Ultra-high precision comb-locked terahertz frequency-domain spectroscopy of whispering-gallery modes

Frequency-domain spectrometers (FDS) based on optoelectronic terahertz (THz) generation driven by the difference frequency of two optical fields transfer the tunability and precision of the lasers to the THz domain.¹ State-of-the-art FDS rely on the frequency stability and phase noise of two independent laser sources due to the difficulty of establishing a lock between sources separated by multiple GHz. Moreover, several tunable distributed feedback (DFB) lasers are combined to take full advantage of the bandwidth of current photomixers.^2,3 Preserving a lock during scanning over hundreds of GHz is even more challenging. The system presented here is based on an external endless frequency shifter of the optical comb spectrum of a short-pulse erbium-fiber oscillator⁴ introduced in Ref. 5 and phase-stable tuning is characterized in Ref. 6. It establishes phase-predictable tuning of a continuous-wave (CW) external-cavity diode laser at speeds up to 1 THz/s and over spectral bands as large as 10 THz suitable for precision broadband IR spectroscopy.⁷ Details on fast tuning, data acquisition, and frequency calibration based on a simultaneously sampled marker can be found in Ref. 7. This optical synthesizer is combined with a second fixed-frequency phase-locked laser, thus enabling comb-referenced generation and coherent detection of CW THz radiation by (difference) frequency mixing.⁸ This system has the potential to enable precision measurement equipment to support the development of 6 G terahertz photonics.^9,10

The laser engine of the THz spectrometer is shown in Fig. 1. Two CW external-cavity diode lasers (ECDLs) are phased-locked to a common frequency comb spectrum generated by a mode-locked erbium-fiber oscillator. Within the locking bandwidth (∼1 MHz), they inherit the phase-noise properties of the comb lines. Since the THz signal is based on difference-frequency generation, it is sufficient to lock the repetition rate f_rep = 200 MHz to a low-noise radio frequency (RF) reference oscillator. By locking the RF oscillator to a GPS-disciplined oscillator at 10 MHz, the repetition rate f_rep is referenced to the SI second, allowing for reproducible measurements across different laboratories. Both CW lasers (ν₁, ν₂) are phase-locked to their neighboring comb mode at a fixed RF offset frequency derived from the repetition rate f_rep. To enable broadband tuning of one of the lasers (ν₁) while preserving the phase lock and realizing an optical-frequency synthesizer (OFS), the comb spectrum is shifted by an external endless frequency-shifter.⁴ The frequency shifter is based on serrodyne shifting of the instantaneous carrier frequency by applying a corresponding time-dependent phase shift with an electro-optic modulator (EOM) synchronized to the pulse train. It takes advantage of the 2π periodicity of the optical phase, applying the 2π fly-backs in between pulses, greatly reducing spurious signals. Details on external endless frequency-shifter can be found in Refs. 6 and 11. The ECDL (ν₁) is continuously tunable without any mode-hops over 100 nm. Adapting the spectral width of the comb allows for tuning over the common spectral range. The tuning speed of up to $>1$ THz/s has been demonstrated for a predetermined frequency sweep limited by the mode-hop-free tuning speed of the CW laser.

We experimentally verify the frequency resolution and absolute frequency stability of the novel FDS by analyzing ultra-high quality THz whispering-gallery modes (WGMs). A schematic of the experimental setup is shown on the right-hand side of Fig. 1. The superimposed light from the FDS based on the OFS is focused onto two photoconductive antennas (PCAs) to generate and detect the coherent THz radiation. Off-axis parabolic mirrors and specifically designed ultra-high weight molecular polyethylene symmetric-pass lenses are used to collimate and focus the THz radiation into a sub-wavelength air–silica step-index waveguide.¹² With a diameter of 200 µm, the dielectric waveguide supports single-mode guidance in the frequency range from about 350 to 700 GHz. More importantly, with an effective refractive index n_eff of about 1.3 at 460 GHz, the coupling waveguide is phase-matched to higher-order radial modes of the investigated whispering-gallery-mode resonator (WGMR).¹³ The WGMR is a 4 mm diameter sphere of high-resistivity silicon with a resistivity $>$10 kΩ cm. These resonators are known to support THz WGMs with extremely narrow linewidths (high quality factor).¹³ The WGMR is mounted on a computer-controlled translation stage to adjust the position of the resonator relative to the waveguide; this allows for precise manipulation of the evanescent coupling efficiency of the WGMs, which is essential to achieve near-critical coupling, as discussed below. Since the WGMs are sensitive to environmental temperature and humidity fluctuations, the entire setup is in a thermally insulated enclosure with a relative humidity of less than 10%.¹⁴ The temperature of the resonator is actively controlled using Peltier elements, reaching typical temperature stability better than ±5 mK over 48 h. The thermal isolation housing sits on an active vibration isolation table to reduce coupling to seismic noise in the lab. Please note that the WGMR is held at the poles of the sphere. Hence, the investigated WGMs confined to the equatorial plane of the resonator are not disturbed by the7 mount.

