## Introduction

Terahertz radiation, typically referred to frequencies from 300 GHz to 10 THz (λ ≈ 30–1000 μm), has spurred remarkable advances in condensed matter physics, biology and medical sciences, global environmental monitoring, metrology, information and communications technology. To achieve high frequency resolution (MHz or lower) and broad bandwidth (100 GHz or more) requirements simultaneously for frequency-agile imaging, communications, metrology, and spectroscopy applications, both widely tunable continuous-wave (CW) terahertz sources as well as terahertz frequency comb sources are demanded extensively. The current frontier techniques for the generation of tunable CW terahertz radiation include millimetre-wave oscillator frequency multiplication1,2, dual infrared wavelengths photomixing3,4,5,6, as well as using quantum-cascade lasers (QCLs)7,8,9, molecular gas lasers10, and free-electron lasers11. Frequency-multiplication approaches can offer milliwatt level radiation power, but hardly reach beyond 1 THz in frequency. In addition, their frequency bandwidths are often limited due to impedance matching and/or waveguide cutoff constraints.

Photonics paves the way to the realization of reliable high-frequency terahertz radiation sources. Notably, much recent progress is offered by the development of terahertz QCLs based on inter-miniband transitions within the conduction band of semiconductor heterostructures. Several realizations of tunable QCLs have been recently demonstrated by intracavity difference-frequency generation12,13, microelectromechanical-based transverse mode manipulation14, and metasurface-based vertical-external-cavity surface emission15. Nevertheless, their practical application is often limited by low duty cycle (quasi-CW) operation, restricted frequency tuning range, and the need for cryogenic cooling. Aside from QCLs, a widely tunable terahertz gas laser based on discrete rotational transitions of molecular gas was recently demonstrated with CW operation10. However, apart from a relatively larger system footprint due to the required vacuum pump and pressure control elements, its emission frequencies have not surpassed 1 THz so far. Photomixing, which is by far the most reliable technique to achieve wide frequency tunability, CW generation and room temperature operation, is the process of converting a pair of beating optical waves into terahertz radiation at their frequency difference16. Unlike nonlinear difference-frequency generation, photomixing in a photoconductor does not require phase matching and its conversion efficiency is not constrained by the Manley–Rowe limit since the terahertz radiation stems from the acceleration of electron-hole pairs rather than direct photon conversion17,18. Therefore, it has been widely recognized in state-of-the-art terahertz generation for frequency-domain spectroscopy19,20 and communication systems21,22. Furthermore, terahertz frequency combs have been explored for many high-resolution, high-accuracy metrology and spectroscopy applications, and the typical techniques for the generation of terahertz frequency combs include direct synthesis within QCLs23, dispersion compensation of QCLs24, down-conversion and frequency division from stabilized femtosecond mode-locked lasers via photomixing25,26. Besides the QCL and mode-locked laser, a semiconductor injection laser with three different active regions is designed to obtain octave-spanning terahertz frequency combs27. The demonstrated terahertz frequency combs can be challenging in terms of the system complexity, small repetition rate, and sizable footprint for cryogenic cooling, limiting some applications outside the laboratory. In parallel, microresonator-based nonlinear processes are proven to provide coherent parametric oscillation generation28,29,30, octave-spanning spectral translation31,32, and precision broadband frequency combs33,34,35,36 via cavity-enhanced four-wave mixing which are advancing research in laser spectroscopy37, optical coherent communications38, optical neural networks39,40, laser ranging41,42,43,44, and optical clocks45,46. High-power and coherent millimeter waves are generated recently with the microresonators combining with optical frequency division, dispersion compensation, and optical spectral shaping techniques47,48, but the frequency tuning range and resolution can be limited.

