2.1.

Schematic and Principle of the Hertz-level Broadband Spectrometer

Figure 1 illustrates the schematic and the working principle of a hertz-level broadband spectrometer based on dual RF modulation. As shown in Fig. 1(a), an external cavity diode laser with a widely tunable frequency range is modulated by two RF signals via an electro-optic intensity modulator (EOM), generating four optical sidebands in the frequency domain. The dual-RF modulated light is used to probe a reference cavity with an FSR smaller than or similar to the modulation frequency, such as fiber loop cavities, fiber linear cavities, and Fabry–Perot cavities. Important is only a high finesse and narrow linewidth of the reference cavity. Note that this reference cavity also exhibits dispersion, such that the FSR changes with wavelength. Depending on the FSR of the reference cavity, the frequency of the two RF signals $f_{\mathrm{mod}1}$, $f_{\mathrm{mod}2}$ ($f_{\mathrm{mod}1}>f_{\mathrm{mod}2}$) can be set to around $(n+1/2)\times \mathrm{FSR}$, where $n$ is an integer number or zero. The frequency difference $f_d$ between the two RF signals can be set to around a few megahertz, depending on the optical linewidth of the dispersive cavity. The transmitted or reflected light of the reference cavity is monitored by a photodiode (PD) and recorded by an oscilloscope. Figure 1(b) shows the principle of time-to-frequency conversion based on the dual RF modulation. As plotted in the frequency domain [upper part of Fig. 1(b)], we determine the moment in time ($t_0$) when the laser carrier frequency $f_{\text{carrier}}$ is exactly in the middle between two reference cavity resonances, which are separated from each other by $(2n+1)\times \mathrm{FSR}$. Just a tiny amount of time before $t_0$, the modulation sidebands (1) and (2) cross a cavity resonance, generating calibration markers (1) and (2) in the time domain. When the tunable laser continues sweeping toward higher frequencies, the modulation sidebands (3) and (4) will generate additional calibration markers. As a result, we measure four calibration marker transmission dips that are symmetric around $t_0$. These are used to determine the local FSR of the reference cavity with ultrahigh precision as well as the scan speed of the laser. In addition to the four calibration markers in the time domain, we also measure additional transmission dips when the tunable laser itself crosses the cavity resonances, as shown in the lower part of Fig. 1(b). This pattern repeats every time the tunable laser crosses one FSR of the reference cavity. An animation of this measurement scheme is available in the Supplementary Material. As shown in Fig. 1(b), the local FSR of the dispersive cavity is determined by $(2n+1)\times \mathrm{FSR}=2f_{\mathrm{mod}1}+f_{?}$ where $f_{?}/2$ is the frequency difference between the optical sideband $f_{\mathrm{mod}1}$ and nearby cavity resonance when the carrier frequency lies in the middle of two cavity resonances. Since we know $f_{\mathrm{mod}1}$ and $n$, we just need to determine $f_{?}$. From Fig. 1(b), we can see that the laser took the time interval $(t_3-t_2)$ to scan across $f_{?}$. In addition, we know a precise value for the current laser scan speed, which is given by $v_{\text{scan}}=f_d/(t_2-t_1)$ or $v_{\text{scan}}=f_d/(t_4-t_3)$, where $f_d=f_{\mathrm{mod}1}-f_{\mathrm{mod}2}$ is the difference between the RF modulation frequencies. Thus, we obtain $f_{?}=v_{\text{scan}}\times (t_3-t_2)$. We can take the average of the two ways to calculate the scan speed $v_{\text{scan}}$ to increase the measurement precision. Note that we can set the RF modulation frequencies close to each other, such that all the calibration markers 1–4 are recorded with little temporal offset. This allows us to achieve a nearly instantaneous calibration of the cavity FSR and the current laser scan speed. Under this setting, the measurement is largely insensitive to fluctuations of the laser scan speed, which we assume to be constant within the tiny time interval $(t_3-t_2)$. In comparison, for a single RF modulation scheme, the calculation of the laser scanning speed would involve averaging the scanning speed over a much longer time duration. During this period, the laser scans across $(n+1/2)\times \mathrm{FSR}$. This would induce a relatively large frequency uncertainty in $f_{?}$. Details on the FSR calculation can be found in Note 1 in the Supplementary Material. As shown in Fig. 1(a), by utilizing the calibrated dispersive cavity and optionally incorporating an absolute frequency reference from atomic or molecular absorption lines, this spectroscopy method can be applied to measure integrated photonic devices like microresonators, for open-path gas sensing or generally optical frequency metrology. Notably, this approach is highly robust for out-of-lab applications, as it does not depend on mode-locked frequency combs or optical stabilization schemes, making it particularly advantageous for use in challenging or rough environments.

