## Timing on a chip

Laser-induced optical frequency combs allow precision measurements and affect a broad range of technologies. Brasch *et al.* generated optical frequency combs on an optical chip (see the Perspective by Akhmediev and Devine). They induced an optical soliton, or optical bullet, and propagated it in an engineered microcavity waveguide. The emitted output light formed a coherent comb of frequencies spanning two-thirds of an octave. Such an on-chip demonstration bodes well for miniaturization of metrological technology and its adaption for widespread application.

## Abstract

Optical solitons are propagating pulses of light that retain their shape because nonlinearity and dispersion balance each other. In the presence of higher-order dispersion, optical solitons can emit dispersive waves via the process of soliton Cherenkov radiation. This process underlies supercontinuum generation and is of critical importance in frequency metrology. Using a continuous wave–pumped, dispersion-engineered, integrated silicon nitride microresonator, we generated continuously circulating temporal dissipative Kerr solitons. The presence of higher-order dispersion led to the emission of red-shifted soliton Cherenkov radiation. The output corresponds to a fully coherent optical frequency comb that spans two-thirds of an octave and whose phase we were able to stabilize to the sub-Hertz level. By preserving coherence over a broad spectral bandwidth, our device offers the opportunity to develop compact on-chip frequency combs for frequency metrology or spectroscopy.

Optical solitons are propagating pulses of light that retain their temporal and spectral shape as the result of a balance between nonlinearity and dispersion (*1*). In the presence of higher-order dispersion, optical solitons can emit soliton Cherenkov radiation (*2*, *3*). This process, also known as dispersive wave generation, is one of the key nonlinear frequency conversion mechanisms of coherent supercontinuum generation (*4*), which allows a substantial increase in the spectral bandwidth of pulsed laser sources. The generation of a coherent supercontinuum from a pulsed laser propagating through a photonic crystal fiber has enabled the first self-referenced optical frequency combs (*5*, *6*) and has given access to coherent broadband spectra for frequency combs with repetition rates up to ~10 GHz.

One route to broadband frequency combs with higher repetition rates was established with the discovery of microresonator (Kerr) frequency combs (*7*, *8*). Since then, the field of microresonator frequency combs has made substantial advances (*9*–*11*), including frequency comb generation in complementary metal-oxide semiconductor (CMOS)–compatible silicon nitride (Si_{3}N_{4}, henceforth SiN) photonic chips (*12*, *13*) and a detailed understanding of the comb formation process (*14*–*17*). However, it has been a challenge to achieve broadband frequency combs that are coherent (*18*, *19*). Recently, temporal dissipative Kerr solitons (DKSs)—analogous to dissipative cavity solitons (*20*, *21*) in fiber loop cavities—have been observed in crystalline microresonators (*19*), leading to coherent frequency combs. These solitons, which balance dispersion and loss via the Kerr nonlinearity, can be generated spontaneously from chaotic Kerr frequency combs when tuning the pump laser through the cavity resonance (*19*). Recent numerical simulations (*16*, *17*, *19*) have predicted that such solitons in the presence of soliton Cherenkov radiation (*3*, *16*, *22*) can provide a path to the reliable generation of broadband and coherent frequency combs, which can even span a full octave.

The resonance frequencies of one mode family in a microresonator can be approximated around ω_{0} as a Taylor series: (1)where μ ∈ Z is the relative mode number, *D*_{1}/2π is the free spectral range of the resonator, *D*_{2} is related to the group velocity dispersion (GVD) parameter β_{2} by , and *D*_{3}, *D*_{4}, … are related to higher-order dispersion. Figure 1E shows the integrated dispersion *D*_{int}(μ) relative to the pump mode at μ = 0; that is,
(2)When pumping a microresonator with a continuous wave (CW) laser with frequency ω_{P} near ω_{0}, the dynamics of this system can be described by a master equation, (3)(*14*, *17*, *22*), where (4)is the slowly varying field amplitude; φ is the azimuthal angular coordinate inside the resonator, corotating with a soliton; is the nonlinear (per photon) Kerr coupling coefficient [where *ħ* is Planck’s constant divided by 2π, *c* is the speed of light in vacuum, *n* and *n*_{2} are the linear and nonlinear (Kerr) refractive indices of the material, and *V*_{eff} = *A*_{eff}*L* is the effective nonlinear mode volume (where* A*_{eff} is the effective nonlinear mode area and *L* is the cavity length)]; κ is the cavity decay rate; η is the coupling efficiency; and *P*_{in} is the pump power inside the bus waveguide. Formally, this equation is identical to the Lugiato-Lefever equation (*16*, *17*) (a damped, driven nonlinear Schrödinger equation). For anomalous GVD and in the absence of third- and higher-order dispersion, approximate solutions can correspond to bright temporal solitons superimposed on a CW background: (5)with φ* _{j}* corresponding to the relative angular position of the