Here we present a coherent frequency-agile terahertz wave synthesizer with 2.8-octave tunability, sub-Hz linewidths, and 10−15 1,000-sec relative frequency instabilities, capable of generating both single-frequency and multi-frequency terahertz oscillators, through a microresonator-based optical parametric oscillator integrated with a plasmonic nanoantenna array photomixer. First, multi-segmented cavity mode dispersion engineering and polarized hybrid mode interactions in the nonlinear microresonator enables the broadband χ(3) optical parametric oscillation over 2.3 THz, including competition between mode-crossing Turing instabilities and Faraday instabilities. Secondly, with plasmonic photomixed optical-to-terahertz frequency division, our chip-scale platform achieves the coherent single-frequency synthesis from 330 GHz to 2.3 THz, the largest range realized to our knowledge, and with fine ≈27 Hz tuning steps on the terahertz carrier. The generated coherent signal has a cavity-mode-limited frequency tuning step of ≈20 GHz and ≈10 MHz intracavity mode continuous frequency tuning range at each step. Our terahertz synthesizer does not require optical spectral shaping and, via control of the microresonator intracavity power, multi-frequency terahertz oscillators are also observed through heterodyne beat between our pump laser and the localized optical sub-comb lines. Third, with feedback stabilization of the microresonator intracavity power, we observed less than 100 Hz linewidths on the terahertz radiation, bounded only by the instrument limits. With dual-stage pump power- and frequency-stabilization, the long-term frequency stability is improved by five orders-of-magnitude, reaching an instrument-limited frequency residual instability of 1.6 × 10−15 at 1000 s averaging. The chip-scale 2.8-octave tunable synthesizer, with coherent sub-Hz linewidths and 1.6 × 10−15 frequency instabilities, at room temperature provides an alternative platform for next-generation terahertz imaging, sensing, metrology and communications.

## Results

### Chip-scale frequency-agile terahertz radiation synthesizer based on a nitride microresonator and a InAs nanoantenna array

A conceptual schematic of the broadly-tunable terahertz radiation synthesizer based on a dispersion-managed silicon nitride nonlinear microresonator and a plasmonic photoconductive InAs nanoantenna array photomixer is illustrated in Fig. 1a. The tunable optical parametric oscillation (OPO) generation setup is shown in inset i and potential applications of the tunable terahertz synthesizer are illustrated in inset ii. The microresonator contains seven tapered waveguide segments to engineer the dispersion with varying widths from 1 to 2.5 µm, five 180° bends and two 90° bends for a fixed silicon nitride height of 800 nm. The curved waveguides serve as mode-filters49,50 to maintain the fundamental transverse mode (TE00 and TM00) operation. The averaged cavity group velocity dispersion (GVD) of the fundamental TE and TM modes are 18.31 and 158.10 fs2/mm extracted from the cold-cavity mode spectra measured by swept-wavelength interferometry36 after the mode frequency calibration (detailed in Methods and Supplementary Information I). The loaded quality factors are 1.55 × 106 (TE00) and 1.23 × 106 (TM00) around the pump laser mode, with mode free spectral ranges (FSR) of 19.87 and 19.72 GHz respectively. The resonant frequencies of the fundamental mode family are $$\omega \left(\mu \right)={\omega }_{0}+{D}_{1}\mu+\frac{{D}_{2}}{2}{\mu }^{2}$$ when just considering GVD, where ω0/(2π) is the pump cavity mode frequency, D1/2π is the cavity mode FSR, µ is the azimuthal mode number with respect to the pump cavity mode, D2 = −c/n0D12β2 is the cavity mode dispersion, c is the speed of the light, n0 is the refractive index of the nitride resonator, and β2 is the GVD. To effectively compensate the phase mismatch β2Ω2 (Ω is the parametric oscillating frequency spacing) and χ(3) Kerr nonlinear phase shift γPin (γ is nonlinear coefficient and Pin is microresonator intracavity power) in our normal-dispersion microresonator, mode-frequency shift induced by the transverse mode interaction between our hybridized polarization orthogonal modes (TE and TM) is utilized. Figure 1b shows the computed mode interaction evolution based on the operation temperature by solving the coupled mode equations (detailed in Supplementary Information I). The target cavity mode ωµ is split into two modes ωµ ± Δω, where Δω/(2π) ≈μ(D1TE − D1TM)/2π + μ2(D2TE − D2TM)/4π ≈ μc/L(T)/(nTE (T) − nTM (T))/2π is the mode-shift frequency, nTE (T) and nTM (T) are the temperature-dependent refractive indexes for TE and TM modes, and L(T) is the physical cavity length. Insets i, ii, and iii show the representative mode spectra at different chip temperatures, fitted to the double-Lorentzian model to obtain the mode linewidth and mode-splitting frequency.