2.2.

Characterization of the Fiber Loop Cavity

As a proof-of-concept demonstration, we verify our scheme by measuring the dispersion of a fiber loop cavity. The fiber cavity is made from a 10-dB fiber coupler and 5 m standard telecom fiber (SMF-28) with a zero-dispersion wavelength of 1310 nm. The FSR of the fiber cavity is 39 MHz, and the mode linewidth is 1 MHz, which limits the measurement speed to around $1\mathrm{THz}/\mathrm{s}$.²⁰ To demonstrate the ultrahigh frequency precision of our scheme, we use a $1.3\text{-}\mu \mathrm{m}$ laser with a tuning range from 1270 to 1330 nm in order to resolve the small FSR variation of the fiber cavity around the zero-dispersion wavelength. In the experiment, the tunable laser is modulated by two 20-GHz signals with a 4-MHz frequency difference. The two RF signals are combined with a power combiner and applied to the EOM. The light transmitted through the fiber cavity is detected by a PD and recorded using an oscilloscope with a memory depth of 62.5 million data points. The frequency resolution is determined by the laser scanning speed ($1\mathrm{THz}/\mathrm{s}$) and the oscilloscope sampling rate ($5\mathrm{MSa}/\mathrm{s}$), resulting in a frequency resolution of 200 kHz. This resolution is presently constrained by the memory depth of the oscilloscope utilized in our measurements, which limits the sampling rate to $5\mathrm{MSa}/\mathrm{s}$. However, using a more advanced oscilloscope with a memory depth of $3\times {10}^9$ points could elevate the sampling rate to $300\mathrm{MSa}/\mathrm{s}$, thereby achieving a frequency resolution of 3.3 kHz. In this case, the laser linewidth of 10 kHz would impose the ultimate limit on frequency resolution. The accuracy of the FSR measurement can be traced to that of the modulated RF signal, which is referenced to a rubidium clock with an accuracy of $5\times {10}^{-11}$.

Figure 2(a) shows the transmission spectrum of the fiber cavity from 1270 to 1330 nm; the inset shows an enlarged section of the data around 1300 nm. The deep transmission dips are fiber cavity resonances seen directly by the carrier, while the four additional small dips within one FSR result from the modulation sidebands crossing resonances. Figure 2(b) shows the measured FSR evolution of the fiber loop cavity as a function of wavelength around the zero-dispersion regime. The blue trace (upper panel) in Fig. 2(b) shows the results calculated from the dual RF modulation scheme. With a sub-10-Hz frequency precision, the blue trace clearly resolves the small FSR variation ($<800\mathrm{Hz}$) for a range of 11 THz and reveals the evolution of the cavity dispersion from normal at short wavelengths (FSR increasing with wavelength), crossing zero (region with nearly constant FSR), and to anomalous dispersion at longer wavelengths (FSR decreasing with wavelength). The black trace is the second-order polynomial fit with the zero-dispersion wavelength at 1315 nm. For comparison, the red trace (lower panel) in Fig. 2(b) shows the calculated FSR evolution based on a single RF modulation sideband (20 GHz modulation frequency), which cannot resolve the small changes of the cavity FSR. This demonstrates that our dual RF modulation, with ultrahigh precision in determining the laser scan speed and cavity FSR, is critical for the effectiveness of this spectroscopy method. Figure 2(c) shows the frequency difference of the measured FSRs by dual RF modulation with respect to the fitted values. Figure 2(d) shows a histogram of the frequency difference based on the data in Fig. 2(c), with a root-mean-square deviation of 8.3 Hz. To further verify our study, additional experiments are performed by using two independent 5-m fiber loop cavities (see Note 2 in the Supplementary Material for details).³¹ These results confirm the ultrahigh-frequency precision of our dual RF broadband spectroscopy.

