Terahertz microcomb oscillator stabilized by molecular rotation Open Access

The scientific community has recently focused its efforts on filling the so-called terahertz gap.^1–4 The gap refers to a lack of mature and user-friendly technologies for generating, manipulating, and detecting radiation in the frequency range of 100 GHz to 30 THz.¹ The terahertz band exists on the electromagnetic spectrum between microwaves and infrared light, both of which have well-developed libraries of components for commercial and scientific use. Much of the motivation for bridging the gap is driven by a number of critical applications. One example is next-generation wireless communications technology. Proposals for the sixth-generation wireless technology standard, 6G, call for carrier frequencies from 300 GHz to many terahertz.⁵ These larger frequencies are required to facilitate greater bandwidths for faster data rates and to relieve congestion in the currently deployed communication channels. In particular, low phase noise and long-term stable terahertz oscillators are required to realize the full potential of this spectral band.^6,7 Another example is very long baseline interferometry for radio astronomy. A large coherence length local oscillator is needed to maintain constructive interference in the phased array antennas.⁸ An additional application is for terahertz radars, the higher frequency of which leads to better angular resolution than microwave radar, while being less susceptible than LIDAR to the weather and ambient light. Stable and low-noise terahertz oscillators can provide a high signal-to-noise detection sensitivity, as well as detection of slow motion targets, enabled by low phase noise close to the terahertz carrier.^9,10 With such applications in mind, the need for high-performance terahertz oscillators is clear.

A common method for producing terahertz radiation is through the electronic frequency multiplication of microwaves. This technique multiplies the microwave phase noise, which degrades the resulting terahertz spectral purity. Photomixing is an alternative approach that utilizes the interference of two or more continuous wave (CW) optical frequencies on a fast photodiode. Specifically, a uni-travelling-carrier photodiode (UTC-PD) can respond up to terahertz frequencies produced by optical interference.^11–14 This approach enables the use of widely available and high performance photonic technologies, including broadly tunable low-noise lasers, optical frequency combs, and fast optical modulators.¹⁵ An example of these technologies coming together to produce terahertz radiation is electro-optic (EO) combs.¹⁶ While there has been recent progress on low V_π voltage electro-optic modulators,¹⁷ an EO comb source requires a microwave reference that employs a radio frequency synthesizer. Scaling down such a synthesizer inevitably impacts its spectral purity, creating challenges to integration when phase noise is of critical importance. As an alternative, the photomixing technique can take advantage of advanced integrated photonic chip technologies, such as dissipative Kerr solitons (DKSs) generated in microring resonators,¹⁸ providing a path for low-size, weight, and power terahertz devices.¹⁹ For example, there are already integrated photonic demonstrations that utilize microcombs to generate low-noise microwaves that do not rely on external frequency synthesis.²⁰ Turing patterns produced in microresonators have also been coupled with plasmonic photomixers to produce high power terahertz radiation.²¹ Importantly, microcombs and photomixing provide the most spectrally pure terahertz radiation. It was recently demonstrated that a pair of highly coherent continuous wave lasers, divided down to the terahertz domain using a microcomb, led to the implementation of a 0.3 THz oscillator with phase noise that outperforms direct generation or multiplication of microwave references (for Fourier frequency above 100 Hz).²² 

The improved spectral purity of photonic terahertz oscillators opens the door to another application: precision molecular spectroscopy. In the terahertz domain, many polar molecules possess unique transitions between quantized rotational states.²³ These levels are populated at room temperature by black body radiation, and transition frequencies can be accurately measured via direct absorption techniques. The phase noise requirement of the probing terahertz oscillator is set by the Doppler limited linewidth of a rotational transition, which is on the order of a megahertz or less. A terahertz source based on an optical microcomb fulfills this criterion and allows for the study of rotational spectroscopy. Precise measurements of molecular energy levels can serve multiple purposes, such as remote identification of gases, unveiling details of molecular structures, and measuring fundamental constants.^24–30 