Three modes (ω0, ω, ω) are involved in the excitation of the parametric oscillation. By directly modifying the frequency of the target mode through mode-shifting, the tunable phase matching condition is obtained over a broad optical spectral range via 2ω0-ωµ-ω = β2Ω2Pin − φ (ωµ, T), where φ (ωµ, T) = β1ω/2π) + β2ΩΔω/(2π) and β1 is the group velocity. The required mode-splitting frequency is Δω/2π = (β2Ω2Pin)/(β1+β2Ω). The target mode frequency and the corresponding splitting frequency both depend on the microresonator temperature, which is related to the intracavity power Pin(δ) based on the pump-resonance detuning δ = (ωpω0)/(2π). Figure 1c shows the measured tunable parametric oscillation generation via continuous pump-resonance detuning of δ = 4 GHz when the resonant mode is pumped at 1588.15 nm. The resultant parametric signal sidebands tunes from 1585.4 to 1571.5 nm, corresponding to a Ω tuning from 370 GHz to 1.7 THz (ΔΩ = 1.33 THz). This specific microresonator thus offers an optical-to-terahertz tuning ratio χ = ΔΩ/δ of 332.5.

Substantiating our measurements shown in Fig. 1, the χ(3) phase matched parametric sideband curves are calculated based on the modified phase matching condition $$L\frac{{\beta }_{2}}{2}{\varOmega }^{2}+2\gamma L{{P}_{{in}}\left(\delta \right)}^{2}-\delta -\varphi \left({\omega }_{\mu },T\right)=0$$ after considering the cavity boundary condition in the Lugiato-Lefever (LL) model and the thermal-mediated mode-splitting as shown in Fig. 1c with orange (idler) and purple (signal) curves. The calculated parametric sidebands are consistent with our measurements. Mode-splitting is a local effect which breaks the phase-matching symmetry of the signal (ω) and idler (ω) leading to asymmetric parametric gain – Fig. 1d shows the resultant measured and calculated signal and idler peak powers. The power difference is attributed to the asymmetric parametric gain, calculated as $$G=\sqrt{{\left(\gamma {P}_{{in}} \right)}^{2}-\triangle k}$$, where Δk is phase mismatch. The computed parametric gain profiles are also shown in Fig. 1d where the blue curve is the calculated parametric gain profile with anomalous dispersion as a reference, and σ = δ + φ represents the extra phase mismatch in the parametric gain symmetry breaking. After adding the extra mode frequency shift Δω/(2π), the parametric oscillation is generated at effective pump-resonance blue detuning. Figure 1e shows the representative tunable parametric oscillation at different normalized pump-resonance detunings [δ0 = δ/D1TE/(2π)], where the power imbalance between signal and idler from 10.6 dB (δ0 = −0.4841) to 0.8 dB (δ0 = −0.0383) originates from the additional mode-frequency shift. The pump-to-signal power conversion efficiency is related to the pump-to-signal power imbalance ratio (denoted in dB in Fig. 1e) determined by the pump resonance detuning and avoided mode frequency shift. A higher pump-to-signal conversion (smaller power imbalance in dB) allows for a higher terahertz wave emission power.

