www.sciencemag.org/cgi/content/full/science.aad4811/DC1

Supplementary Materials for

Photonic chip–based optical frequency comb using soliton Cherenkov

radiation

V. Brasch, M. Geiselmann, T. Herr, G. Lihachev, M. H. P. Pfeiffer, M. L. Gorodetsky, T. J. Kippenberg*

*Corresponding author. E-mail: tobias.kippenberg@epfl.ch

Published 31 December 2015 on Science First Release DOI: 10.1126/science.aad4811

This PDF file includes: Materials and Methods

Supplementary Text

Figs. S1 to S8

References

Materials and MethodsNanofabrication

Starting with a silicon wafer with 4µm of thermal silicon dioxide (SiO2) we depositclose to stoichiometric silicon nitride (Si3N4 or short SiN) for the waveguide cores as a800 nm thick film in a low-pressure chemical vapor deposition (LPCVD) process. Aftersome auxiliary steps the waveguides are patterned in a 100 kV electron beam lithographysystem using ZEP520 as resist. After development the resist is re-flown. The followingreactive ion etch is the critical etch step. It uses CHF3 and SF6 gases and transfers the pat-tern into the SiN. This is followed by a photolithography step to define auxiliary structures.The wafer is thoroughly cleaned before an additional thin layer of SiN is deposited with thesame process as before. Afterwards the wafer is annealed and the 3µm thick SiO2 claddingis deposited with a CVD process on top. The last steps are the definition and separation ofthe chips and a second anneal.

Finite element simulationsUsing the commercial Comsol Multiphysics package for FEM simulations, we imple-

mented a 2D simulation which takes into account the cylindrical symmetry of the systemin the third dimension. Material dispersion is taken into account via an iterative approachwhich takes the values of the refractive index from measured values for our SiN films. Thevalues for dispersion parameters are obtained from fitting appropriate polynomials to theabsolute frequencies of the modes around our pump wavelength of 192.2 THz. The relativemagnitude of the values for the free spectral range in the simulation is used to identify themode families (TE and TM) inside the resonator.

Implementation of the numerical simulationFor the numerical simulation we used coupled mode equations which are propagated in

time using an adaptive step size Runge-Kutta algorithm (32). The calculation of the non-linear mixing terms is efficiently calculated in the Fourier-domain (33). An additional self-steepening term (22) was added to better model the behavior of few-cycles pulses. To allowfor deterministic simulations of required states one, two or three-soliton states were seededas initial waveform, starting the simulation with a detuning that allows for stable soli-tons. The parameters used for the simulation presented in Fig. 2A are: D2/2π = 2.2 MHz,D3/2π = 25 kHz, D4/2π = −300 Hz, detuning ζ = 12, Qint = 1.5 · 106, Qext = 8 · 105,Ppump = 1 W.

Noticeable differences between the experimental spectra and the theoretical simulations(Fig. 2A and Fig. S4K) are the reduced intensity of the soliton Cherenkov radiation andthe absence of an observable soliton recoil in the experiment. We attribute the former toexperimentally confirmed higher optical losses in the setup, which is based to large partson SMF-28 fiber, for these long wavelengths and the latter to nonlinear terms (in particularthe Raman shift (29,34,35) but also the frequency dependence of the nonlinear coefficient),which are not taken into account in our simulation.

2

Multi-soliton and single-soliton spectraWhen N identical solitons circulate around the resonator they produce a frequency

comb spectrum S(N)(µ) with a structured envelope which results from interference of sin-gle soliton Fourier spectra S(1)(µ):

S(N)(µ) =

∣∣∣∣∣F{

N∑j=1

Aµ(φ− φj)

}∣∣∣∣∣2

=

= S(1)(µ)I(µ) = S(1)(µ)

(N + 2

∑j 6=l

cos(µ(φj − φl))

). (1)