The photocurrent at the PCA receiver is recorded using a high-resolution oscilloscope. The signal is frequency-modulated due to the imbalanced THz interferometer and analyzed with a Hilbert transform data analysis procedure. The Hilbert analysis allows for the retrieval of amplitude and phase information from the photocurrent with a frequency resolution only limited by the linewidth of the THz laser source.¹⁵ To characterize the THz WGMs, we normalize the transmission of the waveguide coupled to the WGMR (sample scan) to the transmission without a resonator (reference scan). This allows for full characterization of the WGMs while minimizing the impact of standing waves in the spectroscopy setup.

The zero of the difference frequency is calibrated by sweeping the tunable laser over the fixed laser by identifying a low-frequency beat note. It is sufficient to identify the common mode order since both lasers are locked with a common frequency offset to the neighboring mode. The common frequency offset is derived from f_rep to provide a stable low-phase-noise signal, which is common mode with respect to the difference frequency. Residual frequency errors are well below the stability of the experimental WGMR setup. The absolute stability and calibration of the difference frequency scan achieve a measurement time-limited signal-to-noise ratio (SNR) when averaging multiple scans.^8,11 A typical normalized waveguide transmission when coupled to the WGMR is shown in Fig. 2. Increasing the number of averages clearly improves the achievable SNR.

To verify the long-term stability of the setup, the dataset shown in Fig. 2 is grouped into several blocks of data over time and evaluated independently, to estimate the fluctuations in the system. During the measurement, 100 traces are recorded over 3 h with a fixed coupling between the waveguide and WGMR. Each trace consists of ten scans averaged in the oscilloscope. Afterward, 100 reference traces are measured over the same timeframe. Each of the 100 data traces is Hilbert-transformed, normalized to a respective reference scan and fitted, to retrieve the center frequency, Q-factor, and signal-to-noise ratio of the resonance at 472 GHz. For each block of ten consecutive traces, the fit parameters are averaged, and the standard deviation is determined, to get an error estimation. This results in ten data points for the resonance frequency over time, shown in Fig. 3. Within ∼3 h, the determined frequency deviation of the resonance lies below 250 kHz.

Typical measurements of the normalized transmission and phase of the WGMR in the frequency range from 443 to 479 GHz over several free spectral ranges (FSRs) of the WGMR are shown in Figs. 4(a) and 4(b), respectively. Four main mode families, each with their particular FSR, can be identified by comparison with COMSOL Multiphysics® finite element simulations. The excited WGMs are transverse electric (TE) modes and are labeled according to the index in the azimuthal (m), polar (p), and radial (q) directions. For example, mode TE_24,0,11 has 24 wavelengths around the circumference of the WGMR, a single maximum in the polar direction and 11 maxima in the radial direction.¹⁶ 

The scan in Fig. 4 encompasses 2.5 × 10⁶ points over 40 GHz, leading to a 20 kHz resolution. The exceptional absolute frequency stability of the FDS facilitates highly reproducible scans and allows for the averaging of multiple scans. In Fig. 4, 150 scans are averaged to improve the SNR (as discussed above). The acquisition time per scan is about 5 s corresponding to a scanning speed of 8 GHz per second.

The spacing between the WGMR and the waveguide is chosen such that the modes are under-coupled, as can be seen from their corresponding phase profile.¹³ At critical coupling, the phase profile features a π phase jump at the resonance frequency.¹⁷ Resolving this step is an ideal gauge for the effective frequency resolution of the FDS and the stability of the system. To this end, we optimize the distance between the WGMR and the waveguide to achieve close to critical coupling for the mode at 472.2 GHz (TE_25,0,11; highlighted in red in Fig. 4). The corresponding phase profiles for five different distances are shown in Fig. 5(a) and zoomed in with ±200 kHz detuning around the resonance frequency in Fig. 5(b). The WGM is overcoupled at a relative distance of −2 µm and eventually becomes undercoupled as the distance between the waveguide and the WGMR increases. At a relative spacing of 0 µm, the WGM is very close to critical coupling, with a relative width of the step function of less than 200 kHz, demonstrating the exceptional frequency resolution and stability of the novel FDS. Please note that the expected frequency resolution of the THz system itself is estimated to be several kHz, which can be further improved by locking the frequency comb to an optical reference. This being said, minute fluctuations in the WGMR’s environment can lead to a broadening of the averaged phase transition and impede precise control of the coupling position, for example, due to thermal expansion and the thermo-optical coefficient of silicon.¹⁸ 