### Thermal and local dispersion dependences of the tunable terahertz optical parametric oscillation

Tunability of the parametric oscillation is associated with the differential TE-TM thermo-optic effects in the microresonator, demonstrated by monitoring the parametric oscillation at different operating temperatures from 51.74 to 53.07 °C for a fixed pump laser power and the same resonant mode frequency, as shown in Fig. 2a. The frequency of the initial parametric oscillation pair can be tuned over 12 nm when the temperature changes 1.3 °C. The power variation of the parametric sidebands is related to the magnitude of the Δω/(2π). Maximizing the spectral tuning range of the parametric oscillation requires a careful selection of the pump cavity mode. By pumping the cavity modes located between 1582 to 1590 nm with a 1 or 2 nm spectral step at a fixed on-chip pump power, we examine the parametric oscillation generation dynamics. At the shorter pump wavelengths, the tunable parametric oscillation vanishes but Faraday parametric oscillation51 emerges due to the periodic dispersion design in our microresonator49. When resonant modes at longer wavelengths are pumped, bistable switching dynamic between Faraday and Turing parametric oscillation is observed, which results from parametric gain competition between the two parametric oscillation processes52. The Faraday branch is induced by periodic modulation of the microresonator dispersion. The gain peak sideband frequency away from pump laser frequency is $${\omega }_{f}=\sqrt{\frac{2\left[\left(\frac{\delta }{L}-2\gamma {P}_{{in}}\right)\pm \sqrt{{\left(\gamma {P}_{{in}}\right)}^{2}+{\left(\frac{\pi }{L}\right)}^{2}}\right]}{{\beta }_{2}}}$$, where L is microresonator cavity length. To thoroughly denote the pump wavelength dependence, the high frequency parametric sidebands (signal) are filtered out to highlight the frequency tuning range and parametric oscillation competition dependence on the pump-resonance detuning as shown in Fig. 2b. The frequency tuning range is highly related to the local resonant mode dispersion and thermally-mediated mode-splitting frequency shift resulting from the optical power absorption in the microresonator. Mode-hopping instabilities are also observed but these can be suppressed by optimizing the microresonator operation temperature. In support of the measurements, the LLE numerical modeling of Fig. 2c (detailed in Methods) verifies the comparative deterministic parametric oscillations when the mode frequency shifts are introduced at µ = 15 and 37.

### Tunable radiation and multi-frequency terahertz oscillators generation

With the designed cavity mode dispersion and differential TE-TM thermal dependence, the tunable OPO is facilitated by gradually changing the pump-resonance detuning, where the optical power absorption leads to increase (decrease) of the resonant mode temperature by forward (backward) pump wavelength tuning. The generated tunable OPO is passively down-converted to the terahertz radiation via a plasmonic nanoantenna array built on the InAs substrate53. The incident optical beam excites surface plasmons on the nanoantenna array, providing strong enhancement of the generated near-field photocarriers, subsequently accelerated by the surface built-in electric field in the InAs substrate without any external bias voltage. Since the nanoantenna array operates without any bias voltage, the direct current through the emitter is eliminated, leading to minimal heating. Consequently, the shot noise and thermal noise contributed from the nanoantenna array itself is believed to be negligible. Figure 3a illustrates the terahertz signal generation and detection schematic. By optimizing the intracavity power and detuning, terahertz radiation is generated over a broad frequency range (detailed in Methods and Supplementary Information II). To generate tunable terahertz radiation, two adjacent resonant modes ($${\mu }^{{\prime} }$$= 9500 and 9501, where $${\mu }^{{\prime} }$$ is the absolute cavity mode number) are examined and optically pumped, which is subsequently down-converted to radio frequency (RF) spectral ranges using a harmonic mixer to examine the signal-to-noise ratio (SNR), frequency tunability (intra- and inter-resonant modes), and frequency stability. The detected terahertz radiation spectra span from 330 to 750 GHz (bounded only by harmonic mixer frequency bandwidth) as shown in Fig. 3b and feature an ≈50 dB SNR, apart from the 550 and 750 GHz frequencies where strong water vapor absorption occurs. The fluctuations in frequency and power are included in standard deviation error bars of Fig. 3b as well. Subsequently, in Fig. 3c, we examine the terahertz frequency tuning range over a single resonant mode ($${\mu }^{{\prime} }$$= 9500), with tunable optical parametric sidebands covering from 1586 to 1570 nm where the spectral gaps are related to local phase mismatching of the Turing instabilities. The spectral gaps require more accurate phase matching between the pump and signal by optimizing the dispersion engineering and operating temperature. The corresponding pump laser absolute frequencies are shown in Fig. 3c inset over the thermally broadened cavity resonance. Power equalization between the pump and parametric sidebands are performed to optimize photomixing efficiency. The radiated terahertz power is measured with lock-in detection using a pyroelectric detector at a 20 Hz. Figure 3d shows the detected terahertz power across 2.8-octaves from 330 GHz to 2.3 THz, which decreases at the higher radiation frequencies and then rolls off to the noise floor of the pyroelectric detector. The detected power above 500 GHz agrees well with the simulated magnitude response of the nanoantenna array denoted with the dashed gray curve. Below 500 GHz, the power deviation arises from spectral filtering as well as degraded terahertz beam coupling inside the pyroelectric detector.