If now the Fourier transform of this optical spectrum (sometimes transformed initially bypicking only the frequency comb lines) is taken, it will result in the field autocorrelationfunction of the waveform according to the Wiener-Khinchin theorem. In the case of Nsolitons it contains two peaks at 0 and 2π and in addition N(N − 1) single-soliton au-tocorrelation peaks at positions in the interval (0 2π) symmetric around π, correspondingto pairwise distances between circulating solitons (Fig. 3D). Using simple peak-finding itis possible to obtain these distances and define the trigonometric multiplier of the spec-trum. To reconstruct the single-soliton spectrum we simply divide the initial spectrum byN + 2

∑Nj 6=l cos(µ(φj − φl)). Note that in the degenerate case when all the distances are

related as integer numbers (like in Fig. 3B) the trigonometric sum may turn to zero at somepoints, which should be dropped from consideration. With the measured or reconstructedsingle-soliton spectrum and assuming flat phases it is possible to estimate an approximatesymmetrized form of the soliton and its duration, assuming that the asymmetry of the pulsesis small (Fig. 2F). From the 3-dB width of the spectrum the pulse duration can be derivedvia the time-bandwidth product for soliton pulses of 0.315 (19).

Sample characterizationThe dispersion of our resonators was measured by sweeping a widely tunable external

cavity diode laser (ECDL) over the resonances and recording the transmitted power on aphotodiode. Part of the power of the ECDL is differed before the resonator and used torecord its beat signal with a commercial fiber laser frequency comb (repetition rate around250 MHz). By counting the crossings of this beat at certain frequencies and interpolation,the position of the laser and the relative positions of the resonances can be determined witha precision of a few MHz (36). The average linewidth of 300 MHz of the mode familyused for the generation of the frequency comb is measured by determining the linewidthsof many resonances within the range from 1510 nm to 1580 nm (Fig. S2C). To determinethe linewidth the laser is scanned over each resonance and the polarization is optimized.The laser scan is calibrated by using the same technique that is used to calibrate the laserscan for the dispersion measurement above.

To measure the parametric threshold an amplified diode laser is swept over the reso-nance while the pump light is filtered out from the transmission using a tunable fiber Bragggrating. The remaining converted light which is the light at other frequencies than the pumpis detected on a photodiode. The power of the pump is adjusted until a clear signature of

3

converted light on the photodiode is observed indicating that parametric frequency conver-sion takes place. To take into account asymmetric input and output losses from the chipthis measurement is repeated with changed direction on the chip. The measured thresholdis Pthres = 300 mW in the waveguide. The coupling loss per chip facet is approximately3 dB.

Laser tuning procedure for soliton generation and beat note measurementsIn order to achieve a stable soliton state we have to overcome the transient instability

of the states within the steps. This is achieved by modulating the pump power with a twostep protocol. The first step is to induce the soliton with a quick drop in power. The secondstep is to stabilize the soliton state by increasing the pump power. The two step processto obtain stable soliton states is implemented using one acoustic optical modulator (AOM)and one Mach-Zehnder amplitude modulator (MZM). The only reason for the use of twomodulators is the required speed of the modulation which can not be obtained solely withthe AOM. The first step of a short dip in power is done with the MZM and typically of 100to 200 ns in length. The following increase in pump power is obtained with the AOM as astep function that remains at high power throughout the measurements afterwards.

In order to measure the electronic beat note at 189 GHz with a photodiode, we sup-press the pump by around 30 dB using a fiber Bragg grating and modulate sidebands of40 GHz onto the lines of the remaining frequency comb using a MZM. The correspond-ing modulation sidebands reduce the difference to 109 GHz amenable to direct detectionwith a commercial telecommunication photodiode with an optical power of around 1 mW.The electrical signal of 109 GHz as obtained from the photodiode is down mixed with aharmonic mixer on the sixths harmonic before being detected with an electrical spectrumanalyzer at a frequency of around 1.7 GHz.

The two narrow linewidth fiber lasers used for the heterodyne beat note measurementsare based on NKT Koheras sources and have linewidths below 10 kHz and a very highfrequency stability. We attribute the difference between the widths of the beat at 1552 nmand 1907 nm to the effect of the non-stabilized repetition rate.

