High-efficiency and broadband on-chip electro-optic frequency comb generators

Developments in integrated photonics have led to stable, compact and broadband comb generators that support a wide range of applications including communications1, ranging2, spectroscopy3, frequency metrology4, optical computing5,6 and quantum information7,8. Broadband optical frequency combs can be generated in electro-optical cavities, where light passes through a phase modulator multiple times while circulating in an optical resonator9–12. However, broadband electro-optic frequency combs are currently limited by low conversion efficiencies. Here we demonstrate an integrated electro-optic frequency comb with a conversion efficiency of 30% and an optical span of 132 nm, based on a coupled-resonator platform on thin-film lithium niobate13. We further show that, enabled by the high efficiency, the device acts as an on-chip femtosecond pulse source (336 fs pulse duration), which is important for applications in nonlinear optics, sensing and computing. As an example, in the ultrafast and high-power regime, we demonstrate a frequency comb with simultaneous electro-optic and third-order nonlinearity effects. Our device paves the way for practical optical frequency comb generators and provides a platform to investigate new regimes of optical physics that simultaneously involve multiple nonlinearities. A double-ring-resonator device on thin-film lithium niobate enables the generation of electro-optic frequency combs with a 30% power efficiency and an optical span of 132 nm.

threshold that is determined by both the Kerr nonlinearity and the quality factor (Q) of the resonator. This results in a nonlinear dependence between the comb and pump powers, which leads to saturation effects and limits the absolute comb power.
Electro-optic (EO) modulation provides an attractive alternative to OFC generation [9][10][11][12][21][22][23] . The electrical controllability of EO combs provides not only versatility but also excellent comb stability and phase coherence. EO combs based on conventional (travelling-wave) modulators feature a high pump-to-comb conversion efficiency, but their span is only a few nanometres due to the weak frequency mode interaction during a single pass through the modulator 21,22 . Wider optical combs can be generated using cavity-based EO combs in which light passes through a phase modulator multiple times while circulating inside an optical microresonator (Fig. 1a) [9][10][11][12]23,24 Recent progress on thin-film lithium niobate (TFLN) has enabled on-chip EO combs with a record-high optical span of ~80 nm (ref. 12 ). However, the comb conversion efficiency of this single-resonator EO comb generation is limited to only ~0.3%. Such a low conversion efficiency originates from a strongly under-coupled 'hot' cavity resonance (<1% extinction ratio) when the generator is driven using a strong microwave tone. As a result, most of the light is not coupled into the resonator and is transmitted through the bus waveguide without entering the cavity (Fig. 1c).
Here we address the low conversion efficiency of the cavity-based EO comb and experimentally demonstrate an on-chip EO comb with a line spacing of 30.925 GHz, a pump-to-comb conversion efficiency of 30% and a wide comb span of 132 nm. This is enabled by two mutually coupled resonators (Fig. 1d) realized in the TFLN platform 13,25 . A small resonator (cavity 1) is used to over-couple only the pump mode of a racetrack cavity (cavity 2 for comb generation) while rejecting the other non-pump modes (Fig. 1e). This results in a critically coupled device when the microwave modulation is on (Fig. 1f) and increases the conversion efficiency as theoretically predicted 26,27 . Here in this work, we use the tight-binding model, previously developed for frequency crystals 28 , as well as the generalized critical coupling (GCC) condition, developed for EO frequency shifters and beam splitters 13 , to model our coupled-resonator EO comb generator. Importantly, our theoretical approach enables the derivation of an analytical solution for the system and can be extended to multiple coupled-resonator devices that may provide additional novel functionalities. Our experimental demonstration, along with the previous investigation of coupled-resonator

