatorand circulator technology remains a serious obstacle facingthe field of integrated photonics. By contrast, this siliconBrillouin laser uses a forward Brillouin scattering process,which produces Stokes emission in the forward direction.In this way, the inter-modal forward Brillouin laser geome-try avoids the need for isolators and circulators. Moreover,since the forward Brillouin scattering process is created us-ing structural control, this device concept is quite generaland in principle, can be translated to a number of materialsystems.

Beyond its suitability for integration, this inter-modalBrillouin laser yields flexibility in design and control ofdynamics in ways that are difficult to achieve in Brillouinlasers that use backward stimulated Brillouin scattering.In particular, the multimode nature of the inter-modalBrillouin process permits independent tuning of the dissi-pation rates and cavity enhancements for the two spatialmodes, allowing us to precisely shape the energy transferbetween pump and Stokes waves. This is different fromconventional Brillouin lasers, where the pump and Stokeslight propagate in the same spatial mode and have nearlyidentical dissipation rates. If, through Brillouin laser os-cillation, the Stokes wave becomes strong enough in con-ventional systems, it can reach the lasing threshold to pro-duce cascaded energy transfer to successive Stokes orders.In contrast, this inter-modal system allows us to preciselycontrol the degree to which cascading can or cannot oc-cur. For example, by tailoring the Q-factor of the pumpwave, we can precisely engineer the Stokes-power thresh-old required for cascaded energy transfer. Alternatively,we can completely suppress cascading by engineering themode spectrum such that the 2nd Stokes order wave is notsupported by a cavity mode. This is the strategy used inour current system to suppress cascading.

In addition, the tailorability afforded by the inter-modalBrillouin scattering process allows us to optimize this laserdesign according to any number of constraints. As wehave discussed, this silicon laser has an estimated on-chipslope efficiency of 3%. However, significant improvementscan be achieved, for example, by developing independentlytunable mode-specific couplers; our models indicate thatby increasing the Stokes output-coupling to 16% we canachieve a 10-fold improvement in slope efficiency (see Sup-

plementary Section 5.2). Further performance enhance-ments may also be achieved by reducing the optical waveg-uide losses and increasing the Brillouin gain using alterna-tive waveguide designs. Beyond modifying the waveguideproperties, the multi-mode nature of this inter-modal Bril-louin interaction permits great flexibility in the design ofthe laser cavity. This is because this inter-modal Brillouinlaser topology permits phase-matching to be satisfied overa tremendous range of resonator dimensions. By reduc-ing the footprint and increasing the cavity finesse, we cansignificantly increase the resonant pump-power enhance-ment, thereby dramatically lowering the laser threshold.For example, a 100× size reduction could permit resonantpower enhancements of ∼ 180, pushing the laser thresholdbelow 100 µW.

We have shown that due to the combination of spatialand temporal dynamics, this laser exhibits optical self-oscillation with distinct spectral characteristics. In thissystem, laser oscillation produces coherent unidirectionalfrequency shifting of the incident pump field, behavingmuch like a self-driven acousto-optic frequency shifter oroptical phase locked loop [55]. These laser properties couldbe harnessed as the basis for new photonic technologies, asthis form of self-oscillation may be useful for signal demod-ulation, sensing, or as an oscillator. At the same time, thehighly coherent distributed acoustic field produced by thissystem could become a valuable resource that may enableon-chip phononic applications ranging from acoustic inter-ferometers to coherent information processing [56, 57].

Beyond the regimes of dynamics examined here, sys-tems of this type can be used to access a wide variety oflaser dynamics by tailoring both the optical modes as wellas the strength and character of the Brillouin interaction.In contrast with Raman and Kerr nonlinearities, whoseproperties are principally governed by the waveguide ma-terial, Brillouin nonlinearities are exceptionally tailorablethrough device geometry. In fact, it is possible to engineera range of essential characteristics of this Brillouin nonlin-earity, including resonance frequency, acoustic dissipationrate, and Brillouin gain [31, 32]. Hence, by modifying theBrillouin gain mechanism in concert with the optical prop-erties the laser cavity, an array of laser dynamics can beexplored.

For example, complementary laser dynamics can beachieved if the optical and Brillouin interactions are en-gineered to conform to the standard dissipation hierarchy(γs, γp Γ). This more widely studied regime of dy-namics has proven invaluable for applications includinglow noise microwave sources [25, 26] and precision opticalgyroscopes [23, 24, 58]. It is possible to reach this moretraditional regime of oscillation by simultaneously increas-ing the optical Q-factors and the acoustic dissipation rate(Γ). Prior works have shown that optical Q-factors ex-ceeding 20 million are possible in silicon [59], and phononlifetimes are readily reduced by designing Brillouin-activewaveguides which more quickly radiate acoustic energy.Furthermore, for improved oscillator performance, someapplications will also require higher intra-cavity powers.In this case, it could prove advantageous to operate atlonger wavelengths (> 2.2 µm), where two-photon absorp-

10

tion becomes negligible in silicon. Thus, building on thiswork, a rich landscape of Brillouin laser technologies canbe explored in silicon photonics.

In conclusion, we have demonstrated a Brillouin laserin silicon by leveraging a stimulated inter-modal Brillouinscattering process. This flexible Brillouin laser conceptyields size and frequency scalability, precise control of en-ergy transfer dynamics, and laser emission in the forwarddirection, representing an important step towards integra-tion of Brillouin lasers in complex silicon photonic cir-cuits. Additionally, this system permits access to an un-usual limit of Brillouin laser physics with distinct tempo-ral dynamics and spectral characteristics. In this regime,Brillouin laser oscillation produces Stokes emission thatbecomes exceptionally coherent with the incident pumpfield, reflecting the spectral compression of a distributed,spatially heavily damped acoustic field. Beyond this study,these dynamics may enable a range of on-chip sensingand communications applications for monolithic integra-tion within silicon photonic circuits.

METHODS

Experimental Parameters

Cavity Parameters ValueL 4.576 cm

FSR (symmetric) 1.614 GHzFSR (anti-symmetric) 1.570 GHz

Optical Parameters Valueγ1 2π 83 MHzγ2 2π 481 MHzvg,1 7.385 ×107 m/svg,2 7.163 ×107 m/sGb 400 W−1m−1

Ithreshold 5.4× 106 Wcm−2

Pump linewidth 13 kHzAcoustic Parameters Value

Ωb 2π 6.02 GHzΓ0 2π 13.1 MHzq 4.5× 105 m−1

vb,group 826 m/svb,phase 8.4× 104 m/s

Device fabrication

The Brillouin laser devices were fabricated on a silicon-on-insulator chip with a 3 µm oxide layer beneath a215 nm-thick crystalline silicon layer. The waveguideswere patterned using electron beam lithography on hy-drogen silsesquioxane photoresist. After development inMicropositTM MFTM-312 developer, a Cl2 reactive ionetch (RIE) was used to create the ridge structures. Fol-lowing a solvent cleaning step, slot structures were writtenwith electron beam lithography on ZEP520A photoresistand Cl2 RIE. The device was then wet released via a 49%hydrofluoric acid etch of the oxide layer. The laser struc-ture reported here consists of two regions of 436 suspended

segments each and has a total length of 4.576 cm.

Acknowledgements

We would like to thank Professor Douglas Stone for en-lightening discussions surrounding the nature of laser os-cillation in this system.

This work was supported through a seedling grant underthe direction of Dr. Daniel Green at DARPA MTO andby the Packard Fellowship for Science and Engineering;N.T.O. acknowledges support from the National ScienceFoundation Graduate Research Fellowship under GrantNo. DGE1122492.

Author contributions

P.T.R., N.T.O., and E.A.K. developed inter-modal laserdevice concept and design. N.T.O and E.A.K. fabri-cated the laser devices with guidance from P.T.R. N.T.O.and E.A.K. conducted experiments with the assistance ofP.T.R. and R.B.O. R.B.O. and N.T.O. developed analyt-ical and numerical tools to model laser dynamics with in-put from P.T.R. N.T.O. and R.B.O. developed delayedsub-coherence heterodyne technique for phonon linewidthdetection with assistance from E.A.K. and P.T.R. Z.W.contributed to the understanding of phonon-linewidth nar-rowing in the inverted dissipation hierarchy. All authorsparticipated in the preparation of this manuscript.

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Supplementary Information

Nils T. Otterstrom,1 Ryan O. Behunin,1 Eric A. Kittlaus,1 Zheng Wang,2 and Peter T. Rakich1

1Department of Applied Physics, Yale University, New Haven, CT 06520 USA.2Department of Electrical and Computer Engineering,

University of Texas at Austin, Austin, TX 78758 USA.(Dated: October 29, 2017)

Contents

1. Model of inter-modal Brillouin laser dynamics 21.1 Spatial dynamics and origin of laser oscillation 21.2 Mean-field analysis 31.2.a Validity of mean-field analysis 5

1.3 Inter-modal Brillouin laser noise theory 51.3.a Threshold behavior, lasing, and steady-state phonon amplitude 61.3.b Slope eciency 7

1.4 Phonon power spectrum at high pump powers 71.4.a Laser equations 71.4.b Langevin force 71.4.c Power spectra 8

1.5 Simulations of laser dynamics 81.5.a Model parameters used in laser simulations 9

1.6 Optical parametric oscillators, Brillouin lasing, and phonon lasing 101.6.a Defining characteristics for lasers 101.6.b Singly-resonant OPO physics 111.6.c Brillouin lasing and phonon lasing 111.6.d Properties of the silicon Brillouin laser 111.6.e Optical self-oscillation and phonon linewidth narrowing 12

2. Sub-coherence self-heterodyne measurement 132.1 Self-heterodyne interferometry 142.2 Sub-coherence self-heterodyne measurement using a silicon Brillouin laser 142.2.a Inverted dissipation hierarchy 142.2.b Standard dissipation hierarchy 16

3. Six-port coupler theory and analysis 173.1 Six-port coupler 183.2 Free optical propagation 183.3 Optical response of a multimode ring resonator 183.4 Relation between intracavity pump power and transmitted pump power 193.5 Relation between intracavity laser power to output laser power 193.6 A minimal model of the six-port coupler scattering matrix 203.7 Inter-modal ring resonator characterization 203.8 Laser power conversion factor 22

4. Data analysis 224.1 Slope eciency 224.2 Linewidth measurements 224.2.a Heterodyne measurements 234.2.b Sub-coherence self-heterodyne measurements 23

5. Additional measurements and considerations 245.1 Anti-Stokes suppression 245.2 Output coupling and slope eciency 245.3 Additional sub-coherence self-heterodyne measurements with open-loop laser 24

References 25

2

1. MODEL OF INTER-MODAL BRILLOUIN LASER DYNAMICS

In this section, we present a simple model that describes the salient features of the dynamics of our inter-modalBrillouin laser. In what follows, we first explore the spatial dynamics of this Brillouin laser. This approach allowsus to identify the origin of laser oscillation in this system and derive the threshold condition. We also show that thespatial behavior of this system is identical to conventional Brillouin lasers. After analyzing the spatial dynamics, wemake a series of approximations to develop a mean-field model of the laser physics — a reduction that is adequate tocapture the essential aspects of the temporal dynamics. After discussing the validity of this model, we corroborate thelaser threshold and then derive the slope eciency and noise properties. In particular, we show that in contrast to thetemporal dynamics of conventional Brillouin lasers, optical self-oscillation in this system causes the phononic degrees offreedom to undergo strong spectral compression.