Fig. 2

Mode spectrum measurement of a fiber loop cavity calibrated by dual RF modulation. (a) Transmission spectrum of the 5-m fiber loop cavity. Inset, zoomed-in section showing the calibration markers around 1300 nm. The deep transmission dips are cavity resonances measured by the sweeping carrier laser while the small four dips within one FSR result from the RF modulation sidebands. (b) Measured FSR evolution (blue, upper panel) of the fiber loop cavity interrogated by the dual RF modulation scheme, together with a second-order polynomial fit (black), in contrast to the result (red, lower panel) measured by single RF modulation. (c) Frequency difference between the measured FSR by dual RF modulation and the fitted curve in panel (b). (d) Histogram of the data in panel (c) and a fitted Gaussian curve with an rms deviation of 8.3 Hz.

Figure 3(a) shows the calculated group velocity dispersion ${\beta }_2$ of the 5-m-long fiber cavity, based on the fitted data in Fig. 2(b). The results agree well with the dispersion of standard telecom fiber. The blue trace in Fig. 3(b) shows the corresponding group delay dispersion (GDD) of the 5-m-long fiber loop cavity, which includes the dispersion of a 10-dB coupler that is part of the cavity. We can use this setup to precisely measure dispersion of optical fibers by adding fiber to the cavity and measuring the change in dispersion. The red trace in Fig. 3(b) shows data from another measurement with 3 m fiber removed from the cavity, leading to a 2-m-long cavity with a zero-dispersion wavelength of 1318 nm. By subtracting the GDD of the 2-m-long fiber cavity from that of the 5-m-long cavity, we can get the GDD of the removed 3-m-fiber, which is shown as a black trace in Fig. 3(b) with a zero-dispersion wavelength of 1312 nm. From the measurements, we can see that the zero-dispersion wavelength of longer cavities approaches that of the fiber used.

Fig. 3

Measured dispersion of the fiber cavity based on the fitted trace in Fig. 2(b). (a) Group velocity dispersion of a 5-m fiber cavity. (b) GDD of a 5-m-long fiber cavity (blue), 2-m-long fiber cavity (red), and 3 m of optical fiber.

2.3.

Characterizing an On-chip Optical Microresonator

Instead of measuring dispersion of optical fibers, our method can be easily applied to spectrally characterize the optical properties of different optical components, such as dispersion-engineered broadband mirrors and integrated photonic circuits. We show this by characterizing the transmission spectrum of an optical microresonator.¹⁹ As shown in Fig. 1(a), one part of the tunable CW laser is modulated by two RFs and injected into the fiber cavity to generate precise reference markers that are recorded with an oscilloscope. This allows us to know the current frequency of the laser with high precision at any given time during the sweep. We can now use another part of the laser for spectroscopy by simultaneously measuring a test device and recording the transmission signal on another channel of an oscilloscope, as shown in Fig. 1(a). To demonstrate this technique, we use the resonance frequencies of the calibrated 5-m fiber cavity as frequency markers to measure the transmission spectrum of an in-house fabricated ${\mathrm{Si}}_3{\mathrm{N}}_4$ ring resonator. The ${\mathrm{Si}}_3{\mathrm{N}}_4$ resonator is made from a 750-nm ${\mathrm{Si}}_3{\mathrm{N}}_4$ thin film that is deposited onto a $3\text{-}\mu \mathrm{m}$ ${\mathrm{SiO}}_2$ intermediate layer on a silicon substrate via low-temperature reactive sputtering.^32,33 The ${\mathrm{Si}}_3{\mathrm{N}}_4$ microresonator used in the experiments has a diameter of $200\mu \mathrm{m}$ and a waveguide cross section of $1.8\mu \mathrm{m}\times 750\mathrm{nm}$. The resonator has an FSR of 231 GHz and an intrinsic optical quality factor of 3 million. The resonance frequencies of a mode family in a dispersive resonator can be described by a Taylor series as^34,35