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Kerr comb generation 16 , indicates that the coupled-resonator platform could be a general approach for power-efficient EO and Kerr comb sources.
The origin of the low conversion efficiency for single-resonator EO combs is the strong effective loss rate κ MW for the pump mode, induced by the microwave modulation, which extracts power from the pump mode into other comb lines. The loss rate κ MW is given in equation (1) (see the section 'Theoretical analysis' in the Methods): where Ω is the mode-coupling rate between resonator modes separated by the free spectral range (FSR) and is proportional to the microwave voltage, and κ (= κ e + κ i ) is the cavity loss rate, with κ i and κ e being the intrinsic loss and coupling rate to the bus waveguide, respectively. Here κ MW can be orders of magnitude higher than κ e and κ i (for example, κ MW ≈ 10 GHz and κ e ≈ κ i ≈ 100 MHz for TFLN). Broadband EO combs require a strong microwave driving power (large Ω), which leads to a large κ MW . This, however, results in the cavity being strongly under-coupled (κ e ≪ κ i + κ MW ), reducing the comb efficiency. This trade-off between the EO comb span and the efficiency limits all single-resonator EO comb sources. A coupled-resonator EO comb generator can overcome this trade-off and ensures efficient energy flow into the comb cavity (Fig. 1d). In this case, a critical coupling between the bus waveguide and the device can be achieved under the existence of the strong κ MW , if the following condition, referred to as the GCC condition, is met (Fig. 1e,f):

Bus waveguide
Cavity Fig. 1 | Concept of the coupled-resonator eo comb and GCC condition. a-c, Schematics of the device structure (a), energy-level description (b) and transmission spectrum (c) of the single-resonator EO frequency comb generator. EO modulation creates efficient coupling Ω between adjacent frequency modes of the cavity (b). All the frequency modes are coupled to the input bus waveguide. The rate κ e is the waveguide-cavity coupling rate. When the microwave signal is off, the pump resonance is nearly critically coupled but becomes strongly under-coupled when the microwave signal is turned on. MW, microwave; ω 0 , resonance frequency of the mode pumped by the laser; ω MW , frequency of the microwave signal. d-f, Schematics of the device structure (d), energy-level description (e) and through-port transmission spectrum (f) of the coupled-resonator EO frequency comb generator. The inclusion of cavity 1 effectively causes an over-coupling between the pump mode of cavity 2 and the bus waveguide, while rejecting other frequency modes of cavity 2 (e). Such coupled-ring structures can achieve a GCC condition, resulting in a high extinction ratio of the pump resonance when the microwave signal is turned on (f). The rate κ e1 (κ e2 ) is the coupling rate between the waveguide and cavity 1 (2). The coupling rate between cavities 1 and 2 is μ. The terms κ i1 and κ i2 are the intrinsic loss rates of cavity 1 and cavity 2, respectively.