1.1 Spatial dynamics and origin of laser oscillation

Stimulated Brillouin scattering is a three-wave interaction that produces spatial stimulated optical gain. Thesecharacteristic Brillouin dynamics arise because the spatial decay rate of the phonon field is much larger than the spatialoptical decay rates. In this section, we show that this silicon Brillouin laser exhibits spatial dynamics which are consistentwith other Brillouin lasers. Due to the large acoustic spatial decay rate, the phonon field can be eliminated from thespatial equations of motion, resulting in exponential spatial amplification for the Stokes field while the phonon fieldstrength is given by the local optical beat note between pump and Stokes fields [1, 2]. We also derive the thresholdcondition and show that laser oscillation occurs when round-trip optical Brillouin gain balances round-trip optical lossof the Stokes wave.

We begin our analysis from the envelope equations of motion [3, 4] given by

Ap + i!pAp + vg,pA0p +

1

2vg,p↵pAp = igAsB (1)

As + i!sAs + vg,sA0s +

1

2vg,s↵sAs = igApB† (2)

B + (ib +1

2)B vg,bB0 = igApA†

s . (3)

To simplify our analysis of the threshold condition, we move to the rotating frame, assume an undepleted pump, andsolve for the steady-state spatial dynamics. The resulting coupled slowly-varying envelope equations are

vg,sA0s = igApB† 1

2vg,s↵sAs (4)

vg,bB0 = igApA†s

1

2B. (5)

From these governing equations, the formal solution of B(z) is

B(z) =igAp

vg,b

Z L

z

dz0A†s(z

0)e(zz0)2vg,b . (6)

This expression greatly simplifies by recognizing that the Stokes wave varies very slowly compared to the spatial decaylength of the phonon field. This fact allows us to pull the Stokes envelope out of the integral, resulting in a simplespatial solution of the phonon field:

B(z) =2ig

ApA†

s(z)(1 e(Lz)

2vg,b ). (7)

For (L z) vg,b

(which is very well-satisfied in our device), this expression reduces to

B(z) =2ig

ApA†

s(z). (8)

From Eq. 8, we see that the phonon field strength is given by the local pump-Stokes beat note. This aspect of thespatial dynamics is common to practically all Brillouin systems and thoroughly discussed in standard nonlinear optics

3

textbooks (see [1, 2]). We can now insert our solution for B(z) into the equation of motion for the Stokes field, whichbecomes

A0s =

2|g|2|Ap|2vg,s

As ↵s

2As. (9)

Substituting the known expression for Gb [4] and solving the ODE, we find the following solution for the Stokes field:

As = C1e(

GbPP2 ↵s

2 )z. (10)

We next apply the boundary conditions to determine the value of C1. The Stokes cavity is formed by fashioning amulti-mode waveguide into a racetrack ring configuration with an input/output coupler. Consistent with the physicalparameters of the device under study, here we apply no acoustic feedback (Section 1.6.d). The boundary condition cantherefore can be expressed as

As(0) = iµAs,in + rAs(L). (11)

Here, |µ|2 is the coupling into/out of the ring and r parameterizes the fraction of light that feeds back into the ring

after the coupler (r =p

1 µ2). In our laser, there is no input Stokes light, and thus in this simple analysis, As,in servesas a proxy for the noise that initiates laser oscillation. Applying this boundary condition, we find that the solution forthe Stokes field envelope is given by

As(z) =iµAs,in

1 re(GbPP↵s)L

2

e(GbPP↵s)z

2 . (12)

This is the main result of this section. Here we wish to make a couple of important observations. From Eq. 12, wesee that the field diverges when GbPP = ↵s + 2

L log 1r . In other words, the Stokes field experiences a pole on the real

axis (a lossless state) when round-trip Brillouin gain matches the optical round-trip loss; a condition which pushes thesystem into optical self-oscillation. This is the threshold condition for Brillouin lasing [2].

From Eq. 8, we note that above threshold, optical self-oscillation also dramatically increases the phonon field strength.Since the phonon field is spatially heavily damped, the spatial dynamics are determined by the local optical driving force,which radically increases in magnitude above threshold. Thus, because the phonon field mediates coupling between thepump and Stokes waves, phonon emission can be viewed as a byproduct of Stokes laser-oscillation (much like the idler ina singly-resonant OPO, see Section 1.6). Thus, in general, Brillouin lasers require only an optical cavity for the Stokesfield; no phonon feedback is necessary for laser oscillation to occur.

In subsequent sections, we will see that even though the phonon field is not responsible for the feedback and oscillation,it plays a crucial role in the linewidth narrowing dynamics of this silicon Brillouin laser simply because it is the carrierwith the longest temporal memory.

1.2 Mean-field analysis

We next develop a simple mean-field [5] model that is adequate to capture the basic temporal and noise dynamics ofthis laser. This model can be derived by again starting from the slowly-varying envelope dynamics of these fields [3, 4]given by

Ap + i!pAp + vg,pA0p +

1

2vg,p↵pAp = igAsB +

1pLp(z, t) (13)

As + i!sAs + vg,sA0s +

1

2vg,s↵sAs = igApB† +

1pLs(z, t) (14)

B + (ib +1

2)B + vg,bB0 = igApA†

s +1pL(z, t). (15)

Here, the coupling g in the envelope picture is related to g as g = g/p

L, vg,p (vg,s) is the group velocity of the pump(Stokes) field, vg,b is the group velocity of the phonon field, and ↵p (↵s) is the spatial decay rate of the pump (Stokes)field in the inter-modal waveguide. Above, we have added an explicit spatial dependence for the Langevin forces, whichare assumed to be locally correlated in space, e.g. h†(t, z)(t0, z0)i = Lnth

b (t t0)(z z0) and similarly for the othernoise correlation functions.

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The dynamics of the mean optical and acoustic fields can be obtained by averaging each envelope over the length ofthe ring L. For example, the mean pump field is given by

Ap 1

L

Z L

0

dz Ap. (16)

Applying this averaging procedure to the envelope equation for the pump we find

˙Ap + i!pAp +vg,p

L(Ap(L) Ap(0)) +

1

2vg,p↵pAp = igAsB +

1pLp. (17)

This equation can be further simplified by assuming that the envelopes change very little over a round trip, and that

the pump light in the ring obeys the boundary condition Ap(0) = rpAp(L) iµpAext where rp and µp =q

1 r2p are

the reflection and transmission coecients for the pump in the ring resonator. In addition, the phase accumulated bythe fields after a round trip will select a set of frequencies which are resonant for the ring. In the following, we assumethat this resonance condition is satisfied. In this limit we find

AsB AsB (18)

Ap(L) Ap(0) iµpAext + (1 rp)Ap(L). (19)

Assuming that µp 1 we find

Ap(L) Ap(0) iµpAext ln rpAp, (20)

where we have assumed Ap(L) Ap and we have approximated rp 1 as ln rp. We should mention that we representrp 1 as ln rp in order to make a clear connection between the e↵ective optical decay rate in the mean field approachand the loaded decay rate in ring resonators. By carrying out a similar treatment for the Stokes and the phonon fieldswe find the mean-field equations of motion given by

˙Ap +1

2vg,p(↵p 2

Lln rp)Ap = igAsB i

vg,p

LµpAext +

1pLp (21)

˙As +1

2vg,s(↵s

2

Lln rs)As = igApB† +

1pLs (22)

˙B + (ib +1

2)B = igApA†

s +1pL. (23)

These equations can be written in a form that is reminiscent of normal mode Langevin equations through two steps.First note

h†(t)(t0)i =1

L2

Z L

0

dz

Z L

0

dz0 h†(t, z)(t0, z0)i (24)

=1

L

Z L

0

dz

Z L

0

dz0 nthb (t t0)(z z0) (25)

=n(t t0), (26)

which has the same correlation properties as the Langevin forces of normal modes.For simplicity of notation, we define ap

pLAp (and similarly for the other fields). We also note that the loaded

decay rates of the optical modes are given by p = vg,p(↵p 2L ln rp) and s = vg,s(↵s 2

L ln rs). These notationalsimplifications lead to the coupled mean-field equations of motion, upon which we base our model of the temporaldynamics of laser oscillation. These equations are given by

Mean-field equations of motion =

8><>:

ap = (i!p + p/2) ap igasb + paextp /2 + p

as = (i!s + s/2) as igapb† + s

b = (ib + /2) b igapa†s + .

(27)

Here, the field aextp is an external drive sourcing the pump field. In the absence of Brillouin coupling, this external

drive determines the steady-state intracavity pump power. In addition, the Langevin forces p, s, and that accountfor thermal and quantum fluctuations of each of the fields. We assume that these forces are zero-mean Gaussian randomvariables with white power spectra, yielding correlation properties given by

†p(t)p(t0)

↵= pnth

p (t t0) (28)†s (t)s(t

0)↵

= snths (t t0) (29)

†(t)(t0)

↵= nth

b (t t0). (30)

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The magnitude of these power spectra are set by the thermal occupation numbers nthp , nth

s , and nthb (given by the Bose

distribution).