Full stabilizationTo fully stabilize our frequency comb we implement a phase lock of the pump laser to a

reference frequency comb and a lock of the repetition rate of the frequency comb to an RFreference. The repetition rate is actuated using the pump power (31). The repetition rate isdetected as described above and downmixed to 70 MHz. The 70 MHz signal of the repeti-tion rate and the pump laser offset at the same frequency are fed into two separate digitalphase comparators with the same 70 MHz RF reference signal on both. The phase com-parators provide the error signal for two PID controllers which provide the feedback whichis used to modulate the pump power via an AOM and the laser current for the repetitionrate lock and the laser offset lock respectively. All involved RF equipment in the schemeis referenced to the atomic reference of the fiber frequency comb that provides the absoluteoptical reference for the pump laser offset lock. To measure the modified Allan deviation,three counters of the Λ-type were used to measure the frequencies of the laser offset, therepetition rate and the out-of-loop beat simultaneous. The result therefore does not agree

4

perfectly with the modified Allan deviation but shows the same scaling for different noisesources (37). The slope of the modified Allan deviation for the out-of-loop data in Fig. 4Bis –0.83.

The number of the comb line from the fiber comb that was used for the out-of-loopmeasurements (13613) was determined numerically from the measured frequencies. Theexact frequencies are for the repetition rate of the microresonator frequency comb (locked,frep): 189179.658 MHz, for the repetition rate of the fiber laser frequency comb used as areference (locked, frep,fc): 250.144820075 MHz, for the offset of the pump laser from thereference frequency comb (locked, foff): 70.000 MHz and for the out-of-loop beat betweenthe two frequency combs (unlocked, derived from fit in Fig. 4(a), fol): 57.59168095 MHz.

Supplementary TextTheoretical model

In the frequency domain the Master equation given in the main text is equivalent toa set of coupled mode equations, which directly describes the amplitudes of the modesin the resonator (17, 19). The stable soliton solutions occur in the bistable regime, wheresimultaneous existence of the upper branch (soliton) and lower branch (CW pump) solutionis warranted (19).

Due to the synchronization of the soliton and the Cherenkov radiation, the Cherenkovradiation can become comparable in strength to the soliton itself and can not be treated asa small perturbation (3, 38). In this case another approach based on the linearization of theLLE may be used (23) allowing to get complex valued DW for the position of the solitonCherenkov radiation. Of the complex value the imaginary part is related to the width of thefeature in the spectral domain and the real part gives its position in the spectral domain.

Comparison with conventional high-noise stateIt is insightful to contrast the coherent single-soliton and multi-soliton states to the inco-

herent high noise state that can be generated when tuning the pump laser continuously intothe resonance, a tuning mechanism that has been widely employed in Kerr frequency combgeneration experiments (15). We observe in this case a spectrum that markedly deviates inits shape from the single soliton spectrum (Fig. S4A). While the overall bandwidth is onlyslightly reduced the spectrum is not coherent and a degradation of the beat note to a widthof the order of GHz is measurable in CW heterodyne measurements (Fig. S4B) as well as asignificant increase in the intensity noise of the light at the ouptut of the resonator. This isin agreement with the formation of sub-combs (15). In this high noise regime the height ofthe Cherenkov radiation is significantly lower (by approximately 10 dB) and the spectrumdevelops two characteristic lobes symmetric around the pump. Also this low coherencestate is in good agreement with our numerical simulations (Fig. S4A) and recent theoreticalunderstanding (39).

Signs of solitons in microresonatorsWhen looking for resonators and resonances which support solitons, several signs can

be observed. One sign are discontinuities in the transmission as well as the convertedlight when sweeping the pump laser over the resonance from blue detuning to red detuning

5

(Fig. S5A and S1 for data from several other SiN microresonators). Such steps have beenpreviously identified with temporal soliton formation in crystalline microresonators (19,27), where each step corresponds to the successive reduction of the number of solitonscirculating in the microresonator. The regime where steps are recorded coincides with anarrowing of the recorded repetition rate beat note (Fig. S1C,D) in agreement with the lownoise operation within the soliton formation regime. In silicon nitride resonators these stepscan be as short as around 100 ns (Fig. S5A). We attribute this length to the fast thermaleffect of the waveguide geometry. When one can get a stable state, the characteristic,smooth spectral envelope is a clear sign in particular with the typical interference patternsdue to multi-soliton states (Fig. 3). Low phase-noise performance is not sufficient to claimsolitons as it has been observed in states which clearly differ from soliton states (9, 15, 40,41).