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where κ e1 and κ e2 are the waveguide coupling rates between the bus waveguide and cavity 1 and between the output waveguide and cavity 2, respectively, κ i1 and κ i2 are the intrinsic loss rates for cavities 1 and 2, respectively, and μ is the coupling rate between cavities 1 and 2. The term 4μ 2 κ e2 +κ i2 +κ MW ≡ κ 1eff can be interpreted as an effective loss rate of cavity 1 that is induced by cavity 2, the microwave modulation and the output waveguide (drop port). With the expression for κ MW (equation (1)), the coupled-resonator EO comb generator, which involves hundreds of frequency modes, can be simplified to a two-level system that can be solved for analytically. See the detailed discussion regarding the expression, and the maximum theoretical limitation, of efficiency in the 'Theoretical analysis' section.
To experimentally demonstrate the coupled-resonator comb generator, we fabricated TFLN-on-insulator devices (Fig. 2a). The small ring resonator and the long racetrack resonator are used as cavities 1 and 2, respectively. A microwave signal is sent to the electrode of cavity 2 to provide phase modulation. A thermal heater is used in cavity 1 for efficient resonance tuning.
We first demonstrate high-efficiency and broadband EO frequency comb generation. Continuous-wave light at 1,605 nm is fed into the device through the bus waveguide. A 30.925 GHz microwave signal, which matches the FSR of cavity 2, is used to drive the electrode. By applying microwave modulation, the transmission of the through port changes from over-coupled to nearly critically coupled (inset of Fig. 2c), indicating the efficient flow of pump power into the device. The frequency comb is collected at the drop port of the device (Fig. 1d) and measured using an optical spectrum analyser and photodetectors. The output comb power increases linearly with the pump power (Fig. 2b), indicating a pump-to-comb conversion efficiency of η ≡ P comb Ppump = 30%, where P comb and P pump are the power values on the output and input waveguides, respectively. Using 2 mW of pump power (the same pump power as used in ref. 12 ), the comb spans 132 nm at a −70 dBm power level and features an efficiency η = 30% (Fig. 2c). Our device can be pumped at any of the resonances of cavity 2 by tuning the heater to overlap the resonance of cavities 1 and 2 (see Extended Data Fig. 1). The current pump wavelength is ~1,600 nm. It is chosen to avoid the polarization mode crossing at ~1,400 nm, which can be suppressed via dispersion engineering (see Extended Data Fig. 2). Compared with state-of-the-art integrated EO combs 12 , our method yields a two-orders-of-magnitude improvement in the conversion efficiency and a 2.2 times wider span is measured at the −70 dBm power level with the same pump power (Fig. 2c). It should be noted that in the single-resonator TFLN EO comb generator 12 , the pump intensity is 40 dB higher than the first comb line due to the low efficiency from the strong under-coupling of the resonator (purple trace in Fig. 2c). For our coupled-resonator device, the output spectrum in the through port (residual pump) is discussed later in Fig. 3. . Cavity 1 is controlled using a thermal heater (light yellow) and cavity 2 is modulated by a microwave signal applied to gold electrodes (orange). b, Measured comb power versus pump power, indicating a pump-to-comb conversion efficiency of 30%. The data points labelled as blue circles and the green diamond represent data from two different devices with the same parameters. Data points denoted as blue circles are for comb shapes similar to the comb in c, and the green diamond represents the nonlinear comb in the ultrafast high-power regime (see Fig. 4d). The error bars for each value of the pump power are due to Fabry-Perot fringes induced by the two facets of the chip. Each data point is obtained by averaging the maximum and minimum pump power values in the fringes. Therefore, the pump power with an error range of P in,max − P in,min . The theoretical curve is obtained based on the conversion efficiency formula η = P 2 P in × ξ (see 'Theoretical analysis' section). c, Optical spectra of the coupled-resonator EO comb generator (blue, this work) and the single-resonator structure (purple, ref. 12 ). both spectra show the on-chip power. This is accomplished by taking into account the different in-and out-coupling facet losses to compare the spectra on the same footing, that is, for the same on-chip pump power of 2 mW. At a −70 dbm power level, the span of the coupled-resonator comb is 132 nm, whereas the span of the single-resonator EO comb 12 is 60 nm. The inset shows the transmission spectrum of the through port (see Fig. 1d) when the microwave signal is off (magenta) and on (cyan). The span between two shallow transmission dips (indicated by the grey vertical lines with arrows) is the conventional resonance broadening of the EO comb generator, which gives the coupling rate 2Ω = 2 × 2π × 7.4 GHz as well as the modulation index