1.2.a Validity of mean-field analysis

A few comments are in order about the implications of this approximation. This analysis shows that the validityof our model requires that the optical field envelopes change slowly over the length of the ring resonator, which iswell-satisfied for the Stokes field in our system which has decay length bigger than the size of the resonator. However,this approximation is only marginally satisfied for the pump field which has a decay length just bigger than the ringcircumference.

The validity of the mean field treatment of the phonon field is more nuanced. The decay length for the phonon fieldis 63 µm. Since many decay lengths fit within the racetrack (4.6 cm) it may seem more natural to model the phonons asa collection of oscillators, made up of the di↵erent segments of the resonator. Moreover, the real resonator is composedof two distinct segments that support phononic guidance. However, by examining the phonon dynamics from the pointof view of a driven field, we notice that if the optical fields change slowly over the length of the resonator so does thephotoelastic driving force. From Eq. 8, we see that above threshold the steady-state phonon field strength is given bythe local Stokes-pump beat note, which varies slowly around the device. Hence, under this approximation, we modelthe response of the elastic field as an average field with distributed coupling.

1.3 Inter-modal Brillouin laser noise theory

Having established our laser model, we now calculate the steady-state laser power, theshhold, slope eciency, andpower spectrum of our inter-modal Brillouin laser.

We model the laser physics by using the coupled equations of motion for the mean-field amplitudes of the pump, Stokes,and phonon fields given in Eq. 27. In contrast with conventional Brillouin lasers, the temporal optical dissipation ratesare much greater than the acoustic dissipation rates in our system, and therefore the optical fields can be adiabaticallyeliminated.

When the pump decay rate is large compared to the decay rates for the Stokes and the phonon fields, as well as thedecoherence rate of the external laser, the pump field can be approximated as

ap igasb

i(b ) + p/2+ aext

p igp()asb + aextp , (31)

where = !p !s and p() is the pump susceptibility due to stimulated Brillouin scattering. This solution for thepump can be plugged into the remaining equations for the Stokes and the phonon fields yielding

as = (i!s + s/2) as |g|2p()|b|2as + s igaextp b† (32)

b = (ib + /2) b |g|2p()|as|2b + igaextp a†

s . (33)

Next, we formally solve the equation for the Stokes field

as(t) =

Z t

1d e[(i!s+

12s)(t)+|g|2p()

R t

dt0 |b(t0)|2]s() igaext

p ()b†()

(34)

= as(t) igZ t

1d e[(i!s+

12s)(t)+|g|2p()

R t

dt0 |b(t0)|2] aextp ()b†(), (35)

where as represents the “dressed” annihilation operator of the Stokes field given explicitly by

as(t) =

Z t

1d e[(i!s+

12s)(t)+|g|2p()

R t

dt0 |b(t0)|2] s(). (36)

The correlation properties of as are are implicitly given in Eq. 46. We assume that the pump phase noise and phononamplitude vary slowly in time compared to the decay rate of the Stokes field. This separation of time scales allowsR t

dt0 |b(t0)|2 |b(t)|2(t ) in the exponent and the amplitudes of the external pump and the phonon to be taken

outside of the integral (i.e. by adopting the Markov approximation where b() b(t)eib(t) under the integral). Theseapproximations give

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as(t) as(t) igaextp (t)b†(t)

Z t

1d e[i(!s!p+b)+ 1

2s+|g|2p()|b(t)|2](t) (37)

as(t) igaext

p (t)b†(t)

i(b ) + 12s + |g|2p()|b(t)|2 as(t) igs()aext

p (t)b†(t). (38)

Having derived the dynamics of the Stokes field for our system let us momentarily pause to point out some criticalphysics for our measurements. The heterodyne technique employed in our experiment directly measures the beat noteof the Stokes field and the external laser, which acts as a local oscillator. Ignoring the intrinsic fluctuations of the Stokesfield this beat note takes the form

aext(t)a†s(t) = ig

s ()|aextp (t)|2b(t). (39)

There are few important points to make here. First, our external source laser is phase-noise dominated. Therefore, thebeat note (a) is una↵ected by the external laser noise (i.e. the beat note depends on the modulus square of the aext),and (b) the beat note provides a direct window on the phonon dynamics. In the following we will discuss the dynamicsof the phonon field.

By plugging the approximate solution for the Stokes field into Eq. 33 we obtain the e↵ective nonlinear dynamics ofthe phonon field given by

b = (ib + /2) b + |g|2|s()|2(i(b ) + s/2)|aextp |2b + igaext

p a†s , (40)

where we have used the definition of s. In the following we consider the resonant case ( = b, p(b) = 2/p) wherethe equation of motion for the phonon drastically simplifies to

b = (ib + /2) b +s

2

|g|2|aextp |2

( 12s + 2

p|g|2|b|2)2 b + , (41)

where igaextp a†

s is the e↵ective Langevin forcing for the phonon, including pump noise and quantum fluctuationsof the Stokes field.

1.3.a Threshold behavior, lasing, and steady-state phonon amplitude

These nonlinear dynamics capture the basic the lasing behavior and noise properties of the 3-wave system. For largepump powers, when

2

s|g|2|aext

p |2 > /2, (42)

the mean-field phonon amplitude exhibits a dramatic increase in strength. From this analysis, we see that the thresholdpump amplitude predicted by mean-field is

|aextp,th|2 s

4|g|2 , (43)

which, by converting to optical power P = h!vg|a|2/L, gives the threshold power

Pth h!g,pvg,ps

4|g|2L =s

Gbvg,s=

1

Gb(↵s

2

Lln(rs)). (44)

From Eq. 44, we see that the mean-field prediction is consistent with the threshold pump power required for the round-trip optical SBS gain for the Stokes field to match the round-trip optical loss (See Eq. 12). We note here that becausethe phonon field mediates coupling between pump and Stokes wave, the coherent growth of the mean phonon field issimply a byproduct of the optical self-oscillation of the Stokes field. However, we will see in Section 1.4.b that due tothe temporal dissipation hierarchy, the phonon field plays a dominant role in the linewidth narrowing dynamics.

The magnitude of the coherent amplitude of the phonon field above threshold can be obtained from the time average

of Eq. 41. Assuming that b = eibt, where (q

nthb ) is a time independent amplitude, multiplication of Eq. 41

by eibt and taking a long-time average, where the Langevin force averages to zero, we find

0 = /2 s

2

|g|2|aextp |2

( 12s + 2

p|g|2||2)2

, (45)

which relates the coherent phonon amplitude above threshold to the source power.

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1.3.b Slope eciency

The steady-state phonon amplitude, above threshold, also allows us to estimate the slope eciency for lasing on theStokes mode. Assuming that the pump power is just above threshold we can assume < 1. In this limit, the steadystate amplitude equation Eq. 45 gives the relation Pext Pth 8|g|2Pth

2/(sp). Plugging this relationship into theequation for the Stokes field, and converting amplitudes to power, we find the following relationship between the Stokeslaser power Ps and the external source laser power Pext:

Ps p!svg,s

2s!pvg,p(Pext Pth). (46)

Using the decay rates, frequencies, and group velocities of the optical modes obtained from transmission spectra in Sec.3, Eq. 46 predicts an intracavity slope eciency of 3. By using the pump build up factor of 1.8, and the estimatedring-bus power coupling of 0.075 %, this analytical expression gives an on-chip slope eciency estimate of 4 %, closeto the measured slope eciency of 3% quoted in the main text.

In the following we consider the noise properties of the phonon field. These noise properties, described quantitativelyby power spectra, depend sensitively on the nonlinear dynamics of the system and the source laser power. For thisreason, we obtain the phonon power spectra near threshold numerically through stochastic simulations of the laserdynamics described by Eq. 27, and only at high source powers, well above threshold, will we present analytical resultsfor the power spectra. We will show that our inter-modal laser exhibits power narrowing of the phonon linewidth.

1.4 Phonon power spectrum at high pump powers

1.4.a Laser equations

Above threshold the system undergoes a lasing phase transition where the phonon and Stokes amplitude goes fromhaving zero mean to having a finite expectation value given by . Using this physics, we can find the power spectrumof the phonon field above threshold by reexpressing the phonon amplitude in terms of phase and amplitude

b = ( + )eibt+i', (47)

where is the time-independent mean value of the phonon amplitude, and and ' represent the zero-mean fluctuationsof the phonon amplitude and phase, respectively. The dynamics of and ' can be found by plugging this representationof the phonon field into the e↵ective equation of motion Eq. 41 and taking real and imaginary parts, giving

˙ 1

2RIN + r (48)

' 1

i, (49)

where we have assumed || ||, r(t) Re[(t) expibt i'(t)] and i(t) Im[(t) expibt i'(t)], and wehave used the expression for the steady amplitude given in Eq. 44.

The decay rate for the amplitude fluctuations (or relative intensity fluctuations (RIN)) is given by

RIN 8

p

s|g|4|aextp |22

( 12s + 2

p|g|22)3

(50)

=8

p

|g|22

( 12s + 2

p|g|22)

. (51)

Next, we consider the correlation properties of the Langevin force .

1.4.b Langevin force

In order to compute the power spectrum for the phonon field we need the correlation properties of . The relevantcorrelation function is given by

8

h†(t)(t0)i = h†(t)ei'(t)ei'(t0)(t0)ieib(tt0) (52)

=

h†(t)ei'(t)ei'(t0)(t0)i + |g|2h[aext

p (t)a†s(t)]

†ei'(t)ei'(t0)[aextp (t0)a†

s(t0)]i

eib(tt0) (53)

nthb (t t0) + |g|2|aext

p |2ei(!pb)(tt0)has(t)a†s(t

0)i (54)

nthb (t t0) +

s|g|2|aextp |2

s + 4|g|22/p(nth

s + 1)e12 (s+4|g|22/p)|tt0| (55)

nth

b + 4s|g|2|aext

p |2(s + 4|g|22/p)2

(nths + 1)

(t t0) (56)

(nthb + nth

s + 1)(t t0). (57)

In the third line above we’ve assumed that the correlation time for the Stokes field is much shorter than the externalpump and the phonon phase, allowing the phase noise of the external pump and phonon to be neglected. In the fourthline we have evaluated has(t)a

†s(t

0)i. In the fifth line we’ve assumed that the Stokes correlation function is e↵ectivelylocal in time, allowing us to replace the damped exponential with a delta function. This approximation will yield reliableresults when computing the power spectrum for the phonon field as the decay rate for the Stokes field is much larger.Finally, in the last line we used to relation between the steady-state phonon amplitude and the phonon decay rate.