A transformative application of precision spectroscopy is the use of a transition resonance as a frequency discriminator, exemplified by electronic transitions in atomic species used in microwave atomic clocks.³¹ The concept of molecular frequency references, although older,³² has been much less explored than its atomic counterparts. One reason is the relatively weak interactions between molecules and microwaves. An early attempt to stabilize microwaves with ammonia molecules used an $∼50$ cm long gas cell and only achieved a fractional frequency stability of 5 × 10⁻¹⁰ at 1 s of averaging time, limited by the absorption signal strength.³³ By contrast, rotational spectroscopy of small molecules at terahertz frequencies offers a stronger absorption, only leaving the need for spectrally pure terahertz radiation. A chip-scale terahertz oscillator recently demonstrated molecular frequency stabilization in a compact device, but performance was limited to a fractional stability of 5.37 × 10⁻¹⁰ at 1 s by the multiplied electronic oscillator, ultimately reaching a stability of 2 × 10⁻¹¹ at 10 000 s.³⁴ 

Here, we experimentally demonstrate a 300 GHz oscillator based on a DKS microcomb and a UTC-PD capable of precision molecular rotational spectroscopy. Disciplined to a rotational level of nitrous oxide (N₂O), the oscillator linewidth was reduced by almost a factor of 1000 with a fractional frequency stability of 1.35 × 10⁻¹¹ at 1 s and 5 × 10⁻¹² after 10 s of averaging time. This result corresponds to an unprecedented level of stability using a rotational transition in the terahertz domain.

Figure 1 shows the conceptual progression of the photonic oscillator and the associated performance improvements with each step. Shown in Fig. 1(a) is the basis of a terahertz oscillator using photomixing. A terahertz wave generated from two lasers which are simultaneously incident on a UTC-PD has a stability given by the quadratic sum of the optical waves’ stabilities. For example, two inexpensive laser diodes (LDs) with ∼1 MHz linewidth on a fast photodiode would lead to a similar terahertz linewidth, which is insufficient to perform rotational spectroscopy. There are a few possible ways to improve the stability. One is by using approximately ten times more expensive lasers with narrower individual linewidths. Another way is by phase locking the frequency difference of the two lasers to a microwave reference, although this comes with both costly and complex down-conversion schemes through EO combs or subharmonic mixers.^35–37 

An alternative photonic-based approach to creating phase correlation between two LDs is shown in Fig. 1(b). First, a DKS is generated in a microring resonator. The resulting optical pulse train has a repetition rate set by the size of the ring resonator, which, in practice, can be 10 GHz to a few terahertz. The pulse train corresponds to a frequency comb with coherent comb teeth spaced by the repetition rate. In principle, the comb can be directly incident on a UTC-PD to generate terahertz radiation. However, the dispersion of the comb in fiber leads to dephasing between high-order comb modes and can ultimately cause destructive interference in the UTC-PD. Optical injection locking (OIL) of two neighboring comb modes into two respective LDs avoids the issue of dispersion, transfers the spectral purity of the DKS repetition rate to the differential phase noise of the diode lasers, and also provides optical amplification.^38,39 After photomixing, the result is a terahertz wave with a 1000-fold improvement in linewidth, capable of probing rotational transitions in molecules.

The final step of the oscillator stabilization is shown in Fig. 1(c). Terahertz radiation from the microcomb oscillator in Fig. 1(b) interacts with a gaseous sample of N₂O to perform direct absorption spectroscopy. Interrogating a rotational line acts as a terahertz frequency discriminator, as discussed in the introduction.^33,34 The frequency difference between the two LDs is locked to the rotational frequency, producing an additional 1000-fold improvement in the photomixed linewidth and a dramatic reduction in the absolute frequency drift. In principle, all these components can be contained in a compact architecture due to the chip-scale nature of the microcomb, advances in integrated photonics, and the strong absorption of molecular rotations at terahertz frequencies.^40,41

The experimental demonstration of this concept is shown in Fig. 2. The setup consisted of two distinct parts: the microcomb oscillator and the molecular rotational spectroscopy used for frequency locking. A combined schematic is shown in Fig. 2(a). For the oscillator, we generated a single dissipative Kerr soliton in a silicon nitride (SiN) microring resonator by quickly sweeping the frequency of an amplified diode laser through resonance. This was accomplished by using a carrier suppressed, single-sideband modulator, as demonstrated previously.⁴² This generated a DKS in the microring, which emitted an optical pulse train with a repetition frequency within 100 MHz of a rotational transition in N₂O. The repetition rate was sensitive to thermal fluctuations. These fluctuations can be large enough to destroy the DKS when the microcomb is in a free-running state. To reduce this effect and to maintain a stable DKS over a long period of time, the optical detuning of the laser from a cold-cavity resonance was actively stabilized. Identical to previous reports, Pound–Drever–Hall (PDH) offset sideband locking of the pump-resonance detuning was employed.^42,43 All free-running microcomb data presented in this article include this optical detuning stabilization, which sufficiently reduced the impact of thermal fluctuations on the microcomb stability for long term operation.