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To compare the performance between the existing frequency comb generators, we summarized the metrics of demonstrated EO and Kerr frequency combs in Table 1.
The conversion efficiency η can be understood by decomposing η into two components: P in is the ratio between the optical power P 2 that flows into cavity 2 and the input power P in , and ξ = κ e2 κ e2 +κ i2 with κ e2 and κ i2 being the waveguide-cavity 2 coupling rate and the intrinsic loss rate of cavity 2, respectively (see 'Theoretical analysis' section). The power ratio θ is determined by the GCC condition. When the GCC condition is met and when κ e1 ≫ κ i1 , we have θ ≈ 1. The factor ξ describes how much power from cavity 2 is coupled into the output waveguide, which sets the theoretical upper limit for the conversion efficiency η max for each device. It also generates a fundamental trade-off between the comb span and the conversion efficiency: for large ξ the comb generated inside cavity 2 can be efficiently coupled to the output waveguide, but at the same time the slope of the comb spectrum is increased due to the reduced lifetime of cavity 2, which leads to a lower loaded-quality factor of cavity 2. We theoretically obtained η max and the slope of the comb for a range of κ e2 and intrinsic quality factor Q i2 for cavity 2 (which corresponds to κ i2 ) to show this trade-off (Fig. 3a). Our comb (Fig. 2b) features a slope of −0.7 dB nm −1 and a measured efficiency of 30% (η = 28% in theory and simulation), which is close to η max = 38% of our device.
With an improved quality factor of 1 × 10 7 , it is possible to obtain η max = 83% with a slope of −0.27 dB nm −1 or η max = 50% with a slope of −0.09 dB nm −1 . The difference between the achieved efficiency   Fig. 3 | theoretical analysis of conversion efficiency and the experimental optimization of mode-crossing effect. a, The comb slope versus the conversion efficiency η for different waveguide-cavity 2 coupling κ e2 and intrinsic quality factor Q i2 values. Each curve is generated by calculating η max and the slope with κ e2 varied from 0 × κ i2 to 10 × κ i2 . The intrinsic loss rate κ i2 is determined via the intrinsic quality factor Q i2 . The rate κ i2 (as well as Q i2 ) is fixed for data points in each curve and varied for data points between different curves. Experimental data points in this figure represent the measured efficiency η, while theoretical data points use the theoretical upper limit of the efficiency η max . b, Illustration of the Vernier effect for optimizing mode-crossing-induced loss using an energy-level diagram. Cavity 1 is designed to have its resonance aligned with the pump frequency of cavity 2 and is misaligned across a wide range of other frequencies. This overcomes the losses that would otherwise be induced by mode crossing. The through-port comb line will have a higher power close to the overlap area (grey shaded region in c) and vice versa. c, Output spectrum from the through port and drop port of the device shown in Fig. 2c. The shape of the through-port spectrum reflects how the Vernier effect of the cavity resonances of two rings affects the output comb in both the through port and the drop port. The final optimized device FSr values are 254.54 GHz (cavity 1) and 30.925 GHz (cavity 2).

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η and the theoretical efficiency η max is due to the imperfect GCC condition, the non-zero value of κ i1 and the mode-crossing-induced loss due to cavity 1 (see below).
Cavity 1 has a large number of frequency modes that can lead to mode crossing with the modes of cavity 2, potentially causing comb power loss. To overcome this, we use an optimization algorithm to design the FSR values of cavities 1 and 2 so that they have overlapping resonances at the pump frequency only and misaligned resonances for most other frequencies (Vernier effect, Fig. 3a). To verify this, we collect the generated combs from both the through port and the drop port. The comb profile measured at the through port clearly shows the optimized Vernier effect (top third of Fig. 3b) and the spectrum at the drop port preserves the linear slope (bottom third of Fig. 3b) without cut-off 26 , indicating that the mode-crossing loss is minimized. The profile of the comb collected at the through port is the result of the comb generation inside cavity 2 followed by 'cavity filtering' in cavity 1. This signal can also be useful for, for example, heterodyne measurements, as local oscillators or clock references.
Next, enabled by the high-efficiency and wide comb span, we demonstrate that the device can serve as an on-chip ultrafast pulse source, which is important for nonlinear applications such as broadband parametric frequency conversion and optical atomic clocks. To characterize the time-domain signal of the output frequency comb, the optical output is sent to an erbium-doped fibre amplifier (EDFA) and dispersion-compensating fibre (DCF) followed by a second-harmonic generation-based intensity autocorrelator (see Fig. 4a for the setup). The full-width at half-maximum (FWHM) of the autocorrelator trace is 1.072 ps, which corresponds to a pulse FWHM of 536 fs (Fig. 4b). We infer that the pulse FWHM in the output facet of the chip is around 336 fs after extracting the total dispersion of the fibre output path of 36 fs nm −1 . The two sidelobes near the Lorentzian-shaped pulse are caused by the non-uniform gain coefficient of our EDFA (see the section 'Pulse characterization' in the Methods).
Finally, we demonstrate the observation of EO and χ (3) combined high-power frequency combs in a single device, enabled by the formation of ultrafast pulses with a high circulating peak power inside the resonator. To demonstrate this effect, a 63 mW pump power (in the bus waveguide) is used to feed the device (see the setup in Fig. 4a) and the output frequency comb features a 32% conversion efficiency, a 20 mW on-chip comb power and a broadened span of 161 nm (Fig. 4d). As a result, we infer that an estimated ~85 W peak pulse power (~1 W average power) is circulating inside the comb resonator (cavity 2) (see the section 'Frequency comb characterization' in the Methods), large enough to stimulate the additional Raman 29 and four-wave mixing effects 14 (Fig. 4e). Optimizing the dispersion of the current device from normal dispersion to anomalous dispersion could further enhance the four-wave mixing effect, which could be useful for ultrabroad comb generation. Combining χ (2) and χ (3) nonlinearities also provides an intriguing opportunity for investigating new regimes of nonlinear optical dynamics such Span defined at the −70 dbm power level. c Not reported in ref. 12 , so extracted from the raw data in ref. 12  In summary, we demonstrate high-efficiency and broadband EO frequency combs using a coupled-resonator structure. We show that it can be used as an integrated femtosecond pulse source and can