Using this resulting correlation function we find

hr(t)r(t0)i =

1

2(nth

b + nths + 1)(t t0) (58)

hi(t)i(t0)i =

1

2(nth

b + nths + 1)(t t0) (59)

hr(t)i(t0)i = 0. (60)

1.4.c Power spectra

Now we combine the correlation properties of the Langevin force with the phonon laser equations to derive the powerspectrum for the phonon field. Using the correlation function

hei('(t)'(t0))i = e 1

42 (nthb +nth

s +1)|tt0|, (61)

obtained by solving Eq. 49 [6], we find

hb†(t)b(t0)i = [2 + h(t)(t0)i]hei('(t)'(t0))ieib(tt0) (62)

=

2 +

(nthb + nth

s + 1)

2RINe

12RIN|tt0|

e 1

42 (nthb +nth

s +1)|tt0|eib(tt0) (63)

2e

142 (n+ns+1)|tt0|

| z phase noise

+(nth

b + nths + 1)

2RINe

12RIN|tt0|

| z RIN

eib(tt0). (64)

This result shows that at very high pump powers the phonon field becomes highly coherent. As 1 the powerspectrum is dominated by the phase noise, which exhibits Schawlow-Townes narrowing. In this high-field limit, thecoherence of the phonon oscillation is given by

b =

42(nth

b + nths + 1), (65)

which is the central result of this section. Thus, this mean-field model predicts phonon linewidth narrowing that isreminiscent of the phonon linewidth narrowing in cavity-optomechanical systems [7].

1.5 Simulations of laser dynamics

To understand the noise of our system near threshold, we explored the discrete time dynamics of the laser modeldescribed by Eq. 27. To perform these simulations we modify Eq. 27 in a number of ways. (1) We factored out the

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fast dynamics of the field amplitudes, i.e. we simulated the dynamics of the temporal envelopes of all fields, e.g. for thepump envelope ap defined as ap = apei!pt. (2) We considered the laser dynamics on resonance, i.e. !p = !s + b. (3)We ignored quantum fluctuations of each of the fields, which play a minor role in our system. And, (4) we added thee↵ects of nonlinear optical losses present in our silicon waveguide (these e↵ects are described in detail in [8]).

Our simulations are performed in two steps. First, random noise fields are generated for the phonon field and theexternal source laser over the entire simulation domain, and then the fields are propagated in time using discrete timedynamics. Note that the intrinsic noise of the optical fields is quantum, so these sources of noise are neglected. Inthe discrete time picture the time axis is broken into time steps labeled tj and separated by t (chosen to be 1 ns inour simulations). In this framework, the phonon Langevin force becomes (tj) ! j , where j is a zero-mean Gaussianrandom variable, with variance nth

b /t, assigned to time step j. We assume a phase-noise dominated source laser,which we describe with the phase di↵usion model [9]. These noise fields are generated at the outset of the simulation.Subsequently, the fields are propagated from their initial values (set to 0) to find their time dynamics, at each timestep the Langevin forces produce a random kick of the field amplitude and phase. For example, the phonon field ispropagated by iterating the equation

bj+1 = bj t[(/2)bj + igap,jas,j j ] (66)

which is the discrete time form for the mean-field phonon equation (Eq. 27), and similarly for the other fields. Weobtain a wealth of information from these simulations, ranging from the power spectra and the occupation for all fieldsto the laser slope eciency. In particular, the linewidth of the heterodyne beat note between the pump field and theStokes field is displayed in Fig. 4 of the main text, and the simulated output laser power versus input source laser poweris shown Fig. 3 of the main text.

1.5.a Model parameters used in laser simulations

The model parameters used in these simulations were obtained or corroborated through independent measurements.The optical decay rates are derived from transmission spectra of the ring resonator described in Sec. 3, and theBrillouin coupling, phonon frequency, and phonon decay rate were obtained from Brillouin scattering measurements inlinear waveguides [8].

Brillouin coupling rate: Using the coupled envelope equations given in Eqs. 13-15, the Brillouin gain Gb can bedirectly related to the coupling parameter g, yielding Gb = 4|g|2L/(h!pvg,pvg,s) [4]. The Brillouin gain of 470 (Wm)1,measured in linear inter-modal waveguides with the same cross section, correspond to a coupling rate of g = 12 kHz. Inaddition, the measured intracavity threshold power can be used as a cross check of this estimate. By using Eq. 44 withthe measured laser threshold power of Pth = 19.3 mW, along with the measured optical and acoustic dissipation rates,we find a Brillouin coupling rate of g = 10.6 kHz. Nonlinear optical losses are present in our system and will raise thethreshold power. If we account for nonlinear optical losses the Stokes decay rate is slightly larger 2 90 MHz nearthreshold (compare Tab. I), yielding a slightly larger estimate for the Brillouin coupling rate of g = 11.1 kHz used inour simulations, this coupling rate corresponds with an estimated Brillouin gain of Gb = 400W1m1.

Phonon dissipation rate: The phonon dissipation rate was obtained from spontaneous Brillouin scattering measure-ments in linear waveguides. These measurements give = 2 13.1 MHz.

Linear optical loss: We obtain the linear optical loss parameters by fitting the model presented in section 3 to themeasured transmission spectra.

Nonlinear optical loss: The nonlinear optical parameters were studied in our previous work [8] and are included inthe following table for convenience. From this detailed characterization of both intra- and inter-modal nonlinear loss,we observed that inter-modal scattering greatly reduces the spatial overlap between pump and Stokes waves (becausethey propagate in distinct optical spatial modes), resulting in low inter-modal TPA and FCA nonlinear loss coecients.These nonlinear losses have a relatively small e↵ect on the threshold and slope eciency at low Stokes powers.

These model parameters are summarized in the following table.

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Linear optical properties valuer2 0.54µ2 0.84r1 0.97µ1 0.24↵2 15.8 m1

↵1 6.0 m1

vg,2 7.163 107 m/svg,1 7.385 107 m/s2 2 481 MHz1 2 83 MHz

Nonlinear optical properties valueBrillouin Gain

Gb 400 W1m1

Kerr coecients11k 74 ± 11 m1W1

22k 71 ± 11 m1W1

12k = 21

k 44 ± 7 m1W1

TPA coecients11 34 ± 10 m1W1

22 30 ± 9 m1W1

12 = 21 20 ± 6 m1W1

FCA coecients111 900 ± 400 m1W2

222 720 ± 430 m1W2

211 122 310 ± 200 m1W2

Acoustic properties valueb 2 6.02 GHzg 11.1 kHz0 2 13.1 MHzq 4.5 105 m1

vb,group 826 m/svb,phase 8.4 104 m/s

1.6 Optical parametric oscillators, Brillouin lasing, and phonon lasing

To explore the nature of laser oscillation in this system, we analyze the dynamics in the language of well-establishedoptical parametric oscillator (OPO) physics [1], which provides a general framework for understanding and characterizingthree-wave lasers. The intuition gained from optical parametric interactions is especially useful in the context of Bril-louin scattering, since they have a number of characteristics in common, including spatial gain through traveling-waveinteractions, dependence on phase-matching, and nearly identical spatio-temporal equations of motion.

In this section, we show that this framework allows us to (1) clearly distill the subtle yet fundamental di↵erencesbetween phonon lasers and Brillouin lasers in their many incarnations and (2) discover analogous systems that provideinsight into the novel dynamics of this silicon Brillouin laser. In particular, we analyze our system in the contextof singly-resonant optical parametric oscillators. These optical parametric oscillators exhibit coherent laser-oscillationsimilar to conventional lasers, with the crucial distinction that the optical gain is supplied by a three-wave (or (2))interaction.

1.6.a Defining characteristics for lasers

Before discussing the basic dynamics of OPO physics, we first highlight the essential properties of lasing. In its mostbasic form, a laser requires a cavity (or feedback) and a gain medium. Thus, lasing can occur for a field that (1)experiences feedback, which gives the system a pole, and (2) also experiences round-trip gain that matches round-triploss, putting the pole on the real axis [10].

11

ωp

ωs

ωi

Gain: χ(2)ωp

ωs

ωi

ωs

a b

FIG. 1: Basic operation of a singly resonant OPO [1]. Panel (a) shows the energy conservation of this parametric process. Thepump frequency (!p) is equal to the sum of signal (!s) and idler (!i) frequencies. (b) In a singly-resonant OPO, only the signalwave experiences feedback from a cavity, and consequently self-oscillation. The idler wave is emitted as a byproduct of signalself-oscillation because it mediates the parametric process.

1.6.b Singly-resonant OPO physics

A singly-resonant OPO consists of a nonlinear medium that provides parametric gain which is surrounded by a cavitythat produces feedback for only one of the nonlinearly generated waves [1] (see Fig. 1). Within the nonlinear crystal,three waves interact: pump, the signal, and the idler waves. In this configuration, only the signal wave requires opticalfeedback for laser oscillation to occur. For instance, both the pump wave and idler can pass ballistically through thesystem without any form of feedback (see Fig. 1 ). Within a singly-resonant OPO, the condition for laser-oscillation issatisfied when the parametric gain equals the round-trip loss for the signal wave, resulting in optical self-oscillation ofthe signal wave [1]. The idler is also emitted in tandem with the signal wave, as it mediates the parametric interaction;thus, the idler-wave emission is a byproduct of signal-wave self-oscillation.

1.6.c Brillouin lasing and phonon lasing

We can now understand the basic operation of Brillouin lasing in terms of well-known OPO physics. In Brillouinlasers, high-frequency (10 GHz) phonons are heavily damped (spatially: mean-free path of lc 100µm), preventingthe formation of acoustic modes in systems with large dimensions (L lc). In contrast, the optical fields propagatewith low spatial losses, permitting the formation of optical spatial modes within optical cavity resonators. Analogousto a singly-resonant OPO, the Stokes wave possesses a cavity while the acoustic field does not; thus, the Stokes waveperforms the function of the signal wave while the acoustic field plays the role of the idler field, resulting in Stokesself-oscillation. As the phonon field mediates coupling between pump and Stokes waves, phonon (idler) emission can beviewed as a byproduct of Stokes laser-oscillation. Thus, in general, Brillouin lasers only require an optical cavity for theStokes wave; no phonon feedback is necessary for laser oscillation to occur.

