Nature - USA (2020-05-14)

Nature | Vol 581 | 14 May 2020 | 167

detuning between the cavity resonance and the continuous-wave pump
laser^26.
We next perform numerical simulations based on the Lugiato–Lefe-
ver equation^27 ,^28 , which demonstrate the ability of the DKS state to
transfer the chirp from the pump laser to all comb teeth (see Fig. 2c).
The numerical laser scan is started at Δ = −0.4 GHz and the detuning Δ is
subsequently increased with a linear chirp rate of |dΔ/dt| = 4 × 10^15  Hz^2 ,
tuning past the modulation instability region and exciting a single
soliton. Hereafter, the linear laser scan is inflected and a symmetric
triangular frequency modulation with equal chirp rate is contin-
ued. If stimulated Raman effects^25 ,^29 and higher-order dispersion are
neglected, the repetition rate remains almost perfectly constant and
the frequency chirp is faithfully transduced to each comb line. Even
more surprisingly, the inclusion of stimulated Raman scattering and
third-order dispersion effects induces only a small repetition rate mis-
match Δfrep of 20.6 MHz per 1.7 GHz of laser tuning, which is observed
as acceleration and deceleration of the soliton in the cavity (see Fig. 2d,
bottom). The fundamental limit for the tuning speed dΔ/dt < (κ/2π)^2  is
set by the cavity photon decay rate κ.
The linear dependence dfrep/dΔ ≈ (ΩR/2πΔ)(D 2 /D 1 ), where D 1 /2π is the
cavity free-spectral range and D 2 /2π is the second-order dispersion,

results in a channel-dependent frequency excursion Bμ and, hence,

a constant rescaling factor of the measured lidar distance, which we

can determine during calibration. Only nonlinear dependencies of

the pulse repetition rate frep on the detuning Δ, from either the Raman

shift^30 or multimodal interactions^29 , actually degrade the linearity of

the transduced chirp. The maximum detuning, which still supports

stable DKS generation, is determined by the input pump power^10 , which

in turn is fundamentally limited by a Raman instability^31.

Characterization of parallel FMCW lidar source

Next, we experimentally demonstrate the ability to faithfully transfer

the pump laser chirp to the individual comb teeth (see Fig.  3 ). Details of

the experimental setup for heterodyne characterization, linearization

of the triangular frequency-modulation patterns, and transduction data

analysis are described in the Methods. Results for the comb tooth at

195 THz (μ = +20) and modulation frequencies 1/T from 100 kHz to 10

MHz are depicted in Fig. 3b and in Extended Data Fig. 3. The frequency

excursion bandwidth Bμ increases linearly with the channel number μ

(see Fig. 3e) at a rate of dBμ/dμ = 22.15 MHz in agreement with the predic-

tions from numerical simulations including the Raman self-frequency

FPC

CW reference

50/50 DSO

FPC Si 3 N 4

EDFA

AFG

PMSSB

CW pump DEMUX

OSA

VNA

Modulated microcomb generator Heterodyne measurements

–50– 40 –30– 20 –1 00

Relative intensity (dBc Hz–1)

–2

0

2

–2

0

2

–2

0

2

Frequency (GHz)

–2

0

2

Time

–2

0

2

Normalized modulation response (10 dB per div)

104 105 106 107 108

Modulation frequency (Hz)

192.1 THz

192.5 THz

193 THz, Pump

193.5 THz

194 THz

194.5 THz

195 THz

195.5 THz

196 THz

192 193 194 195 196

1.2

1.6

2.0

2.4

192 193 194 195 196

0

5

Frequency

excursion (GHz)

Frequency (THz)

DeviationRMS(MHz)

Frequency (THz)

Frequency (GHz)

Deviation (MHz)

–1.0

–0.5

0

0.5

1.0

1.5

0102030

Time (μs)

–5

0

5

192.1 THz 193 THz, Pump 194 THz

40 50

a

cd

b

e

B 20 = 2.196 GHz

100 kHz

300 kHz

1 MHz

3 MHz

10 MHz

T = 100 ns

T = 333 ns

T = 1 μs

T = 3.3 μs

T = 10 μs

RBW 9.6 MHz

RBW 19 MHz

RBW 39 MHz

RBW 77 MHz

RBW 157 MHz

Fig. 3 | Time–frequency analysis of a chirped soliton microcomb.
a, Experimental setup. An amplified external cavity diode laser laser at 193 THz
generates a soliton microcomb on the photonic chip. Frequency modulation is
applied with a single sideband modulator (SSB). The time-dependent sideband
frequencies are detected by beating with a second tuneable amplified external
cavity diode laser. The optical spectrum analyser (OSA) and the vector network
analyser (VNA) are used for soliton state characterization (see Fig.  2 ), only.
DSO, Digital sampling oscilloscope; EDFA, erbium-doped fibre amplifier;
FPC fibre polarization controller; PM, phase modulator; RBW, resolution
bandwidth. b, Time–frequency maps of 1.6-GHz pump laser chirps at
modulation frequencies from 10 kHz to 10 MHz, detected at the μ = +20 comb
sideband (195 THz); B 20 is the frequency excursion at sideband μ = +20. c,

Instantaneous frequency of the heterodyne beat note (top) determined by

short-time Fourier transform. Deviation from a perfect triangular scan

calculated by least-squares fitting (bottom) at modulation frequency 100 kHz.

d, Pump to sideband frequency-modulated transduction determined from the

frequency-modulated amplitude of the first nine harmonics of the modulation

frequency of linearized frequency-modulated traces between 10 kHz (dark

markers) and 10 MHz (light markers). Sideband values (colours as in e) are

offset by 10 dB and normalized with respect to the modulation amplitudes of

the pump (grey marker). e, Channel-dependent frequency excursion

bandwidth at 100-kHz modulation frequency. The inset shows the

root-mean-square (RMS) deviation from the perfectly triangular modulation

pattern.