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Nature PhotoNics stimulate combined second-and third-order nonlinear processes in the ultrafast high-power regime. In addition, we provide a theoretical model that simplifies the coupled-resonator system supporting hundreds of energy levels as a two-level system. Simultaneously achieving a high-efficiency and wide span can enable a broad range of applications. For example, an already demonstrated 100-fold improvement in comb efficiency can lead to a 20 dB increase in the signal-to-noise ratio of frequency-multiplexed applications such as optical communications 1 . These advances can also reduce the optical pump power needed for the realization of optical neural networks 5,6 . In addition, the microwave-power consumption could be reduced by integrating on-chip microwave resonators with our comb source. Furthermore, the ability to generate on-chip femtosecond pulses is important for nonlinear photonics, optical atomic clocks, optical sensing and time-bin-encoded optical computing. Finally, the high conversion efficiency of our device opens the door for a generation of broad EO combs for entangled photons, broadly enabling frequency-domain quantum information processing 28,31 .

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Any methods, additional references, Nature Research reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at https://doi.org/10.1038/ s41566-022-01059-y.

Methods
Device fabrication. Our devices are fabricated from a commercial X-cut lithium niobate on insulator wafer (NanoLN), with a 600 nm LN layer, 2 μm buried oxide (thermally grown), on a 500 μm silicon handle. Electron-beam lithography with a hydrogen silsesquioxane resist followed by argon-ion-based reactive ion etching (350 nm etch depth) are used to pattern the optical layer of the devices, including the rib waveguides and micro-ring resonators. The devices are cleaned and the microwave electrodes (15 nm of titanium, 300 nm of gold) are defined using photolithography followed by electron-beam evaporation and a bilayer lift-off process. One layer of silica (SiO 2 ) (800 nm) using plasma-enhanced chemical vapour deposition is used to clad the devices. The heater (15 nm of titanium and 200 nm platinum) for cavity 1 is defined by photolithography followed by electron-beam evaporation and lift-off. The heater is designed as a short metal strip (5 μm wide) that is placed 3 μm away from the resonator on top of the SiO 2 cladding. The resistance of the heater is ~140 Ω (including parasitic resistance from routing strips). Tuning of the ring resonance via the FSR is achieved using a current of ~50 mA.
Frequency comb characterization. The measurement setup is illustrated in Extended Data Fig. 3. Telecommunication-wavelength light from a fibre-coupled tunable laser (TSL-510, SANTEC) passes through a polarization controller and is coupled to the LN chip using a lensed fibre. The output is collected using a lensed fibre and then sent to an optical spectrum analyser with a spectral resolution of 0.02 nm for characterization of the frequency comb. The microwave signal is generated from a synthesizer followed by a microwave amplifier and delivered to the electrodes on a device using an electric probe. The microwave driving power is 2.2 W. The circulating power in Fig. 4d is inferred from the comb power P comb = 20 mW in the bus waveguide, which for cavity 2 gives an intra-cavity power of: Pintra = P comb × Finesse π . The Finesse is calculated using Finesse = κ2 FSR2 = 222 MHz 30.925 GHz = 139.3. Therefore P intra = 0.9 W and the peak power of the pulse can be inferred via P peak = tRT τ Pintra = 85 W in which t RT = 32 ps and τ = 336 fs are the roundtrip time of cavity 2 and the pulse duration, respectively.