This OPO framework also allows us to explore essential characteristics of optomechanical self-oscillation, also collo-quially referred to as phonon lasing. In a phonon laser, the phonon field plays the role of the resonant signal field, whilethe Stokes wave performs the function of the idler. Thus, phonon lasing requires a phonon cavity (or acoustic feedbackto produce a pole) and round-trip acoustic gain that can balance round-trip loss. In general, phonon lasers do not re-quire optical feedback (for example, see the system proposed in Ref. [11]). In cavity-optomechanical systems, however,optical feedback is typically added to reduce the threshold of mechanical self-oscillation, (or phonon lasing). In this way,optomechanical self-oscillation shares many similarities with doubly resonant OPOs (where both signal (phonon) andidler (Stokes) waves possess a cavity [1]), but lacks the extended spatial characteristics (i.e. phase matching) typical ofoptical parametric oscillators.

1.6.d Properties of the silicon Brillouin laser

Within the framework described above, we note that, despite the unusual temporal dynamics, this silicon laser sharesall the same basic characteristics of Brillouin lasers. As diagrammed in Fig. 1 of the main text, optical feedback isproduced by a long (4.6 cm) race-track resonator for the Stokes-wave. By comparison, the 6 GHz phonons that propagatein the Brillouin-active segments of this racetrack are heavily damped (with lc = 63µm and group velocity of 826 m/s),preventing the formation of discrete acoustic cavity modes. We note that, in addition to severe acoustic propagationlosses, the device geometry prohibits any acoustic feedback. This is because, as shown in Fig. 2 of the main text, the

12

Brillouin-Active Waveguide

63 μm

L = 2.2 cm

Phonons radiate to substrate

Phonons radiate to substrate

Intrinsic phonondecay length

FIG. 2: Diagram illustrating the spatial acoustic properties of this Brillouin laser. As with other Brillouin lasers, there is noacoustic feedback or phonon cavity in this system. The phonons are spatially heavily damped (lc i.e., intrinsic decay length= 63µm), preventing the formation of phonon cavity modes in this large racetrack resonator (circumference = 4.6 cm). Moreover,around the bends the waveguide is not suspended, prohibiting acoustic feedback. Instead, the phonon field eciently radiatesinto the substrate.

acoustic phonons are not guided around the waveguide bends. As a result, the phonons are radiated to the substrate asseen in Fig. 2. Since only the optical Stokes wave can play the role of the signal wave, the phonon field must play therole of the idler wave. Therefore, this silicon laser produces optical self-oscillation, not phonon self-oscillation.

1.6.e Optical self-oscillation and phonon linewidth narrowing

While this system yields optical lasing, we have shown that it exhibits linewidth narrowing dynamics that moreclosely resemble the dynamics of a phonon laser. We again leverage the framework of OPO physics to illustrate how,even though the phonon field is not responsible for the feedback and laser oscillation, it governs the linewidth narrowingdynamics simply because it is the carrier with the longest temporal memory.

In what follows, we explore the dynamics of a generalized singly-resonant OPO that mirrors the behavior of this siliconBrillouin laser. As we have discussed, this and other Brillouin lasers are nearly direct analogues to a singly resonantOPO, in which the optical Stokes wave plays the role of the signal and the phonon field performs the function of theballistic (i.e., cavityless) idler wave. However, this analogy becomes nuanced because a traveling acoustic wave, whichhas a velocity that is 105 slower than light, is taking the place of the optical idler wave. This distinction has profoundconsequences that permit the non-oscillating field to produce Schawlow-Townes-like linewidth narrowing.

To complete this analogy, let us consider a singly-resonant OPO in which the idler wave has an especially high groupindex, resulting in slow optical propagation. In this nonlinear optical system, phase matching plays a crucial role inthe dynamics (see Fig. 3). As the velocities of the other waves are so much larger than that of the idler, the pumpand signal waves essentially dictate the wavevector of the slow idler wave through phase matching. If this slow groupvelocity idler experiences no propagation losses, then the gain bandwidth (associated with phase-matched coupling) ofthis system is simply determined by the length of the gain medium and the group velocity of the idler wave (see Fig.3b). Note that the smaller the group velocity the narrower the gain bandwidth.

However, any dissipation will e↵ectively broaden the idler dispersion curve by the decay rate of the idler field. Inother words, damping causes exponential decay; this produces Lorentzian broadening of the dispersion relation as seenin Fig. 3 [11]. We can conceive a scenario in which the dissipation rate of the idler field is much larger than thedissipation-less phase matching bandwidth. In this heavily damped scenario, instead of the conventional sinc-like phasematching bandwidth, the phase matching condition yields a Lorentzian gain bandwidth determined by the lifetime ofthe slow idler field (see Fig. 3c).

This system arrangement allows us to easily conceive of a condition in which the signal wave produces self-oscillationwhile the idler dictates the linewidth narrowing dynamics. As we have discussed, because there is no feedback for theidler wave, the idler wave cannot self-oscillate. Instead, a non-zero expectation value of the idler can only be achievedas a byproduct of signal-wave self-oscillation. However, because of the disparate velocities of the signal and idler waves,it is apparent that even though the signal wave has a cavity and the idler does not, the idler wave can easily havelonger temporal memory as it ballistically (yet ever so slowly) passes through the nonlinear medium. In this temporaldissipation hierarchy, we have seen that, compared to the idler, the signal wave is temporally overdamped, resulting

13

Gain: χ(2)

ωp

ωs

ωi

ωs

ω

kΓ

ω

Δk = 2 Pi/L

kvgΔk

a

b cL

Δk = 2 Pi/L

vg,idler << vg,signal

FIG. 3: Generalized optical parametric oscillator with a slow group velocity idler field. (a) Diagram illustrating the basic deviceoperation of a singly-resonant OPO with a slow idler field. Here, only the signal wave experiences feedback from a cavity. Theidler field is emitted as a byproduct of signal-wave self-oscillation. (b) Dispersion relation of the idler wave that illustrates thephase-matching condition. The gain medium length and group velocity dictate the Sinc-like phase matching bandwidth. (c)Propagation losses e↵ectively broaden the idler dispersion relation, resulting in a Lorentzian response with a width given by thedissipation rate ().

in dynamics that adiabatically imitate those of the idler field. Thus, this OPO resonator produces laser-oscillation ofthe signal wave and linewidth narrowing of the non-resonant idler wave, in direct analogy to the silicon Brillouin laserstudied in this work.

2. SUB-COHERENCE SELF-HETERODYNE MEASUREMENT

We use self-heterodyne detection to measure the linewidth of the pump laser (Agilent 81600B) and the phononlinewidth of the silicon Brillouin laser. This technique employs a modified Mach-Zehnder interferometer that possessesboth a delay arm and a frequency shifted reference arm (See Fig. 4). The self-heterodyne method typically uses a delayline that far exceeds the expected coherence time. In this way, the output light fields from the two arms are no longermutually coherent, and the resulting optical beat note has a half-width at half-maximum (HWHM) of the intrinsiclaser linewidth. However, this measurement can often be misleading when trying to determine the linewidth of externalcavity diode lasers whose center frequency is unstable. Center frequency fluctuations caused by mode hopping or otherinstabilities have been shown to artificially broaden the self-heterodyne spectrum at long delay lengths [12].

Alternatively, we can experimentally probe the noise properties of the pump laser using a sub-coherence delay arm[12, 13], a method that is much less sensitive to large center frequency shifts. With sub-coherence delay times, themeasured spectral density is composed of a delta function and a Sinc-like background. By comparing the relativemagnitude of each component, one can determine the coherence time of the actual laser. Using this sub-coherencetechnique, we measure the linewidth of the pump laser and show that the silicon Brillouin laser behaves analogously toa self-driven acousto-optic frequency shifter.

14

Theory Δωp= 2 π × 15 kHz

Pump

1550 nm

AOM

RFSA

PD

τd

P1

P2ωp + Ωaom

ωp

Arm 1

Arm 2

a bRBW = 300 kHz

- 5 0 5

- 50

- 40

- 30

- 20

- 10

0

Frequency (MHz)

Self-

Het

erod

yne

Spec

trum

(dB)

FIG. 4: a) Experimental setup for self-heterodyne interferometry. The first arm of the interferometer provides a delay while thesecond arm frequency shifts the light using an acousto-optic frequency shifter (AOM). The two waves are combined and the beatnote is detected with a fast photodiode. b) Self-heterodyne spectrum of the Agilent 81600B diode laser with a delay length of48.25 m.

2.1 Self-heterodyne interferometry

In this section, we discuss the essential elements of the sub-coherence self heterodyne measurement. Figure 4 a)depicts a typical self-heterodyne interferometer. For an arbitrary delay length, the self-heterodyne spectral density isfound to be [12]

S[!] / P1P2

e!pd(! aom) +

!p

(! aom)2 + !2p

(1 e!pd(cos((! aom)d) +sin((! aom)d)!p

! aom))

.

(67)Here, P1 and P2 are the powers in arm 1 and arm 2 of the self-heterodyne interferometer, !p is the linewidth of the

laser in units of angular frequency, aom is the driving frequency of the acousto-optic modulator, and d is the transittime of the delay arm.

Note that in the limit that d 1/!p, the spectrum becomes a simple Lorentzian with a HWHM of the intrinsiclaser linewidth. For this reason, the self-heterodyne technique typically involves a delay line that far exceeds the expectedcoherence time. However, as with other external cavity diode lasers, the low frequency fluctuations of the center laserfrequency of the Agilent 81600B obscure the traditional measurement [13]. For this reason, we employ a sub-coherencedelay arm to measure the linewidth of the pump laser. Figure 4 b) shows a comparison of a typical self-heterodynesignal of the pump laser (Agilent 81600B) with the theoretical spectral density of a laser with a linewidth of 15 kHz.