*j*th soliton. Amplitude

*A*

_{1}, phase ψ

_{0}, and background

*A*

_{CW}are determined by the system’s parameters. The minimal pulse duration is given by (6)(

*19*), where is the resonator finesse and γ = ω

*n*

_{2}/

*cA*

_{eff}. These temporal dissipative Kerr solitons have been generated in fiber cavities (

*21*) and have been observed in crystalline microresonators recently (

*19*). When higher-order dispersion terms are present, the shape and velocity of the stationary solitons change as they develop radiative tails (

*3*,

*23*,

*24*). The spectrum of such a perturbed soliton becomes asymmetric, with its maximum shifted away from the pump frequency and an additional, local maximum (Fig. 1E) is generated (also called a dispersive wave). Because the radiative tail is emitted from the soliton, an analogy to Cherenkov radiation can be drawn (

*3*).

The spectral position of the Cherenkov radiation is approximately given by the linear phase-matching condition (*2*, *25*) *D*_{int}(μ_{DW}) = 0 at μ_{DW} = (–3*D*_{2}/*D*_{3}) for *D*_{4} = 0. In the presence of *D*_{4}, two peaks of Cherenkov radiation may occur at (7)Our experimental platform is based on silicon nitride optical microresonators, which are very suitable for nonlinear optical applications (*13*, *26*). We used SiN ring resonators (thickness 800 nm, diameter 238 μm) embedded in SiO_{2} (Fig. 1, A to D), resulting in anomalous GVD for wavelengths around 1.5 μm. The microresonator fabrication was optimized so as to mitigate avoided crossings of different mode families that can locally alter dispersion (*27*, *28*). Measurements of the dispersion (*28*) revealed that around the pump wavelength, the mode structure closely approaches a purely anomalous GVD (Fig. 2B and fig. S2), with a measured *D*_{2}/2π = 2.4 ± 0.1 MHz, in close agreement with finite element method modeling that yields *D*_{2}/2π = 2.6 MHz (*28*). When pumping the resonator’s TM_{00} mode family at 1560 nm via the bus waveguide, we observed discontinuities in the cavity transmission and converted frequency comb light (figs. S1 and S5A) as well as a narrowing of the repetition rate beat note (fig. S1, C and D), signatures previously associated with dissipative Kerr soliton formation (*19*).

To access the soliton states in a steady state, we developed a laser tuning technique to overcome instabilities associated with the discontinuous transitions of the soliton states (*28*), allowing stable soliton operation for hours (fig. S7). The optical single-soliton spectrum with *P*_{in} ≈ 2 W shown in Fig. 2A has several salient features: (i) It covers a bandwidth of two-thirds of an octave. (ii) It exhibits the characteristic sech^{2} spectral envelope near the pump that is associated with temporal solitons. The 3-dB bandwidth of 10.8 THz corresponds to 29-fs optical pulses. (iii) The sharp feature around 1930 nm (155 THz) corresponds to the soliton Cherenkov radiation (*16*, *22*). Figure 2B shows the measured and simulated dispersion. The spectral position of the Cherenkov radiation at μ *= –*195 is in good agreement with the linear phase-matching condition that occurs for *D*_{int}(μ_{DW}) = 0 at μ_{DW} = −200 with the simulated parameters *D*_{2}*/*2π *=* 2.6 MHz, *D*_{3}*/*2π *=* 24.5 kHz, and *D*_{4}*/*2π *= –*290 Hz (fig. S8).

Also shown in Fig. 2A is a numerically simulated spectrum [based on coupled mode equations (*28*)]. It shows only small deviations from the experimental spectrum, caused by effects that are not included in the simulations (*28*). In particular, the absence of the soliton recoil (Fig. 1E), which is associated with the formation of a dispersive wave (*23*, *24*), is attributed to the cancellation via the soliton Raman self-frequency shift (*28*, *29*). The good agreement with the experimental data establishes numerical simulations as a powerful predictive tool for soliton dynamics in microresonators.