The device parameters are extracted based on measurement on the transmission spectra (Extended Data Fig. 4 and Extended Data Table 1). The parameters of cavity 1 (κ e1 and κ i1 ) and cavity 2 (κ e2 and κ i2 ) are obtained from the linewidth and extinction ratio of the resonances. The coupling μ between the two cavities is extracted by tuning the two resonances of the cavities to a degenerate point and measuring the mode splitting, which gives μ = √ ( is the coupling strength required to reach the exceptional point of the system. Pulse characterization. The output frequency comb is sent to an EDFA and a DCF followed by a second-harmonic generation-based intensity autocorrelator. The total dispersion of the optical path from the chip output facet to the autocorrelator (without the DCF) is characterized to be equivalent to the dispersion of a 12-m-long single-mode fibre (SMF-28, Thorlabs). The added DCF offsets the total dispersion to be equivalent to the dispersion of a 2 m SMF-28 fibre (36 ps nm −1 km −1 ). The FWHM of the autocorrelator trace is 1.072 ps, which corresponds to a pulse FWHM duration of 536 fs (Fig. 4b). Therefore, the pulse FWHM duration at the output facet of the chip is inferred as 336 fs via extracting the total dispersion of the optical path (2 m SMF-28).
We note that the comb spectrum changes after the light passes the EDFA (Extended Data Fig. 5) due to the non-uniform gain spectrum of the EDFA, which might change the pulse duration and pulse shape. To confirm that our measurement reflects the true pulse duration, we analysed the spectrum before and after the EDFA. In this measurement, we used a device with similar parameters and performance to the device shown in Fig. 2. The slope of the spectrum is 1.0 dB nm −1 before the EDFA. After passing the EDFA, it is difficult to get an accurate spectral slope due to the non-uniform gain of the EDFA. However, the 3 dB bandwidth remains the same (3.75 nm). Since the pulse duration is mainly determined by the 3 dB bandwidth of the frequency comb, we infer that the pulse duration should be the same for the comb before and after the EDFA. To verify this, we first calculate the Lorentzian pulse duration for our pulse with the centre wavelength of 1,553.6 nm and the 3 dB bandwidth of 3.75 nm and obtain a pulse duration of 304 fs. We then perform a numerical simulation using the Heisenberg-Langevin equations to simulate the pulse duration for the EO comb with a slope of 1.0 dB nm −1 and find a simulated pulse duration of 316 fs. Both results are consistent with the measured pulse duration of 336 fs.
We also note that the non-uniform gain of the EDFA creates a bump on the left side of the comb. However, this bump is below the 9 dB bandwidth of the comb, so it should have little effect on the pulse duration. Nonetheless, this bump could distort the shape of the pulse from a perfect Lorentzian, and in the experiment we indeed observed a distortion of the pulse shape from the Lorentzian shape (sidelobes in Fig. 4b). We also measured the dual-pulse dynamics of the EO comb with varied optical detuning (Extended Data Fig. 6).