2.2 Sub-coherence self-heterodyne measurement using a silicon Brillouin laser

2.2.a Inverted dissipation hierarchy

In this section, we demonstrate the unique noise properties that result from the inverted dissipation hierarchy ofthis SBS laser by using the Brillouin laser as a frequency-shifter and show that it behaves in complete analogy to anacousto-optic modulator. We perform this measurement using the experimental setup shown in Figure 5.

Pump

1550 nm

AOM

PC

RFSA

PD

EDFAEDFA

PC

PC

ap(t+τd)~

as(t)

τd

ω - Ωb

ωp + Ωaom

Arm 1

Arm 2

FIG. 5: Self-heterodyne interferometer using the silicon Brillouin laser as the frequency-shifter. An auxiliary AOM is employedhere only to spectrally discriminate the light passing through arm 2 from the pump light ballistically passing through the chip.

In the discussion to follow, we present a simple model that captures the e↵ect of phonon noise in the self-heterodynemeasurement. Using the Wiener-Khinchin theorem and assuming stationarity, the spectral density of a heterodyne beat

15

note can be written as

S[!] =1

2

Z 1

1d ei! hELO()E

s ()ELO(0)Es(0)i, (68)

where Es and ELO represent the signal and local oscillator waves, respectively. Applying Eq. (68) to the measurementpresented in Fig. 5, the spectral density becomes

S[!] =1

2

Z 1

1d ei! hap( + d)a†

s()a†p(d)as(0)i, (69)

where as, ap, and ap will denote the Stokes, input pump, and AOM-shifted pump waves, respectively. Here, d representsthe transit time of the delay arm. Through separation of time scales, the Stokes field can be written as

as =2igapb†

s(1 + 4|g|2||2sp

). (70)

Hence, the spectral density becomes

S[!] =2|g|2

(s(1 + 4|g|2||2sp

))2

Z 1

1d ei! hap( + d)a†

p()b()a†p(d)ap(0)b†(0)i. (71)

Due to the separation of time scales, the phonon field and pump fields are uncorrelated. Therefore the expectationvalue in Eq. 71 simplifies to

S[!] =2|g|2

(s(1 + 4|g|2||2sp

))2

Z 1

1d ei! hap( + d)a†

p()a†p(d)ap(0)ihb()b†(0)i. (72)

We will now focus on the first expectation value. We can write the phases of ap and ap as

(t) = i!pt + in(t)

(t) = i(!p + aom)t + in(t).(73)

Here, we assume n obeys the phase di↵usion equation such that

(t) = (t)

h(t)(t0)i = !p(t t0).(74)

Applying Eq. 79 to the first expectation value in Eq. 72, we find

hap( + d)a†p()a†

p(d)ap(0)i = |ap|2|ap|2hein(,d)ieiaom

n(, d) = n( + d) n() n(d) n(0).(75)

By direct integration, we compute n(, d):

n(, d) =

Z +d

dt0 (t0) Z d

0

dt0 (t0). (76)

As n(, d) can be described by Gaussian statistics, we can rewrite the expectation value for hein(,d)i [6], yielding

hein(,d)i = eh[n(,d)]2i

2 . (77)

Evaluating Eq. 77, the spectral density becomes

S[!] =2|g|2|ap|2|ap|2

(s(1 + 4|g|2|b|2sp

))2

Z 1

1dei(!aom)e!p(d(d| |)(d| |))hb()b†(0)i. (78)

Now we use the same process (Eqs. 79-77) to determine the phonon correlation function.

(t) = ibt + in(t)

(t) = b(t)

hb(t)b(t0)i = b(t t0)

hb()b†(0)i = ||2hei()ieib

() =

Z

0

dt0b(t0)

hb()b†(0)i = 2eibb||2

(79)

16

Here, b is the phonon center frequency and b is phonon linewidth (units of angular frequency, b = 2b).Inserting the phonon correlation function into Eq. 78 yields

S[!] =2|g|2|ap|2|ap|2||2

(s(1 + 4|g|2||2sp

))2

Z 1

1dei(!aom!b)e!p(d(d| |)(d| |))b||

2 . (80)

Note that in the limit that d ! 0, heterodyne detection directly measures the phonon power spectrum. Thus, thespectral density becomes

S[!] =2|g|2|ap|2|ap|2||2

(s(1 + 4|g|2||2sp

))2

Z 1

1dei(!aom!b)e

b||2 . (81)

Performing the integration of 80 with an arbitrary delay length, we arrive at the main result for this section:

S[!] =2|g|2|ap|2|ap|2||2

(s(1 + 4|g|2||2sp

))2

4 (b + 2!p)

(b + 2!p) 2 + 4w2

32w!p (b + !p) sin (wd) e12 d(!b+2!p)

(2b + 4w2) ((b + 2!p) 2 + 4w2)

+8!p

2

b + 2b!p 4w2cos (wd) e

12 d(b+2!p)

(2b + 4w2) ((b + 2!p) 2 + 4w2)

.

(82)

Here, w ! aom b. If we instead evaluate Eq. (80) with b = 0 (zero phonon phase noise), the spectrummatches that of the simple sub-coherence self-heterodyne measurement (refer to Eq. (67)). Performing the integrationwe find

S[!] =2|g|2|ap|2|ap|2||2

(s(1 + 4|g|2||2sp

))2

2e!pd(w) +

2!p

w2 + !2p

(1 e!pd(cos(wd) +sin(wd)!p

w))

/ P1P2

e!pd(w) +

!p

w2 + !2p

(1 e!pd(cos(wd) +sin(wd)!p

w))

.

(83)

Here P1 and P2 are the powers in arm 1 and arm 2 of the self-heterodyne interferometer. Figure 6a plots Eq. 82 witha 15 kHz linewidth pump and various phonon linewidths. Note that as the phonon linewidth increases, the interferencefringes wash out and the pedestal becomes more Lorentzian-like.

Using the experimental setup depicted in Fig. 5, we measure the self-heterodyne spectrum with the silicon Brillouinlaser as the frequency-shifter and compare it to that of traditional measurement using an acoustic optic modulator.Fig. 6b shows the two spectra superimposed. Strong qualitative agreement between the two experimental spectrademonstrates that the SBS laser behaves analogously to a self-driven acousto-optic frequency shifter. More specifically,the emitted Stokes light from this silicon Brillouin laser is a frequency-shifted copy of the pump light with a small degreeof phase noise imparted by the phonon field. Fig. 6c shows the range of the spectrum that is most sensitive to thephonon linewidth and the theoretical comparison. The spectrum plotted (Stokes power of 0.025 mW) in Fig. 6c showsgood agreement with the theoretical spectrum with a phonon linewidth of 3 kHz.

We also perform these measurements using an open-loop laser that exhibits exceptional agreement with the sub-coherence spectrum. For these measurements, see Section 5.3.

2.2.b Standard dissipation hierarchy

The self-heterodyne measurement performed with the silicon Brillouin laser demonstrates the unique phonon linewidthnarrowing characteristics of of this device. By contrast, in standard dissipation hierarchy — that is, — thepredicted spectrum is very di↵erent from both Eq. 82 and the experimentally measured spectrum. In this regime, theStokes field experiences Schawlow-Townes narrowing, resulting in a ultra-narrow-linewidth Stokes wave whose phase isnow uncorrelated with that of the pump. In this limit, Eq. 69 becomes

S[!] =1

2

Z 1

1dei! hap( + d)a†

p(d)iha†s()as(0)i. (84)

17

- 5 0 5

- 50

- 40

- 30

- 20

- 10

0

Frequency (MHz)

Self-

Het

erod

yne

Spec

trum

(dB)

0 Hz3 kHz20 kHz40 kHz80 kHz160 kHz

Theory ΔΩb/2π

- 7 - 6 - 5 - 4- 55

- 50

- 45

- 40

- 35

Frequency (MHz)

Self-

Het

erod

yne

Spec

trum

(dB)

Theory ΔΩb/2π3 kHz0 Hz

a b

c

SBS LaserAOM

Frequency shifterRBW = 300 kHz

- 5 0 5

- 50

- 40

- 30

- 20

- 10

0

Frequency (MHz)Se

lf-H

eter

odyn

eSp

ectr

um(d

B)

FIG. 6: a) Theoretical trends showing the e↵ect of phonon phase noise on the self-heterodyne measurement. Observe that as thephonon linewidth increases, the fringes lose contrast. b) Comparison of the self-heterodyne spectrum using the Brillouin laseras the frequency-shifter (Figure 5) with the traditional spectrum using an AOM as the frequency-shifter (Figure 4). Identicaldelay lengths are used in each experiment. c) Theory-experiment comparison in the most sensitive portion of the spectrum to thephonon linewidth. Note that there is some residual phase noise that reduces the contrast of the fringes from the 0 Hz theoreticaltrend (evident from the spectra using the AOM as the frequency shifter). We attribute this noise to small variations in the delayarm of our fiber interferometer.

Applying the same technique as that of the previous section, the spectral density becomes

S[!] =|ap|2|as|2

2

Z 1

1dei(!baom)e

(!p+!s)||2 . (85)

Here, !p and !s are the linewidths for the pump and Stokes waves, respectively (angular frequency units). EvaluatingEq. 85, the spectral density in the conventional Brillouin limit is a simple Lorentzian, completely devoid of coherentinterference. Thus, the spectrum becomes

S[!] / PsPp4(!p + !s)

(!p + !s)2 + 4(! b aom)2. (86)

In Brillouin lasers studied to date, !s !p, and the observed Lorentzian has full width at half maximum (FWHM)of !p. Note that Eq. 86 is independent of d, as there is no phase coherence for interference. Hence, in the traditionalBrillouin limit, the self-heterodyne spectrum is bereft of the characteristic fringes, in sharp contrast to the phononlinewidth narrowing regime explored by the silicon Brillouin laser.

3. SIX-PORT COUPLER THEORY AND ANALYSIS

In this section we derive the optical response of a multimode optical resonator that we use to determine the inputparameters of our laser model. By fitting this theory to measured transmission spectra we obtain a wealth of informationabout the ring resonator physics. We determine the free-spectral range, giving the group velocity, and the width ofthe transmission dips gives the loaded dissipation rates for the each of the optical modes. Once these parameters arespecified this theory can be used to determine the intracavity pump and Stokes powers from the transmitted pump andthe output Stokes powers, and provides all of the input parameters for our lumped element model.