6

−0.4 −0.2 0 0.2

0

0.2

0.4

0.6

0.8

1

time ( s−0.4 −0.2 0 0.2

0

0.2

0.4

0.6

0.8

1

time ( s

norm

aliz

ed in

tens

ity

−0.4 −0.2 0 0.2

0

0.2

0.4

0.6

0.8

1

time ( s

norm

aliz

ed tr

ansm

issi

on

0 2 4 6 8 10

0.7

0.9

time (ms)

38GHz 70GHz 190GHz

−8 −7 −6 −5 −4 −3 −2 −1 0 1 20

0.5

1

scanning time (ms)

norm

aliz

ed in

tens

ity

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80.7

1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8

35

36

37

38

39

40

41

freq

uenc

y (G

Hz)

scanning time ( s)

A

B

C

D

Fig. S1: Signs of solitons in SiN microresonators. (A) The typical triangular shapeof the transmission for higher power laser sweeps from the blue side to the red side of aresonance. (B) The steps in the converted light at the end of the triangle (marked with ared box in (A)) are strong signs for soliton states. Red and green are two traces from twoconsecutive laser sweeps, highlighting the changes in the step patterns. Resonators withdifferent repetition rates, as noted inside the figures, show these steps. (C) The step in thetransmission of a 38 GHz sample with high time resolution at the position of the red boxin (A). (D) The collapse of the beat note at 38 GHz from around 1 GHz width to a widthlimited by the measurement technique within the short step of (C).

7

1620 1600 1580 1560 1540 1520 wavelength (nm)

186 188 190 192 194 196 198

−100

−50

0

50

DIn

t/2π

(GH

z)

186 188 190 192 194 196 198−1

0

1

2

3

frequency (THz)

DIn

t/2π

(GH

z)

186 188 190 192 194 196 1980

200

400

600

800

1000

frequency (THz)

κ/2π

(MH

z)A

B

C

Fig. S2: Sample characteristics. (A),(B) Dint of resonances in the used sample. The red,horizontal mode is the pumped TM11 mode (enlarged in (B)). The blue TE11 mode has adifferent FSR and therefore is a tilted line. The solid, green line in (B) is a quadratic fityielding a value for D2/2π of 2.4 MHz. (C) The full linewidth (κ/2π) of some resonancesof the TM11 mode family. The green vertical line in all plots represents the position of thepump laser for the frequency comb generation.

8

2000 1900 1800 1700 1600 1500 1400 1300 wavelength (nm)

150 160 170 180 190 200 210 220−80

−60

−40

−20

0193.16THz / 1552nm157.2THz / 1907.1nm

frequency (THz)

inte

nsity

(dBm

)

−40 −20 0 20 40

−100

−80

−60

frequency − o�set (MHz)

Beat at 157.2THz/1907nmRBW: 100 kHz

inte

nsity

(dBm

)

−20 0 20

−100

−80

−60 Beat at 193.16THz/1552nmRBW: 50 kHz

frequency − o�set (MHz)−0.4 −0.2 0 0.2

−130

−120

−110

−100

−90 Rep rate beatRBW: 1 kHz

frequency − 189 GHz (MHz)

A

B C D

Fig. S3: Coherence of a two-soliton state in a SiN microresonator. (A) Optical spectrumof a two-soliton state measured under very similar conditions and in the same resonance asthe data of Fig. 3. (B) The beat note of a narrow linewidth fiber laser at 1907 nm with thenearest line of the generated frequency comb. (C) The beat note of a narrow linewidth fiberlaser at 1552 nm with the nearest line of the generated frequency comb. (D) The repetitionrate beat note of the generated frequency comb.