To investigate a key property of a frequency comb, its coherence, we first measured the repetition rate beat note of 189.2 GHz on a photodiode by means of amplitude modulation downmixing (*28*, *30*) (fig. S5B). Figure 2C shows the resulting beat note, which exhibits a narrow linewidth and a signal-to-noise ratio of 40 dB in 100 kHz bandwidth, demonstrating the coherent nature of the spectrum. We also recorded the low-frequency intensity noise of the transmitted light of the soliton state and found no excess noise relative to the pump laser noise (fig. S4). To locally investigate the coherence of the Cherenkov radiation, we carried out additional CW heterodyne beat note measurements at 1907 nm, which exhibited a narrow linewidth around 1 MHz (Fig. 2E). Simultaneously with the beat at 1907 nm, we measured the beat with a laser at 1552 nm. The resulting beat note was similar in width to the in-loop beat of the frequency-stabilized pump laser (~300 kHz). These measurements prove that the entire spectrum is coherent, in contrast to earlier reports (*18*). It is useful to contrast the single-soliton state to the incoherent high-noise state; we observed in the high-noise case a spectrum that markedly deviates from the single-soliton spectrum in terms of the shape of the Cherenkov radiation peak and the shape of the spectrum around the pump (fig. S4K).

Our system also allows us to access states with multiple solitons in the resonator. Figure 3, A to C, shows the optical spectra of three multisoliton states, which are coherent (fig. S3) and stable for hours (fig. S7). The generated spectra show pronounced variations in the spectral envelope that arise from the interference of the Fourier components of the individual solitons, as described by the spectral envelope function
(8)The insets of Fig. 3, A to C, show the reconstructed relative positions of the solitons inside the resonator for the different spectra (*28*). Figure 3B shows the case where two solitons are almost perfectly opposite to each other in the resonator, resulting in an effectively doubled line spacing. Figure 3C shows that a higher number of cavity solitons (*N* = 3) results in a spectrum with more complex spectral modulations.

To prove the usability of our system for metrological applications, we implemented a full phase stabilization of the spectrum by phase-locking the pump laser and the repetition rate of the SiN comb to a common radio-frequency reference. For absolute frequency stabilization of the pump laser, we used an offset lock to a self-referenced fiber laser frequency comb (*28*, *31*). In Fig. 4B, we show the modified Allan deviation of the in-loop signals and an out-of-loop signal that consists of the beat of one comb tooth of the SiN comb (mode number –18) with one tooth of the reference comb. For all three signals, the modified Allan deviation averages down with increasing gate time.

The out-of-loop measurement also allows us to compare the absolute frequency accuracy of the soliton Cherenkov radiation–based comb state with the fiber laser reference comb. Taking into account all locked frequencies as shown in Fig. 4C, and extracting the center frequency of the out-of-loop signal from frequency counter measurements shown in Fig. 4A, we derive a frequency difference of Δ = (18 × *f*_{rep})_{}– (13,613 ×* f*_{rep,fc}) – *f*_{off} +* f*_{ol} = 25 ± 558 mHz for the 1000-s measurement. We therefore validate the accuracy of the SiN soliton frequency comb to the sub-Hz level and verify the relative accuracy (with respect to the optical carrier) to 3 × 10^{−15}.

The observation of soliton Cherenkov radiation in a photonic chip–based microresonator provides a path to numerically predictable, fully coherent frequency comb spectra, with increased bandwidth that extends into the normal GVD regime. The currently achieved coherent two-thirds of an octave can be self-referenced by doubling and tripling the high and low end of the spectrum, respectively (*2f*-*3f* technique), and the bandwidth can be extended to a full octave with modified dispersion designs.

## Supplementary Materials

www.sciencemag.org/content/351/6271/357/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S8

## References and Notes

**Acknowledgments:**Supported by European Space Agency contracts ESTEC CN 4000108280/12/NL/PA and ESTEC CN 4000105962/12/NL/PA, the Swiss National Science Foundation, and contract W911NF-11-1-0202 from the Defense Advanced Research Projects Agency, Defense Sciences Office. This material is based on work supported by the Air Force Office of Scientific Research, Air Force Material Command, under award FA9550-15-1-0099. M.L.G. and G.L. were supported by Russian Foundation for Basic Research grant 13-02-00271 and the Ministry of Education and Science of the Russian Federation project 4.585.21.0005. M.G. acknowledges support from the Hasler foundation and the MSCA-COFUND program at EPFL. All samples were fabricated at the Centre for MicroNanotechnology (CMi) at EPFL.