We undertake this analysis by using general properties of lossless six-port couplers of the type pictured in Fig. 7, andby leveraging the known free propagation of the optical fields in each waveguide type.

18

FIG. 7: Illustration of a six-port coupler. In general, the propagation velocity and dissipation is distinct in each waveguide.

3.1 Six-port coupler

To begin, consider the lossless six-port coupler seen in Fig. 7, which can be described by a symmetric 33 unitarymatrix U. In such a coupler, light injected into any one input port can be scattered out of all output ports; the relativephases of all output ports are in general arbitrary, but the total power is conserved through the coupler. Symmetry ofthe scattering matrix ensures that the coupler is reciprocal. Mathematically, this scattering is described by the relation

Aout = U · Ain, (87)

where A(z) = (Abus(z), A2(z), A1(z)) is a vector composed of the field amplitudes in the respective bus, first excited,and fundamental modes at position z along the composite ‘waveguide’ (i.e. the three waveguides). Ain being the valueof this vector just before entering the coupler, and Aout representing the resultant field amplitudes exiting the coupler.By virtue of unitarity (i.e. U† · U = 1) the total photon flux, or the total optical power, is conserved through thiscoupler, i.e. |Aout|2 = |U · Ain|2 = |Ain|2.

3.2 Free optical propagation

To describe the physics of the ring resonator we need to understand the free propagation of the fields in each waveguide.We assume that the dynamics of optical field is distinct to each waveguide, and can be described by a spatial dampingfactor and a phase. Hence, as the optical field propagates a distance ` the field transforms as

A(` + z) = T(`) · A(z), (88)

where the propagation matrix is given by

T(`) =

0@

e(ikbus↵bus/2)` 0 00 e(ik2↵2/2)` 00 0 e(ik1↵1/2)`.

1A (89)

Here, kj = !/vg,j is the propagation constant; vg,j being the group velocity in the jth waveguide, and ! being theangular frequency. The factor ↵j is the spatial decay rate unique to each waveguide.

In the next section we put these ingredients together to find the transmission spectrum of a ring resonator formed byclosing waveguides 1 and 2 into closed loops of length L.

3.3 Optical response of a multimode ring resonator

Consider a ring resonator formed by closing waveguide 1 and 2 into closed loops of length L (see Fig. 8). These twoclosed loops will each form a distinct set of optical modes that can be characterized by observing the transmission ofthe optical field in the bus waveguide.

First, we assume that the power is conserved through the coupler, giving Aout = U·Ain. Secondly, we use the fact thatthe input fields in waveguide 1 and 2 are related to the output fields by free propagation, i.e. A1,in = e(ik1↵1/2)LA1,out.

To compactly describe this physics we express the field as Ain = (Ain,AC,in), where AC = (A2, A1) are the two modesin the ring, we introduce a propagation matrix TC for the modes in the ring cavity only i.e.,

TC(`) =

e(ik2↵2/2)` 0

0 e(ik1↵1/2)`

, (90)

and we express the scattering matrix as

U =

r iµT

iµ R

, (91)

19

FIG. 8: Multimode ring resonator formed by closing two ports into loops.

where µ is the multi-mode generalization of the ring-bus coupling [14].Using this notation we find that the relationship between the input and the output field in the bus waveguide is given

by

Aout

AC,out

=

r iµT

iµ R

Ain

AC,in

, (92)

and the relationship between the cavity fields is

AC,in = TC(L) · AC,out. (93)

Solving these coupled equations we find the relationship between the input and the output fields in the bus, we find

Aout =r µT · (1 R · TC(L))1 · µ

Ain (94)

AC,out = i(1 R · TC(L))1 · µAin. (95)

Once the scattering matrix for the six-port coupler, the group velocities, and the spatial decay rates are known, theseresults provide the power transmission spectrum given by

Pout

Pin=

r µT · (1 R · TC(L))1 · µ2. (96)

3.4 Relation between intracavity pump power and transmitted pump power

Using the relations above we can find a ratio of the intracavity pump power to the transmitted power.

Pout =r µT · (1 R · TC(L))1 · µ

2 1

|µT · µ|(1 R · TC(L)) · Aout

2. (97)

Using this relation the transmitted pump power can be used to determine the amplitude of the pump field inside theresonator which is a key parameter for our numerical modeling.

3.5 Relation between intracavity laser power to output laser power

In order to fully understand Brillouin lasing in a ring resonator we establish the connection between the intracavitylaser power and the output laser power coupled to the bus waveguide. This connection can be made using the ringresonator theory given above.

When operating as a Brillouin laser there is no Stokes light supplied to the resonator via the bus waveguide. Rather,pump light at a di↵erent frequency is injected into the resonator, and through Brillouin scattering leads to the generationof Stokes light in the resonator at a di↵erent frequency. By setting Ain to zero we find the

Aout,laser = iµT · AC,in,laser. (98)

Here, it is important to stress a crucial point. Ideally, our Brillouin laser operates by scattering injected pump light fromthe first excited mode in the ring to the fundamental mode, and the light in these two fields, of distinct frequencies, isconfined to each of these respective modes. Hence, one may conclude that relation between the out coupled laser powerand the intracavity power is simply given by Aout = iµ · (0, A1). However, unitary scattering through a three portcoupler necessarily couples the first excited and the fundamental modes, and therefore Stokes light will be distributedin both modes. Therefore, the accurate connection between the measured laser power and output power necessarilycontains the coherent addition of the amplitudes of the Stokes fields in both modes as given above.

20

FIG. 9: Minimal model of general reciprocal six-port scattering matrix. A subset of such scattering matrices are described bythree sequential unitary transformations: a) scattering between the bus and port 2, and b) scattering between the bus and port1. c) Resultant six-port scattering from scattering in b), a), and b) sequentially.

3.6 A minimal model of the six-port coupler scattering matrix

In this section we discuss the scattering matrix for the six-port coupler. The most general 3 3 unitary andreciprocal scattering matrix is specified by 6 degrees of freedom. Given such a large parameter space, it is ideal todetermine the scattering matrix through independent transmission measurements, or by finite di↵erence time-domain(FDTD) simulations. However, these two approaches o↵er challenges in many instances. It is often not possible todirectly measure the scattering matrix in on-chip systems (such as ours), and FDTD is limited by non-idealities in realsystems such as sidewall roughness.

To circumvent these challenges we adopt a simple model of our six-port coupler. This model is constructed from threesequential unitary scattering matrices (i.e. U = U1 ·U2 ·U1), the first coupling light between the bus and mode 1 of thering U1, the second scattering the bus to mode 2 U2, and a third scattering the bus to mode 1 U1 (Fig. 9). Unitarityof each matrix conserves power through the coupler and symmetry ensures reciprocity. These two scattering matricesare given by

U2 =

0@

r2 iµ2 0iµ2 r2 0

0 0 1

1A U1 =

0BBB@

q1+r1

2 0 i µ1p2(1r1)

0 1 0

i µ1p2(1r1)

0q

1+r1

2

1CCCA U1 · U1 =

0@

r1 0 iµ1

0 1 0iµ1 0 r1

1A , (99)

where unitarity requires that µ1 =p

1 r21 and µ2 =

p1 r2

2, and where we have factored the “rotation by µ1” fromthe bus to mode 1 into two.

While this approach fails to capture the most general six-port scattering physics, it does capture the salient featuresof six-port couplers and o↵ers a simple tractable model described entirely by two free parameters.

Before moving on, we mention that we have explored the most general model to describe a lossless reciprocal 6-portcoupler. However, the large number of parameters (5 + global phase) are underconstrained by our transmission spectrameasurements, with families of parameters giving equally valid descriptions of our data. Furthermore, we find that theoutput of this general model yield properties of the optical modes in the ring in quantitative agreement with the simplemodel given above.

3.7 Inter-modal ring resonator characterization

Having established the basic physics of multi-mode ring resonators, we analyze transmission spectra measurements ofour system to obtain the group velocities, frequencies, and loaded optical decay rates.

We model the transmission spectra using Eq. 96 and using the minimal mode for a three port coupler described bythe scattering matrix given in Eq. 99. In addition, a slow wavelength dependent shift in the average transmitted poweris added to our model to account for changes in the coupling eciency as a function of wavelength [8]. The result offitting this model to the measured transmission spectra are shown in Fig. 10.

21

FIG. 10: Transmitted pump power as a function of wavelength (open red circles). The black line is a fit of our transmissionspectrum model to these data.

This fitting procedure, in addition to SIMS measurements in linear waveguides, yields the model parameters describingour multimode optomechanical resonator given below.

model parameters valuer2 0.54µ2 0.84r1 0.97µ1 0.24↵2 15.8 1/m↵1 6.0 1/mvg,2 7.163 107 m/svg,1 7.385 107 m/s2 2 481 MHz1 2 83 MHzb 2 6.02 GHzg 11.1 kHz0 2 13.1 MHz

The decay rates above were obtained by an eigenfrequency analysis of the ring resonator, using the fitted scatteringmatrix. Equations 92 and 93 imply that the optical field in the ring is described by a set of lossy modes, characterizedby complex eigenfrequencies. A subset of these eigenfrequencies are separated by 6.02 GHz, the frequency of the phononthat participates in Brillouin scattering. These pairs of modes are responsible for Brillouin lasing. Once these modesare identified, the decay rate can be obtained from the imaginary parts of the eigenfrequency.

The obtained decay rates given above can be checked by assuming negligible crosstalk between the optical modesin the ring, which is approximately valid in our system. When this is true the parameters r1 and r2 can be added tothe bare decay rates, i.e. the decay rates of the fields in a linear waveguide, to estimate the loaded decay rate whichaccounts for coupling loss. These relations are given by

p vg,p

2(↵p 2

Lln r2) = 489 MHz (100)

s vg,s

2(↵s

2

Lln r1) = 86 MHz, (101)

showing the self-consistency of our analysis. The small di↵erence between the tabulated decay rates and the decay ratesgiven above is due to a small amout of coupling between the ring resonator modes.