Two neighboring comb modes were selected out of the microcomb via optical bandpass filters and optically injected into two discrete-mode LDs through three-port fiber circulators. The two LDs were tuned (via pump current) to a frequency near their respective injection comb lines. The direct output of the microcomb and injected LDs is shown in Fig. 2(b). There was an improvement of $∼40$ dB in the optical signal-to-noise ratio (SNR) of the comb lines as a result of the optical injection while also preserving phase coherence. Quantitative analysis of phase transfer fidelity from microcomb modes to LDs can be found elsewhere.^39,44 We employed reflection mode injection, as opposed to the transmission mode, as discussed by Liu et al.²⁰ As implemented, a portion of the injected light was reflected along the path of the LD output. This necessitated pre-filtering of the desired comb modes prior to optical injection. Without filtering, an interferometer for the raw comb modes is built by the split and recombined paths of the oscillator [see Fig. 2(a) for reference]. The interference led to uncontrolled optical power fluctuations over long time scales. We used dense wavelength-division multiplexing (DWDM) fiber components to prevent this interference. The DWDMs had to be specified for the desired comb-mode wavelengths, but they had considerably less loss than a tunable optical bandpass filter.

For molecular spectroscopy, the frequency difference between the LD lines was tuned to be resonant with the J = 11 → J = 12 transition in N₂O (301.4427 GHz). Figure 2(c) shows the calculated rotational ladder of transitions in N₂O at room temperature, with the chosen transition highlighted. There is a rotational line roughly every 25 GHz, with the strongest absorption at 600 GHz. We were practically limited to $∼300$ GHz due to the commercial availability of terahertz components for the waveguide spectrometer (discussed below). An acousto-optic modulator (AOM) was used on one LD line to adjust the frequency difference between the LDs. The AOM produced a shift f_AOM = 80 ± 5 MHz generated by a voltage-controlled oscillator (VCO). Thus, the VCO could be used to tune the laser frequency difference on and around resonance. The other LD line was phase modulated to f_EOM = 900 kHz using an electro-optical phase modulator (EOM) to add sidebands for spectroscopy. The amplitude of phase modulation (PM) was optimized following the standard procedures for PM spectroscopy.⁴⁵ In principle, both frequency tuning and modulation sidebands can be achieved through pump-resonance detuning of the microcomb. However, doing so leads to technical challenges that involve a lack of frequency tuning actuators and residual amplitude modulation (RAM). For this demonstration, we opted to use an AOM for frequency tuning as the pump-resonance detuning was utilized for Pound–Drever–Hall (PDH) offset locking, and a DC-coupled EOM was utilized for PM with RAM compensation (detailed below).

Both optical lines were combined and split between two paths: the upper path [see Fig. 2(a)] contained an optical amplifier, followed by a UTC-PD and then the N₂O absorption cell, which consisted of a rough vacuum chamber with a temperature stabilized waveguide spectrometer. The entire spectrometer, including the terahertz emitter, isolator, and detector, was operated inside the absorption cell to reduce the presence of etalons. The primary source of etalons in terahertz spectroscopy comes from windows that separate the terahertz source and detector from the gas being studied. As discussed by Wineland et al.,³³ an etalon acts as a low-finesse cavity (resonator) coupled with the molecular rotational resonance. The end effect is a slow variation of the spectrometer’s measured frequency, which spoils long-term measurement precision. With that in mind, we placed our spectrometer inside the gas cell to avoid the use of windows. Instead, a UTC-PD radiated the terahertz wave into a single-mode, rectangular waveguide, followed by a waveguide isolator, which further mitigated etalon-like effects caused by standing waves in the waveguide. The terahertz wave then propagated down 20 cm of the waveguide and finally reached a Schottky barrier diode (SBD) intensity detector. The vacuum chamber and, thus, the waveguide were filled with 100 mTorr of N₂O, which absorbed terahertz radiation resonant with the rotational transition.