Theoretical analysis. Effective loss rate induced by microwaves for the EO comb.
To obtain the effective loss rate that is induced by microwave modulation, κ MW , we consider the case that our cavity 2 (the comb cavity) is driven by a continuous microwave signal without the existence of the cavity 1. Then, the Hamiltonian of the system follows a single-resonator EO comb model (we set ℏ = 1 in which ℏ is the Planck constant): in which a j (a j † ) is the annihilation (creation) operator of each optical mode j, t is time, ω j represents the frequency of each frequency mode, Ω is the coupling rate due to microwave modulation and ω MW is the frequency of the microwave signal. We also assume that the frequency modes range from j = −N to j = N. Implementing the Heisenberg-Langevin equation gives a set of equations of motion for each mode a j : in which i is the imaginary unit, κ e2 , κ i2 and κ 2 (= κ e2 + κ i2 ) are the coupling rate between cavity 2 and the output waveguide, the intrinsic loss rate of cavity 2, and total loss rate of a j , respectively. The pump power and frequency are denoted by α in and ω L , respectively. We use the Kronecker delta function δ j,0 to indicate that only the zeroth mode is pumped. Therefore, we write ω j as ω j = ω 0 + j × FSR in which ω 0 is the resonance frequency of the zeroth mode. By changing rotating frames for each mode aj → aje −iωLt e −ijωMWt , we obtain the simplified equations of motion:ȧ in which Δ = ω L − ω 0 and δ = ω MW − FSR are the laser detuning and the microwave detuning, respectively. The steady state of such a system can be analytically solved. Considering the case that Δ = δ = 0, the equations of motion become Note that the equation of motion for the mode with the largest mode number is which gives a relation aN = −i Ω κ2 aN−1. As a result, the equation for a N−1 becomes ) aN−2. By iterating the above steps, we obtain the relation for an arbitrary mode a l with l representing an arbitrary mode number.
where the total number of the factor Ω 2 κ 2 2 is N − l. As a result, the equation of motion for the zeroth mode is Simplifying this equation gives .
Conversion efficiency analysis using the generalized critical coupling condition. With the expression for κ MW , the single-resonator EO comb system can be simplified to a single mode, which has three loss rates κ e2 , κ i2 and κ MW . Therefore, the steady-state equations for a coupled-resonator EO comb are in which d is the pump mode of cavity 1, μ is the evanescent coupling rate between cavities 1 and 2, and κ e1 , κ i1 and κ 1 (= κ e1 + κ i1 ) are the coupling rate between cavity 1 and input waveguide, the intrinsic loss rate of the cavity 1, and the total loss rate of d, respectively. Note that in the above equation, cavity 1 is pumped instead of cavity 2. Hence, the effective loss rate κ 1eff for cavity 1 that is induced by cavity 2 and the output waveguide of cavity 2 is and a critical coupling condition occurs when κe1 = κi1 + 4μ 2 κ2+κMW . The output in the through-port waveguide is dout = αin + ) . The amplitude that is lost in the intrinsic loss of cavity 1 is di = √ κi1d = αin 2 √ κe1 κi1 κ1eff+κi1+κe1 . Therefore, the power that flows to cavity 2 can be obtained from Finally, the output comb power is dominated by the factor ξ = κe2 κe2+κi2 , which quantifies the energy going to the output comb channel when light is circulating inside cavity 2. Note that ξ does not include the losses of other frequency modes except for the pump mode back to cavity 1 and the input waveguide, which is negligible due to the off-resonant condition between cavity 1 and cavity 2. For some frequency modes at which cavity 1 and cavity 2 are resonant because of the Vernier effect, the loss is minor since the power of those frequency modes is much lower than the modes close to the pump mode. Thus, the conversion efficiency is Therefore the parameter ξ sets the fundamental limit of the conversion efficiency. For example, in this work, cavity 2 is nearly critically coupled to the output comb waveguide, leading to a ~50% theoretical limit of the conversion efficiency. With the expression for P 2 , we obtain the final efficiency:

Data availability
The datasets generated and analysed during this study are available from the corresponding authors upon reasonable request.

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Extended Data Fig. 3 | Measurement setup for Figs. 2 and 3. The coupled-resonator device is characterized using the above setup. In the experiment of Fig. 2b, an EDFA is used to obtain higher pump power. In the experiment of Figs. 2c and 3b, the EDFA is not used. PC, polarization controller; DUT, device under test; EDFA, Erbium-doped fiber amplifier; OSA, optical spectrum analyzer; PD, photodetector. Fig. 4 | Device parameter analysis. a, b, Transmission spectrum of a single cavity 1 (a) and 2 (b) with the same fabrication parameters as the coupled-resonator device. c, Transmission spectrum of a coupled-resonator device on the through port. d, Transmission spectrum of a coupled-resonator device when microwave is on. (c) and (d) are measured on two different coupled-resonator devices with the same fabrication parameters. The background oscillation is due to the Fabry-Perot resonance formed in the bus waveguide due to the reflection at the two facets of the chip. The extracted parameters give a theoretical conversion efficiency of 28%.