Fig. 4

${\mathrm{Si}}_3{\mathrm{N}}_4$ resonator mode spectrum and dispersion measurement. (a) Normalized transmission spectrum of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ on-chip resonator with red markers on the fundamental TE mode family. (b) Zoomed-in spectrum around 1270.6 nm together with fiber cavity resonance markers. Inset, scanning electron microscope image of the $200\text{-}\mu \mathrm{m}$-diameter ${\mathrm{Si}}_3{\mathrm{N}}_4$ resonator used in the experiments. (c) Measured integrated dispersion profile (blue circles) at pump wavelength of 1310 nm together with a second-order polynomial fit (red trace). (d) Optical spectrum of a bright soliton generated in the ${\mathrm{Si}}_3{\mathrm{N}}_4$ resonator pumped at 1310 nm and a ${\mathrm{sech}}^2$ envelope fit (blue dashed trace). A dispersive wave (marked with an arrow) is observed at 1326 nm.

For such measurements, a referenced scanning across multiple FSRs of the reference cavity is used to determine the microresonator mode spacing. The same applies to the measurement of the spacing between hydrogen fluoride (HF) gas absorption lines, which will be shown later. This raises a question: Does the FSR error of the reference cavity accumulate linearly with the number of FSRs, thus degrading the measurement precision when increasing the laser frequency tuning range? In our measurement, we find that this is not the case, since the FSR of the reference cavity versus wavelength is fitted by a polynomial, and the fitted FSR values are used to determine the frequency distance between features under test. To experimentally validate this claim, we employ two independent 5-m fiber cavities to calibrate the frequency scanning of the tunable laser. The transmission of the two fiber cavities is recorded and used to convert the laser sweeping time to the optical frequency offset based on the FSR of each cavity independently. The obtained laser frequencies based on the two independent cavities are compared against each other, and the result shows excellent agreement with an rms value of 220 kHz for a scan from 1270 to 1330 nm. This rms frequency difference remains constant independent of the scan range. Thus, we achieve a relative precision of $2.0\times {10}^{-8}$ for a frequency range of 11 THz, which is limited by the maximum tuning range of the laser used in the experiments. This surpasses existing tunable diode laser spectroscopy methods^15–20,36 by 2 orders of magnitude (see Note 2 in the Supplementary Material for details).

The in situ measured FSR of the fiber cavity and the corresponding reference markers are used to convert the time axis of the oscilloscope into optical frequencies. This allows us to measure the mode structure of the ${\mathrm{Si}}_3{\mathrm{N}}_4$ resonator, including the evolution of its mode spacing, resonance linewidth, and dispersion. Figure 4(c) shows the measured integrated dispersion profile (blue circles) at a central mode at 1310 nm together with a second-order polynomial fit (red), depicting the deviation of resonance frequencies from the equidistant grid spaced by the FSR at the central mode, based on the resonances with red markers in Fig. 4(a). The dispersion profile exhibits anomalous dispersion with mode crossings at 1280 and 1326 nm.³⁷ To verify the dispersion measurement, we test this ${\mathrm{Si}}_3{\mathrm{N}}_4$ resonator for the generation of a soliton frequency comb by pumping an optical mode at 1310 nm.^38,39 Figure 4(d) shows the optical spectrum of a single bright soliton with a fitted sech² envelope (blue dashed line),^39,40 which further verifies the anomalous dispersion regime for the pump mode. In addition, the mode crossing at 1326 nm in Fig. 4(c) induces a dispersive wave (marked with an arrow) at 1326 nm in Fig. 4(d).³⁴

2.4.