9

150 160 170 180 190 200 210 220

−60

−40

−20

inte

nsity

(dB)

0 5 10 15−10

0

10

20

30

rela

tive

inte

nsity

(dB)

0 500 1000 1500−10

0

10

20

30

40

150 160 170 180 190 200 210 220

−60

−40

−20

inte

nsity

(dB)

0 5 10 15−10

0

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30

rela

tive

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nsity

(dB)

0 500 1000 1500−10

0

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−20

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(dB)

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tive

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nsity

(dB)

0 500 1000 1500−10

0

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40

150 160 170 180 190 200 210 220

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−40

−20

inte

nsity

(dBm

)

0 5 10 15−10

0

10

20

30

rela

tive

inte

nsity

(dB)

RBW: 20kHz

0 500 1000 1500−10

0

10

20

30

40

rela

tive

inte

nsity

(dB)

RBW: 50kHz

150 160 170 180 190 200 210 220

−60

−40

−20

inte

nsity

(dBm

)

4.73 4.74 4.75

−120

−100

−80

−60in

tens

ity (d

Bm)

RBW: 20kHz

0 500 1000 1500−10

0

10

20

rela

tive

inte

nsity

(dB)

RBW: 50kHz

150 160 170 180 190 200 210 220

−60

−40

−20

frequency in THz

inte

nsity

(dBm

)

−0.0001 0 0.0001−130

−120

−110

−100

−90

−80

frequency (GHz)

inte

nsity

(dBm

)

RBW: 1kHz

0 500 1000 1500−10

0

10

20

frequency (MHz)

rela

tive

inte

nsity

(dB)

RBW: 10kHz

RBW: 20kHz

RBW: 20kHz

RBW: 20kHz

RBW: 50kHz

RBW: 50kHz

RBW: 50kHz

Hig

h N

oise

Low

Noi

se S

olit

on S

tate

s

rela

tive

inte

nsity

(dB)

rela

tive

inte

nsity

(dB)

rela

tive

inte

nsity

(dB)

A B C

D E F

G H J

K L M

N O P

Q R S

Fig. S4: Comparison of a high noise state and solition states in a SiN microresonator.(A), (D), (G), (K) The optical spectra of the high noise state for the same resonance as allother spectra and with similar pump power when tuning the pump further into the resonance(from blue to red). The spectrum in (K) shows a clearly flattened maximum of the solitoninduced Cherenkov radiation at around 156 THz and two maxima to the left and right of thepump at 192.2± 3 THz. The green solid line outlines the spectral envelope derived fromsimulations. Reasons of the deviation between simulation and experiment are explained inthe text. (B),(E),(H),(L) The beat note of a comb line with a narrow linewidth fiber laserat 1552 nm (green vertical ine in (A)) in the RF domain (shifting from around 2.5 GHz in(E) to around 11 GHz in (L)) is very broad and structured. (C),(F),(J),(M) The amplitudenoise of the light after the chip measured up to 1.5 GHz is elevated. (N) Optical spectrumof a four soliton state in the same resonance. The state is low noise. (O) The beat notewith a narrow linewidth fiber laser at 1552 nm is well defined. (P) Amplitude noise. (Q)The spectrum for a two-soliton state shows a clear pattern and is also of low noise. (R) Therepetition rate beat note is narrow and well defined. The data is offset by 189.2 GHz. (S)The amplitude noise in the transmission only shows technical noise at certain frequencies.For all axis labeled “relative intensity” the background has been subtracted. The absolutelevels of the background are (M): –99.6 dBm, (P): –98.5 dBm, (S): –110.8 dBm.