By optimizing the microresonator intracavity power and pump-resonance detuning, the sub-comb lines around the parametric sidebands can be locally excited. Figure 3e shows two example optical cluster frequency comb spectra with the signal-pump frequency spacing of 553 and 651 GHz. The generated frequency comb clusters are subsequently down-converted to the terahertz spectral range to generate multi-frequency terahertz oscillators with line frequency spacing of 19.86 GHz at the 553 and 651 GHz center frequencies. Figures 3f, g show the generated multi-frequency terahertz signal after down-conversion to the RF spectral range via changing the harmonic mixer reference frequency to down-convert all frequency tones simultaneously.

### Active feedback stabilization of the generated terahertz radiation

Tunability of the generated single and multi-frequency terahertz wave highly depends on effective microresonator temperature which, in turn, introduces thermal stochastic frequency fluctuations of the generated terahertz signal induced by thermodynamical fluctuations of the microresonator mode. The small mode volume of the microresonator physically leads to high thermal sensitivity to ambient temperature fluctuations. The thermal instability originates from thermal energy exchange between the microresonator and ambient background or laser heating from the intracavity optical power absorption. The thermal fluctuations, fundamentally mediated by thermal expansion and thermo-refractive index effects, impose phase decoherence of the generated terahertz signal.

The microresonator mode optical frequency is denoted as $${v}_{{\mu }^{{\prime} }}={\mu }^{{\prime} }{f}_{{FSR}}+{f}_{0}$$ where $${f}_{{FSR}}$$ is the mode free spectral range, and $${f}_{0}$$ is the offset frequency. To improve the instantaneous linewidth and frequency stability of the generated terahertz signal and further increase its frequency tuning resolution, microresonator intracavity power stabilization (main stabilization loop) and the pump-resonance detuning stabilization (auxiliary stabilization loop) are engaged, as illustrated in Fig. 4a. The frequency tuning resolution is fundamentally limited by the frequency fluctuations of the microresonator FSR, ΔfFSR(T). With the microresonator thermodynamics54, the frequency fluctuations variance behaves as $$\left\langle \triangle {f}_{{FSR}}^{2}\right\rangle={\left|{\eta }_{{FSR}}\right|}^{2}\frac{{k}_{B}{T}^{2}}{\rho {CV}}$$ where $${\eta }_{{FSR}}=d{f}_{{FSR}}/{dT}$$ is the thermal transduction coefficient, ρ is the material density, C is the specific heat, and V is the microresonator mode volume. Thermal transduction coefficient can be decomposed via $${\eta }_{{FSR}}=\frac{d{v}_{{\mu }^{{\prime} }}}{{dT}}\left(\frac{\partial {f}_{{FSR}}}{\partial {v}_{{\mu }^{{\prime} }}}+\frac{2}{\varepsilon }\frac{\partial {f}_{{FSR}}}{\partial \delta }\right)$$, where ε is the mode linewidth55. The measured temperature-dependent mode frequency change is $$d{v}_{{\mu }^{{\prime} }}/{dT}=39.2{GHz}/K$$ and $$d{f}_{{FSR}}/d{v}_{{\mu }^{{\prime} }} \, \approx \, 1/{\mu }^{{\prime} }$$. To mitigate the thermal noise transduction, $$\frac{\partial {f}_{{FSR}}}{\partial {v}_{{\mu }^{{\prime} }}}+\frac{2}{\varepsilon }\frac{\partial {f}_{{FSR}}}{\partial \delta }=0$$ should be satisfied which requires the stabilization the intracavity power via controlling the pump laser power and frequency.