The terahertz wave also carried the PM that was added in the optical domain. Demodulation, at f_EOM, of the SBD signal produced an error signal from molecular absorption. The error signal shown in Fig. 2(d) was measured by scanning f_AOM across the absorption resonance. The choice of waveguide length and pressure of N₂O was experimentally determined to optimize the signal-to-noise ratio (SNR) of the error signal. The amplitude of the 70 mV error signal was divided by the white noise floor of $2.5μV/Hz$ to obtain an SNR of ∼89 dB in a 1 Hz bandwidth.

The zero crossing of the error signal was coincidental with the peak of the rotational absorption feature. Subsequently, the error signal was fed back to the AOM through an FPGA-based PID loop filter (with a bandwidth of 100 kHz) to lock the frequency difference in the LD lines to the rotational transition of N₂O. The physical location of the AOM, and thus the loop feedback, was chosen as close to the LD line combination point as possible. This choice helped minimize the amount of non-common fiber, which causes degradation of phase coherence between the LD lines.

A few auxiliary servo loops were implemented to minimize slow frequency variations of the oscillator. One servo was used for RAM compensation. The RAM that came from the EOM was demodulated to a time-varying DC signal that corrupted the error signal baseline used for frequency locking. This resulted in slow oscillations of the locked oscillator fractional frequency at the 10⁻¹⁰ level [see Fig. 3(c)]. We compensated for RAM using the same techniques and understanding as presented in previous reports.^46,47 The RAM was detected optically by a photodiode placed immediately before the UTC-PD. Demodulation of the photodiode signal produced a RAM error signal proportional to the RAM intensity. A PID loop (1 kHz bandwidth) fed the RAM error signal back to the DC-coupled EOM that suppressed the RAM at the point of generation.

Optical power fluctuations led to terahertz power fluctuations, which caused frequency fluctuations via the AC stark effect (light shift).²³ We observed fractional frequency shifts on the order of 10⁻¹⁰ for a THz power fluctuation of 1 μW. Thus, it was necessary to stabilize both the relative power between the optical lines and the overall combined power sent to the UTC-PD. These two goals were achieved by two separate servo loops. The first loop generated an error signal from an auto-balanced subtraction photodiode, illuminated by a one-percent sample of each optical tone. The difference signal was processed by a PID loop filter and fed back to the amplitude of the oscillator driving the AOM (loop bandwidth: 1 kHz). This ensured that the optical powers of the two lines fluctuated together in the long term. The overall combined power was stabilized by a digital loop that read the current that was applied to the UTC-PD from its DC bias voltage power supply. Corrections to the overall optical power were made by modulating the current supplied to the final optical amplifier (EDFA) before the UTC-PD (loop bandwidth: 10 Hz). This ensured that the UTC-PD photocurrent remained constant. Although this did not guarantee that the radiated terahertz power was constant, it significantly reduced the effect of optical power fluctuations.

Fluctuations in the terahertz power can also come from temperature dependencies of both the photomixing efficiency and the terahertz detection efficiency. After observing both these effects experimentally, the in-vacuum waveguide and the attached components (UTC-PD, isolator, and SBD) were stabilized at the same temperature (297.150 ± 0.001 K). This reduced the terahertz frequency fluctuations caused by the waveguide components. This also has the added bonus of thermalizing N₂O to a constant temperature. However, we did not find an indication that the temperature of the molecule contributed to the frequency variations.

Figure 3(a) shows the phase noise as a function of the offset frequency, f, of the oscillator in both the free-running and locked states. When locked to the N₂O rotational transition, the oscillator gained a servo bump near the PID loop bandwidth. An f⁻² dependence developed for frequencies below the PID bandwidth, a characteristic of white frequency noise expected from a quantum reference. This dramatically reduced phase noise at lower carrier offset frequencies and led to a $>40$ dB reduction in phase noise, extrapolating to the offset frequency 1 Hz. The spurs at higher offset frequency were caused by PM. The ∼−100 dBc/Hz floor was caused by the shot noise of the photodiode used in the EO-comb downconversion and detection scheme.

We compare the SNR result with the intermodulation limited stability calculated from the free-running oscillator phase noise at 2f_EOM.⁴⁹ To determine the phase noise of −114 dBc/Hz, we fit the f⁻³ slope and extrapolated below the measurement noise floor to 1.8 MHz offset frequency. This yielded $8.4×10−12/τ$⁠. The lower intermodulation limit exemplifies the spectral purity advantage of a microcomb terahertz oscillator.