10

Frequency Comb Generation

Pump Laser Lock

Transmission Monitor

Heterodyne Measurement

RF Beatnote Measurement

ESA

ECDL

Fiber LaserComb

Laser Lock

OM & EDFA

AFGFPC

ESA

Local Oscillator17.9 GHz

VFPDRF Ref FPC

40 GHz HighPower RF

MZAMHM

×6

TLF

1907nmFiber Laser

1552nmFiber Laser

OBPFESA

FBG

Scope

ESA

FBG

OSA

PDs

SiN Microresonator

OBPF

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

norm

aliz

ed g

ener

ated

ligh

t int

ensi

ty

time in μs

A

12

34

56

78

B

Fig. S5: Soliton steps and schematic experimental setup. (A) A color-coded histogram(red denotes high probability, dark blue denotes zero probability) of the recorded steps inthe intensity of the converted light (light that is not at the wavelength of the pump) ver-sus laser detuning, revealing steps between different soliton states inside the cavity. Thewhite numbers indicate the number of solitons for each state. The transmitted light fromthe resonator is filtered for the pump and detected, while the laser is scanned through thecavity resonance of the SiN microresonator. (B) The setup for microresonator based soli-ton generation. The microresonator is pumped with CW laser light from an external cavitydiode laser (ECDL) that is amplified and modulated in power (bright red box). The remain-ing setup is for characterization and stabilization only. AFG, arbitrary function generator;EDFA, erbium-doped fiber amplifier; ESA, electrical spectrum analyzer; FBG, fiber Bragggrating; FPC, fiber polarization controller; HM, harmonic mixer; MZAM, Mach-Zehnderamplitude modulator; OBPF, optical band-pass filter; OM, optical modulators; OSA, opti-cal spectrum analyzer; PD, photodiode; TLF, tapered-lensed fiber; VFPD, very fast photo-diode.

11

10−3

10−2

10−1

100

101

10−3

10−1

101

103

105

gate time (s)

mod

ified

Alla

n D

evia

tion

Out−of−loopRepetition ratePump offsetReference

−40 −20 0 20 400

50

100

150fool: 32.01250317 MHz ± 484mHz

Std dev: 8.2Hz

deviation (Hz)

coun

ts

A B

Fig. S6: Full stabilization of a single dissipative Kerr soliton in SiN. (A) Same dataas in Fig. 4A but for a single soliton state. Histogram of the counter measurement for theout-of-loop beat of the stabilized microresonator frequency comb with a commercial fiberlaser frequency comb. The Gaussian fit gives the exact frequency of the beat. (B) Themodified Allan deviation of the out-of-loop beat as well as the in-loop signals for the twolocks of the repetition rate and the pump laser offset of the microresonator frequency comb.All signals average down over the gate time as it should be for coherent signals.

Time (min)

10 20 30 40 50 0 0.5 1

FFT

(ps)

5

7

9

11

A B

AU

Fig. S7: Passive longterm stability of a two-soliton state. (A) Color coded field autocor-relation of a two soliton state over time. Each vertical slice is the Fourier transform of oneintensity spectrum taken every minute. The peaks in the autocorrelation (bright blue andred horizontal lines) do not drift over time and jitter only very little. (B) One individualautocorrelation at 50 min, marked in (A) with a white line.

12

−200 −100 0 100 200−10

0

10

20

30

40

50

60

relative mode number µ around 192.2 THz

Din

t (G

Hz)

810nm height − interpolated, simulated data810nm height − 4th order polynomial �t800nm height − interpolated, simulated data800nm height − 4th order polynomial �t

−200 −100 0 100 200−20

−15

−10

−5

0

5

relative mode number µ around 192.2 THz

D/2π

(MH

z)

directly from simulationsfrom D2 , D3 and D4 from �t above

150 160 170 180 190 200 210 220 230

frequency (THz)

2

A

B

Fig. S8: Simulated dispersion and fitted dispersion parameters. (A) The dispersion astaken from finite element simulations for two waveguide geometries with a base width of1.77µm, a sidewall angle of 9.44◦, a radius of 119µm and a waveguide heights as indicatedin the legend. Also shown are the fits with fourth degree polynomials in order to derive thedispersion parameters D2, D3 and D4. The exact values of the fit for the height of 800 nmare: D2 = 2.5943 MHz, D3 = 24.4751 kHz and D4 = −289.5820 Hz which we also usein the main text. The data for the 800 nm heigh waveguide is the same as the one shownin Fig. 2B. (B) The dispersion parameter D2 for the waveguide height of 800 nm plottedover mode number µ and frequency. One curve is taken directly as a numerical differencefrom the interpolated simulated data, the other curve uses only the dispersion parametersfrom the fit shown in (A). In our simulations the variation of D2 over the mode number iscaptured in D3 and D4.

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Brasch-SM.cover.page.pdfPhotonic chip–based optical frequency comb using soliton Cherenkov radiation

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