## Discussion

With the terahertz signal generation and active feedback stabilization setup (Fig. 4a), the linearly polarized and power equalized optical beam is injected into the plasmonic nanoantenna array. The generated terahertz signal is down-converted to the RF domain by a harmonic mixer. The detected RF signal is managed by a series of electronics. An electrical spectrum analyzer and a signal source analyzer are used to characterize instantaneous linewidth and frequency noise of the down-converted signal at the intermediate frequency. The measured frequency noise of the down-converted signal is limited by the reference signal (LO1). To facilitate feedback locking, electrical frequency division method is utilized to further decrease the carrier frequency of the down-converted signal which subsequently is referenced to another source (LO2). The timing referenced frequency counter continuously samples the divided signal to examine the frequency stability of the terahertz signal at the free-running and stabilized conditions. The extracted error signal is managed by a proportional-integral-derivative (PID) controller to control the pump laser polarization to stabilize the microresonator intracavity power after polarization demodulation with a linear polarizer. Meanwhile, the integral error signal auxiliary controls pump laser frequency to equivalently stabilize the pump-resonance detuning (δ). We realize a dual-step stabilization protocol including pump laser power locking followed by pump laser frequency locking. Figure 4b shows the free-running and resultant stabilized instantaneous linewidth of the generated terahertz signal. After pump laser power and frequency stabilization, the Gaussian-fitted linewidth is improved ≈105,000× for a 39-ms sweep time.

After terahertz signal frequency stabilization, the frequency noise power spectral density (PSD) is measured as shown in Fig. 4c along with the frequency noise of the reference source (LO1), the β separation line, and the flicker frequency noise power law (1/f) where the 100-Hz linewidth is observed. The low Fourier frequency flicker noise originates from the pump laser amplifier intensity noise conversion56. The instantaneous linewidth and frequency noise PSD characterize the short-term frequency stability. The long-term frequency stability is examined with a referenced frequency counter for a 300-ms gate time (detailed in Methods and Supplementary Information III). Figure 4d shows the Allan deviation of the free-running signal and residual signal at 651.5 GHz. The Allan deviation of the reference signal (LO2) at 4.44841 MHz is included as well. The residual frequency stability is improved by 400,000× for a 1-second averaging time and reaches the instrumental limit of the frequency counter. The stabilized terahertz signal has a frequency tuning range and resolution of 14 kHz and 27 ± 1.5 Hz, respectively (Fig. 4e). The stabilized frequency tuning range is dependent on the feedback locking bandwidth. The frequency tuning resolution is determined by the frequency tuning step of the reference source 2 and bounded by the reference source linewidth and the residual noise of the active feedback locking loop.

We report a chip-scale coherent frequency-agile terahertz radiation synthesizer operating at room temperature based on an integrated microresonator parametric oscillator and a plasmonic nanoantenna array, capable of providing both single-frequency terahertz radiation as well as multi-frequency terahertz oscillators in the same system platform. Through optical-to-terahertz frequency division and without requiring optical spectral shaping, the achieved 2.8-octave frequency tunability is, to our best knowledge, the largest octave range realized in chip-scale semiconductor devices with a single pump laser. Moreover, the ≈27 Hz frequency tuning resolution for the 651 GHz carrier frequency is one of the highest resolution achieved to date. With the deterministic pump-resonance laser detuning and hybridized modes with thermal control, we report less than 100 Hz linewidths of the terahertz radiation with feedback stabilization of the microresonator intracavity power. The dual pump power- and frequency-stabilization improves the long-term frequency stability by five orders-of-magnitude, reaching an instrument-limited relative frequency residual instability of 1 × 10−15 at the averaging time of 1000 s. With our nonlinear parametric oscillation and plasmonic array photomixing, the high-spectral-purity frequency-agile terahertz synthesizer in single-frequency, tunable and multi-frequency realizations enable spectroscopy and waveform synthesis in the terahertz regime, supporting advances in metrology, sensing, frequency-agile imaging and wireless communications.

## Methods

### Integrated dispersion-managed microresonator fabrication

A 3 μm thick SiO2 layer is deposited via plasma-enhanced chemical vapor deposition (PECVD) on a p-type 8” silicon wafer serving as an under-cladding. Then an 800 nm nitride layer is deposited via low-pressure chemical vapor deposition and patterned by the optimized 248 nm deep-ultraviolet lithography and etched down to the buried oxide cladding via optimized reactive ion dry etching. The etched sidewalls have an etch verticality of 88° characterized by a scanning electron microscope. The nitride rings are then over-cladded with a 4.5 μm thick oxide layer.