Looking at longer averaging times, the oscillator reached a floor of 5 × 10⁻¹² after only 10 s. This floor was ultimately caused by the terahertz power instability, which stemmed from the photomixing of optically injected light. A major contributor to the instability was the process of injection locking. As described above, a small portion of the injection comb light is reflected from the laser output facet and co-propagates with the light from the diode. There is a phase difference between the two fields that changes with the incident (injection) power, temperature, and frequency of the injected light. This fluctuating phase difference results in uncontrolled interference between the reflected comb light and the LD light and, therefore, causes fluctuations in the terahertz power. In turn, this leads to fluctuations in absorption frequency through the AC stark effect (discussed above). Despite conventional methods of optical power stabilization, frequency fluctuations caused by the reflected comb light remained.

Photonic terahertz oscillators will pave the way for future terahertz applications, enable fundamental studies of molecular physics, and promise the development of novel quantum sensors. We constructed and characterized an oscillator consisting of a pair of diode lasers simultaneously incident on a UTC-PD. To enable precision spectroscopy, the differential phase noise between the two diode lasers was minimized by optically injecting two modes of a low-noise DKS in each diode. The resulting terahertz wave was emitted into a waveguide filled with gaseous N₂O. PM spectroscopy produced an error signal to lock the frequency difference between the two LDs to the center of a rotational transition. This demonstration led to a terahertz oscillator with a sub-10 Hz linewidth and a fractional frequency stability of 5 × 10⁻¹² at 10 s of averaging time, which corresponds to an unprecedented level of stability using a rotational transition in the terahertz domain.

In the future, we intend to improve the oscillator until the frequency stability is limited by the fundamental properties of the molecule. Since the short-term performance is described by the SNR of the error signal, we will explore options to produce more terahertz power. Waveguide terahertz amplifiers are now becoming commercially available. Future work will also include testing the design impacts and applications of other molecules. For example, carbonyl sulfide (OCS) has much more favorable absorption properties around 300 GHz. Stronger absorption allows for a similar performance with a shorter absorption path length. As a result, we expect oscillator stabilization with OCS to have improved SNR in a much smaller physics package than demonstrated here, similar to previous work.⁴¹ 

Determining the fundamental limits of molecular stabilized oscillators will require improvements in the long-term stability, currently limited by terahertz power fluctuations. In principle, the phase drifts responsible for the terahertz power changes in the microcomb oscillator can be stabilized by advanced photonic techniques,⁵⁰ and thus, decrease the floor caused by light shifts. We would also like to consider new feedback techniques that cancel the light shift altogether.⁵¹ 

The frequency discrimination of a microcomb oscillator with molecular rotational levels is a first look at a much broader field of applications. Further measurements of energy level frequency shifts and absorption line shapes can reveal interactions between molecules and their environment. Quantifying these interactions is the operating principle of molecular quantum sensors. Molecules are particularly desirable for sensor design due to the enormous diversity of species and their associated properties, even when limited to just a few atoms. Depending on the environmental variable of interest (e.g., temperature, pressure, and magnetic field), one could tailor a theoretical quantum sensor to be particularly (in)sensitive through careful choice of molecule(s).

Finally, the operation of the terahertz oscillator is considerably simplified since it only requires compensation for the repetition rate drift of the DKS, eliminating the need for self-referencing. This simplification could be beneficial when the oscillator is prospectively used as a terahertz molecular clock, especially since it produces an optically carried terahertz wave. A two-point lock of a microwave-rate optical frequency comb can achieve frequency division in the terahertz frequency into baseband.⁵² This approach avoids the complexities associated with intricate optical amplification and frequency doubling that are inherent in self-referencing a comb. In contrast to other architectures that utilize self-referenced DKSs as the clockwork of optical transitions,⁵³ this approach offers the advantage of realistic photonic integration of terahertz local oscillators in an ultracompact package. Terahertz oscillators can be easily interfaced with miniaturized frequency combs in their simplest form. This implementation provides a practical application of DKS comb technology.

We thank Mark Yeo and Tomohiro Tetsumoto for their initial input on this project. Furthermore, we thank Tomohiro Tetsumoto for his work in designing the SiN microresonator utilized in this study.

The authors have no conflicts to disclose.

James Greenberg: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Visualization (lead); Writing – original draft (lead); Writing – review & editing (equal). Brendan M. Heffernan: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – review & editing (equal). Antoine Rolland: Conceptualization (equal); Formal analysis (equal); Methodology (equal); Project administration (lead); Supervision (lead); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.