Hydrogen Fluoride Gas Spectroscopy

To highlight another application of this spectroscopy method, we demonstrate the measurement of nonperiodic spectra, such as the absorption spectrum of a gas. A fiber-coupled HF gas cell with 50-Torr pressure and 2.7-cm optical path length is used for this demonstration. Just as before, the output light of the sweeping $1.3\text{-}\mu \mathrm{m}$ CW laser is split into two paths; one part is referenced to the 5-m fiber cavity to generate calibration markers, while the other part is used for simultaneously probing the absorption spectrum of the gas cell. Figure 5(a) shows the strong HF molecular absorption lines (P and R branches) in the O-band range. Figure 5(b) shows the zoomed-in spectrum of the P(2) absorption line (solid black line) with fiber resonances (blue line) as frequency references. A Voigt function [dashed red line in Fig. 5(b)] is used to fit the spectral profile. For the fit, we use the calculated Gaussian full width at half-maximum (FWHM) linewidth ($631.0\pm 0.2\mathrm{MHz}$) based on the Doppler broadening.^41,42 The maximum transmission through the HF cell has been normalized to one. The corresponding fit residuals are less than 0.01 across the fitting range, as shown in Fig. 5(c). The fitted Lorentzian FWHM linewidth, including pressure broadening, is $\sim 3150\mathrm{MHz}$, which is higher than the value from high-resolution transmission molecular absorption database (HITRAN). This discrepancy is attributed to a slightly higher than specified gas pressure in the cell. Table 1 shows the measured position of the HF absorption lines in comparison to the HITRAN database.⁴³ Column 2 is the absorption line position calculated from the HITRAN database, corrected with the pressure shift from the vacuum transition wavelength. Columns 3 and 4 are the theoretically calculated Gaussian and Lorentzian FWHM linewidths, respectively. The uncertainty in the Lorentzian FWHM linewidth is calculated based on 20% uncertainty of the gas pressure, as specified by the gas cell manufacturer. The Gaussian linewidth and its uncertainty are calculated at a temperature of $(21\pm 0.2)°\mathrm{C}$. Column 5 shows the measured line positions. Here, we use the measured R(2) line as absolute frequency reference and set its frequency equal to the value calculated from the HITRAN database. The measurement uncertainty of 0.02 pm is calculated based on statistical analysis of multiple measurements acquired over a month. More details on the measurement uncertainty can be found in the Note 3 in the Supplementary Material. Column 6 is the wavelength difference between the measured results and the HITRAN database, showing that the measured line positions are at slightly shorter wavelengths than the calculated ones. Using the calculated Gaussian linewidth in column 3 and the Voigt fitting function, the last column shows the measured Lorentzian FWHM of the absorption lines. Considering that the measured Lorentzian (pressure broadened) linewidth is larger than the calculated numbers, we speculate that the pressure of the HF gas cell used in the experiments is higher than the 50 Torr specified by the manufacturer. The linewidth measurement of the absorption lines is verified through a modulation sideband spectroscopy technique. In this measurement, we modulate 10 GHz sidebands onto the tunable laser with an EOM. Then the laser with the modulation sidebands is scanned across each individual HF resonance. In the measurement, each HF resonance is recorded 3 times: one time by the laser and one time by each modulation sideband. To determine the Lorentzian FWHM linewidth of each absorption line, we use the known 10 GHz spacing between the measured absorption lines as reference. More details can be found in Note 4 in the Supplementary Material.

Fig. 5

Absorption spectrum of a gas cell filled with HF. (a) Transmission spectrum of the HF gas cell between 1270 and 1330 nm. (b) Spectral profile (solid black line) of the HF P(2) line together with the frequency markers (blue) from the 5-m fiber cavity. The dashed red line is a Voigt fit. (c) Fit residuals of the HF P(2) absorption line. The jumps in the fit residual are due to the discrete voltage resolution of the oscilloscope.

Table 1

Measured wavelengths and linewidth of HF absorption lines and comparison with values from the HITRAN database.