### Microresonator group velocity dispersion and avoided-mode-crossing characterization

A tunable laser (Santec, TSL-510) is swept over a wavelength range from 1520 to 1610 nm at a sweeping speed of 30 nm/s with an output power of 2 mW to measure the cold cavity mode spectra of the microresonator. The 1% of the laser output power is injected into a fiber coupled hydrogen cyanide gas cell (HCN-13-100, Wavelength References Inc.) to calibrate the cavity mode frequency for extracting the cavity mode group velocity dispersion and wavelength-dependent cavity loaded quality factor. The transmissions of the microresonator and the gas cell are recorded during the laser sweep by a data acquisition system with sampling frequency of 1.7 MHz determined by an unbalanced fiber Mach-Zehnder interferometer (MZI).

### Simulated parametric oscillation with Lugiato–Lefever equation

Taking the normal group velocity dispersion (GVD) and avoided mode-crossing into consideration, the tunable optical parametric oscillation dynamics are numerically modeled with the Lugiato-Lefever equation (LLE) written as:

$${t}_{R}\frac{\partial E\left(t,\, \tau \right)}{\partial T}+i\left(\frac{{\beta }_{2}}{2}\frac{{\partial }^{2}}{\partial {\tau }^{2}}-\gamma {\left|E\right|}^{2}\right)E\left(t,\, \tau \right)+\left(\alpha+i\delta \right)E\left(t,\, \tau \right)=i\sqrt{T}{E}_{{in}}$$
(1)

where tR = 50.3 ps, β2 = 18.31 fs2/mm, γ = 1 W−1m−1, Q = 1.55 × 106, α = 0.018, and T = 0.009. The pump power Pin and wavelength λp are set at 1.4 W and 1588 nm. We numerically solve the equation with split-step Fourier method initiating from quantum noise. The tunable parametric oscillation can be excited at the specific microresonator modes by properly setting the avoided-mode frequency shift.

### Tunable terahertz wave and multi-frequency terahertz oscillators generation

The generated parametric oscillation with an average power of ≈20 dBm is injected into an optical preamplifier after an optical bandpass filter. The amplified optical beam is focused by a biconvex lens with a focus length of 10 cm into the plasmonic nanoantenna array. The generated terahertz signal is collimated and focused by a pair of gold-coated parabolic mirrors. The signal is then received by a harmonic mixer (VDI MixAMC with WR 1.5 and WR 2.2 horn antennas) to down-convert into the RF domain for linewidth and frequency stability examination or by a pyroelectric detector (QMC Instruments Ltd) with a lock-in amplifier (SR830) for terahertz power measurements.

### Active stabilization of the terahertz signal

The generated terahertz signal is down-converted to an intermediate frequency (IF) of 1.154 GHz with the harmonic mixer (VDI MixAMC with WR 2.2 horn antenna) referenced to a local oscillator (LO1, Gigatronics 905) at a frequency of LO1 = 12.043 GHz. The IF signal is then amplified by an electrical amplifier (Mini-circuits, ZVA-443HGX+) to ≈−10 dBm and monitored by an electrical spectrum analyzer (ESA, Agilent MXA9020A). The amplified signal is frequency divided with a 260× RF frequency divider (RF bay, FPS-260-4) to the frequency of 4.44 MHz, which is subsequently filtered out by a low-pass filter (LPF, Mini-circuits, BLP-5+). The divided signal is referenced to the local oscillator 2 (LO2) to obtain phase error signal. The error signal is filtered by another low pass filter (LPF) with a cutoff frequency of 1 kHz and routed to a proportional-integral-differential controller (Vescent photonics, DS-125) to generate feedback signal. The feedback signal controls the pump laser polarization via an electrical-controlled polarization controller (EPC, EOSPACE) combined with a polarization beam splitter to stabilize the intracavity power. Furthermore, the auxiliary of the PID controller is used to control pump laser frequency to equivalently stabilize the pump-resonance detuning.