22

3.8 Laser power conversion factor

Our scattering theory treatment of the ring resonator allows us to estimate the ratio of intracavity laser power tooutput laser power, the amount exiting the resonator on the bus waveguide. This parameter is necessary to compare oursimulations of the laser physics to our measurements. For the modes responsible for lasing we find a power conversionfactor of 0.03. However, this parameter is relatively unconstrained by our transmission spectra data as it dependsstrongly on the relative phase between the two optical spatial modes exiting the 6-port coupler (as discussed in section3.5). By fitting various forms of the scattering matrix to the transmission spectrum, we have found a range of conversionfactor values that are consistent with the measured transmission spectra. For this reason, we will treat this conversionfactor as a fit parameter in our theory-experiment comparisons. We find that our measurements of slope eciency andthe pump-Stokes beat-note linewidth are best described for a power conversion factor of 0.0075.

4. DATA ANALYSIS

In this section, we describe the methods used to analyze the experimental data.

4.1 Slope eciency

In Fig. 3c of the main text, we plot the Stokes power as a function of injected pump power. The output (on-chip) Stokes power is determined through a calibrated heterodyne measurement. While performing this heterodynespectroscopy, we simultaneously measure the input and transmitted pump powers through the chip. By monitoringthe input and transmitted pump powers and using the racetrack resonator properties analyzed in Section 3 (includingcoupling and linear loss), we infer the detuning of the pump from resonance and estimate the intra-cavity pump powers.The data presented in Fig. 3c represent the measured Stokes powers (left y-axis) as a function of intracavity pumppower (top x-axis). For comparison, the bottom x-axis assumes a zero-detuned resonant enhancement factor ( 1.8),showing the e↵ective input pump power. The right y-axis indicates the estimated intracavity Stokes power assumingthe intracavity to bus waveguide power conversion factor discussed in Section 3.8 (0.0075).

10- 15 10- 13 10- 11 10- 9 10- 7

1

10

100

1000

Peak Stokes Spectral Density (W/Hz)

Phon

onLi

new

idth

(kH

z)

Resolution bandwidth

Pump laser linewidth

Laser thresholdSelf-heterodyne power limitation

Beat note linewidth

Self-heterodyne

FIG. 11: Details of linewidth measurements. The standard heterodyne detection scheme becomes resoluation bandwidth limitedat a minimum phonon linewidth of 20 kHz. The pump laser linewidth is determined (through sub-coherence self-heterodynemeasurements) to be around 13 kHz. The sub-coherence self-heterodyne detection scheme for phonon linewidth measurementsrequires enough Stokes power to resolve the fringes that are produced by pump-Stokes phase coherence. An approximate limitationfor our measurements is indicated by the blue line.

4.2 Linewidth measurements

This section discusses the analysis of the data presented in Fig. 4e of the Letter.

23

4.2.a Heterodyne measurements

Below threshold, we use standard heterodyne spectroscopy to determine the linewidth of the phonon field (see Fig.4c,e of the letter). To avoid spectral distortion due to pump mode hopping, we acquire rapid spectral traces at di↵erentoutput Stokes powers. We then apply a moving average (with a bin size smaller than linewidth to avoid distortion) tospectrally smooth the data for consistent Lorentzian fits. The data presented in Fig. 4e are acquired using a resolutionbandwidth of 20 kHz. Above threshold, the phonon linewidths are resolution-bandwidth limited. To simplify therepresentation of the data in Fig. 4e, we only include spectral measurements below threshold. In Fig. 11, we plot all ofthe heterodyne spectroscopy measurements, including above threshold.

4.2.b Sub-coherence self-heterodyne measurements

We determine the phonon linewidth above threshold using the sub-coherence delayed self-heterodyne technique dis-cussed in Section 2. To do this, we fit the theoretical spectrum (Eq. 82) to the measured spectra. We extract thephonon linewidth by fitting the power, pump linewidth, and phonon linewidth to the peak spectral density and thefringe-like portion of the spectrum (the feature highlighted in Fig. 6c), which is most sensitive to phonon phase noise.This method is only possible at large Stokes powers, where there is sucient signal to noise so that the fringes do notbecome obscured by the noise floor. This sets a practical or experimental limitation of the minimum powers needed tomake these measurements. An approximate value for this power limitation is illustrated by the blue dashed line of Fig.11.

As discussed in the Fig. 6 caption, instabilities in the fiber delay line of the interferometer add approximately 500Hz of phase noise to the sub-coherence self heterodyne measurements (obtained from a self-heterodyne measurementusing an AOM as the frequency shifter). Therefore, to get an accurate measurement the phonon phase-noise within thesilicon Brillouin laser, we normalize our sub-coherence self-heterodyne measured linewidths by subtracting the residualphase noise.

For a comparison with the spectral measurements below threshold (not resolution bandwidth limited), we convertthe beat-note power into a peak spectral density (x-axis of Fig. 4e) by using the fact that the integral of a Lorentzianis proportional to the peak height multiplied by the width. In this way, we convert the sub-coherence spectrum intoan equivalent Lorentzian that has a FWHM given by the measured linewidth and a peak height given by the peakspectral density. Thus, by dividing the Stokes power by the measured linewidth (and multiplying by 2/), we obtainthe equivalent peak spectral density used for this comparison (plotted in Fig. 4e of the main text).

As discussed in Section 2.2, these sub-coherence self-heterodyne measurements demonstrate the pump-Stokes coher-ence produced by optical self-oscillation in this system. While these data permit a measurement of the phonon linewidthand corroborate the regime of dynamics (presence of fringes and measurements of phonon linewidths below that of thepump, see Section 2.2.a-b), the data do not yield sucient precision to clearly determine the linewidth narrowing trendas function of total Stokes power. To attain the necessary level of precision, this type of analysis would likely requiremore sophisticated control of all variables that influence laser oscillation. Nevertheless, these data establish an upperbound on the phonon linewidths above threshold and clearly confirm the pump-Stokes phase coherence unique to thislimit of Brillouin laser oscillation.

- 60

- 50

- 40

- 30

- 20

- 10

0

- 60

- 50

- 40

- 30

- 20

- 10

0

- 6.05- 6.055- 6.06 - 6.045 - 6.04 6.05 6.0456.04 6.055 6.06

38 dB

Frequency (GHz)

Stokes Spectrum

Frequency (GHz)

Anti-Stokes Spectrum

Sign

al a

mpl

itude

(dB)

Sign

al a

mpl

itude

(dB)

FIG. 12: Above threshold Stokes and anti-Stokes spectra demonstrating 38 dB of anti-Stokes suppression.

24

5. ADDITIONAL MEASUREMENTS AND CONSIDERATIONS

5.1 Anti-Stokes suppression

The inter-modal Brillouin scattering process provides intrinsic Stokes-anti-Stokes symmetry breaking and single-sideband amplification. This unique property of the SIMS gain mechanism permits a large degree of anti-Stokes sup-pression (38 dB) of the output spectrum. As discussed in the main text, the Stokes phonons (generated from thepump and Stokes waves propagating in distinct optical spatial modes) cannot mediate the anti-Stokes process. Abovethreshold, the strong and mutually coherent pump and Stokes waves set up an acoustic grating, which—if the pumppropagates purely in the anti-symmetric mode—phase matches only to the Stokes process. However, in practice a smallfraction of the pump light couples into the wrong (symmetric) spatial mode; pump light propagating in this spatialmode can interact with the strong acoustic grating and phase match in an anti-Stokes process, thus resulting in a smallanti-Stokes sideband.

5.2 Output coupling and slope eciency

0.000 0.005 0.010 0.015 0.020

0.0000

0.0005

0.0010

0.0015

0.0020

Input pump power (mW)

Out

putS

toke

spow

er(m

W)

0.00750.050.10.2

Output Stokes coupling

FIG. 13: Theoretical slope eciency (Eq. 46) with various output Stokes couplings. The blue trace (output coupling = 0.0075)represents the predicted slope eciency for the laser studied in this work.

Increasing the output Stokes coupling can enhance the slope eciency, but this will also degrade the optical Q-factorand consequently raise the threshold pump power. However, because this Brillouin laser is under-coupled for the Stokeswave, the dominant source of loss limiting the Q-factor is the intrinsic propagation loss of the symmetric mode. Inthis Brillouin laser, therefore, increasing the output coupling can dramatically enhance the slope eciency while onlymarginally raising the threshold power. Using Eq. 46, we estimate that increasing the output coupling to 16% couldincrease the slope eciency by 10 fold. To illustrate this dependence, Fig. 13 shows a comparison between the slopeeciencies with various output Stokes couplings.

5.3 Additional sub-coherence self-heterodyne measurements with open-loop laser

We found it most practical to use the Agilent 81600B as the pump laser for our measurements due to its frequencyagility and wavelength sweeping features, which allow us to quickly and systematically characterize the system. It does,however, exhibit additional phase-noise characteristics (slight deviations from theoretical sub-coherence spectrum, seeFig. 6); these features are due to low-frequency locking electronics and are common in lasers that use active lockingfor stabilization. To crosscheck these measurements, we have also used a thermally-tunable, open-loop diode laser thatexhibits superb agreement with sub-coherence spectrum. Using this laser, we observe exceptional agreement betweenthe measured and predicted spectrum (see Section 2.2.b), corroborating the results obtained from our initial analysis.

In Fig. 14, we have included an example sub-coherence self-heterodyne spectrum of our silicon Brillouin laser usingan open-loop pump laser (Pure Photonics PPCL300). This particular spectrum was measured using a laser device witha slightly di↵erent length (3.06 cm circumference). Note that sidebands due to frequency stabilization electronics areabsent from this measurement. The fitted spectrum in Fig. 14 corresponds to a phonon linewidth is 1.73 kHz (andpump linewidth of 32 kHz), revealing similar phonon linewidths under similar conditions using the Agilent 81600B.

25

- 2×107 - 1×107 0 1×107 2×107

- 90

- 80

- 70

- 60

- 50

- 40

- 30

Frequency (Hz) + 6.05 GHz

Self-

hete

rody

nesig

nal(

dB)

Spectral data

Theory

FIG. 14: Typical self-heterodyne spectrum and a fitted theoretical spectrum (see Section 2.2.a) of Brillouin laser using an open-loop pump laser (Pure Photonics PPCL300) . This theoretical fit corresponds to a pump laser linewidth of 32 kHz and an excessphase noise linewidth (phonon linewidth) of 1.73 kHz, demonstrating (1) the pump-Stokes phase coherence unique to this limitof Brillouin dynamics and (2) a phonon linewidth far below that of the pump laser.

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