In addition to the exceptional frequency resolution, the FDS also has an outstanding frequency stability. To further experimentally verify the performance of the FDS, we measure the frequency shifts of the WGMs as the relative spacing between the WGMR and the waveguide is altered. The waveguide’s presence in the evanescent field of the WGMs is sufficient to shift the resonance frequencies by a few MHz. This is due to the change in the refractive index in the WGMs’ environment. Figures 6(a)–6(c) show exemplary resonance frequency shifts of modes TE_22,0,12, TE_24,0,11, and TE_27,0,10 for a relative waveguide WGMR spacing of about 0–140 µm, with 0 µm arbitrarily defined as the closest measured position. Note that each measurement is repeated for six different temperatures (vertically offset curves), which will be discussed in detail below. The resonance frequencies at each position are extracted by simultaneously fitting an analytical model to the normalized transmission and phase profile of the WGMs.¹⁸ The relative frequency shifts (black dots) follow an exponential trend, which can be seen from the very good agreement with the fit (color-coded lines). Interestingly, mode families TE_m,0,12 and TE_m,0,11 experience a blue shift, while mode family TE_m,0,10 experiences a red shift. Notably, there is a transition from a strong blue shift (TE_22,0,12) to a red shift (TE_27,0,10) with mode TE_24,0,11 experiencing a slight blue shift. A comparison with 3D finite element simulations reveals that mode families TE_m,0,12 and TE_m,0,11 have a lower effective refractive index n_eff and mode family TE_m,0,10 has a higher n_eff than the coupling waveguide. In addition, albeit larger, mode family TE_m,0,11 has an n_eff very similar to the waveguide. Accordingly, a mode with lower n_eff than the coupling waveguide is blueshifted, while a mode with an n_eff larger than the coupling waveguide is redshifted. To the best of our knowledge, this is the first time this behavior of WGMs has been observed.

As discussed above, we also analyze the temperature tuning of the WGMs, where each frequency shift measurement due to the presence of the waveguide is repeated for six temperatures from 31.6 to 33.6 °C in 0.4 °C steps. At each temperature, the intrinsic resonance frequency of the WGMs is extracted from the exponential fits (resonance frequency at an infinitely large waveguide resonator distance). The corresponding intrinsic resonance frequencies for WGMs TE_22,0,12, TE_24,0,11, and TE_27,0,10 are plotted in Figs. 6(d)–6(f), respectively. As expected, the intrinsic resonance frequencies follow a linear trend and are redshifted with an increase in WGMR temperature.¹⁸ The linear fits (red solid lines) show an excellent agreement with the data. The corresponding residuals are plotted below each graph. They are in the order of a few tens of kHz, clearly highlighting the frequency-domain spectrometer’s exceptional frequency stability and reproducibility. Those results are at least two orders of magnitude better than typical existing commercial systems based on optical difference frequency generation and pave the way for highly sensitive sensors.¹⁷ 

The presented results clearly highlight the potential of the novel THz frequency-domain spectrometer based on a comb-locked frequency synthesizer. The spectrometer provides unprecedented opportunities for THz spectroscopy applications due to its capacity for kHz-level resolution and frequency stability while maintaining a bandwidth of several THz with scanning speeds of $>1$ THz/s. In particular, in combination with the ultra-high-Q THz WGMRs, a highly selective and sensitive spectrometer for THz sensing applications can be easily envisioned.^19,20 Moreover, referencing the novel spectrometer to a GPS-disciplined oscillator allows for reproducible results across different laboratories. This provides a significant advantage for high-precision THz spectroscopy. Our results also revealed a blue and a red frequency shift of the WGMs due to the presence of the dielectric coupling waveguide. This previously unobserved behavior provides novel insights into the intriguing physics of THz WGMs.

The authors have no conflicts to disclose.

Sebastian Müller: Data curation (lead); Investigation (equal); Methodology (equal); Writing – review & editing (supporting). Kane Hill: Formal analysis (equal); Visualization (equal); Writing – review & editing (supporting). Dominik Walter Vogt: Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Thomas A. Puppe: Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Writing – review & editing (equal). Yuriy Mayzlin: Writing – review & editing (supporting). Rafal Wilk: Project administration (equal); Supervision (equal); Writing – review & editing (equal).

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