Modulated ringdown comb interferometer and associated methods

Abstract

Modulated ringdown comb interferometry (MRCI) combines optical-frequency-comb spectroscopy with massively parallel optical heterodyne detection to simultaneously detect all of the comb lines of the optical frequency comb without spectral overlap. To perform MRCI, an optical frequency comb is coupled into an optical cavity. The optical frequency comb or optical cavity is modulated, at a modulation frequency, such that the optical cavity, in response to each comb line of the comb being swept across a resonance of the optical cavity, transmits one of a sequence of bursts. The sequence of bursts is combined with a multichromatic reference beam and photodetected to generate a sequence of interferograms having an intensity spectrum comprising several radio-frequency (RF) combs. The RF combs are all uniformly spaced by the modulation frequency but have unique offset frequencies. The RF combs are then fit to obtain best-fit decay times that collectively form a ringdown spectrum.

Description

PATENT Client Ref.2024-224-02 Attorney Docket No. UOCO.P2093WO / 00637650 MODULATED RINGDOWN COMB INTERFEROMETER AND ASSOCIATED METHODS CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims priority to U.S. Provisional Patent Application No. 63 / 649,842, filed on May 20, 2024, which is incorporated herein by reference in its entirety. STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[0002] This invention was made with government support under grant number FA9550-19-1-0148 awarded by the Air Force Office of Scientific Research, grant numbers OMA2016244 and PHY2317149 awarded by the National Science Foundation, and grant number 70NANB118H006 awarded by the National Institute of Standards and Technology (NIST). The government has certain rights in the invention. BACKGROUND

[0003] Gas samples relevant to health and environment typically contain many molecular species with concentrations spanning several orders of magnitude. Mid-infrared frequency-comb spectroscopy with high-finesse cavity enhancement has achieved the most sensitive multi-species trace gas detections to date. SUMMARY

[0004] Frequency-comb spectroscopy of a gas sample inside a high-finesse optical cavity benefits from the strong enhancement of absorption path length over a broad spectral range. However, it is challenging to experimentally achieve this strong enhancement over the full spectral range of the optical frequency comb, which may be several hundred nanometers, or more. This challenge is caused by a comb-cavity frequency mismatch that arises, in part, from strongly absorbing compounds in the gas sample. Specifically, the optical frequency comb has comb lines that are uniformly spaced in frequency, where the uniform spacing is usually the repetition rate of a femtosecond-pulsed laser that generates the optical frequency comb. By contrast, the resonant frequencies of an optical cavity are not uniformly spaced in frequency due to the dispersion (i.e., the wavelength-dependent refractive index) of the gas sample and the mirror coatings forming the optical cavity.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0005] Due to the comb-cavity frequency mismatch, it is not possible for all of the comb lines to simultaneously resonate with the optical cavity. For this reason, some prior-art demonstrations of frequency-comb spectroscopy with optical-cavity enhancement have operated over a limited spectral range that is much less than the full spectral ranges of the optical frequency comb and optical cavity. For example, in some prior-art demonstrations, the optical frequency comb is “tightly” locked to the optical cavity so that the comb lines have temporally constant transmission through the optical cavity. Not only does tight locking limit optical spectral coverage, but residual frequency offsets between the comb lines and cavity resonances causes frequency-to-amplitude noise conversion that limits sensitivity.

[0006] The present embodiments solve this challenge with modulated ringdown comb interferometry (MRCI), a technique that overcomes the limited spectral coverage that results from strong intracavity absorption and dispersion. In MRCI, ringdown dynamics are measured from massively parallel comb lines that are transmitted through an optical cavity. The optical cavity or optical frequency comb is modulated such that the comb-line frequencies are swept back-and-forth relative to the resonant frequencies of the cavity. As a result, different comb lines resonate with the optical cavity at different times during each sweep cycle. MRCI combines this modulation with interferometric or frequency-comb-based detection of the comb lines. These detection schemes can advantageously be implemented with high-detectivity photodetectors and without lossy, bulky diffractive optical elements (e.g., diffraction gratings and virtually imaged phase arrays).

[0007] As an experimental demonstration, a modulated ringdown comb interferometer was used to measure both highly dispersive human breath samples and ambient air in the mid-infrared. The optical cavity had a finesse up to 23,000 and optical spectral coverage spanning 1,010 cm-1. This product of finesse and optical spectral coverage is several orders of magnitude higher than all prior-art demonstrations of frequency-comb spectroscopy with optical-cavity enhancement. In the demonstration, twenty distinct molecular species were identified and quantified with sensitivities as low as 8 parts-per-trillion (ppt). These twenty molecular species had concentrations varying by seven orders of magnitude.

[0008] In some embodiments, a method for modulated ringdown comb interferometry includes coupling an optical frequency comb into an optical cavity forming a plurality of cavity resonances. The method also includes modulating, at a modulation frequency, the optical frequency comb or the optical cavity such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts. The method also includesPATENT Attorney Docket No. UOCO.P2093WO / 00637650 combining the sequence of transmission bursts with a multichromatic reference beam to generate a combined optical beam. The method also includes detecting the combined optical beam with a photodetector to generate a sequence of interferograms. The sequence of interferograms has an intensity spectrum comprising a plurality of radio-frequency (RF) combs. Each of the plurality of RF combs has a plurality of RF components that are uniformly spaced by the modulation frequency. The plurality of RF combs have a respective plurality of RF-comb offset frequencies that are unique. The method further includes fitting the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times. The plurality of best-fit decay times form a ringdown spectrum.

[0009] In other embodiments, a modulated ringdown comb interferometer includes an optical cavity forming a plurality of cavity resonances and an input coupler configured to couple an optical frequency comb into the optical cavity. The modulated ringdown comb interferometer further includes a modulator configured to modulate, at a modulation frequency, the optical frequency comb or the optical cavity such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts. The modulated ringdown comb interferometer further includes a beam combiner configured to combine the sequence of transmission bursts and a multichromatic reference beam to generate a combined optical beam. The modulated ringdown comb interferometer further includes a photodetector configured to detect the combined optical beam to generate a sequence of interferograms. The sequence of interferograms has an intensity spectrum comprising a plurality of RF combs. Each of the plurality of RF combs has a plurality of RF components that are uniformly spaced by the modulation frequency. The plurality of RF combs have a respective plurality of RF-comb offset frequencies that are unique. The modulated ringdown comb interferometer further includes a signal-processing circuit configured to fit the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times. The plurality of best-fit decay times forms a ringdown spectrum. BRIEF DESCRIPTION OF THE FIGURES

[0010] FIGs. 1A–1D illustrate modulated ringdown comb interferometry (MRCI), in embodiments.

[0011] FIGs. 2A–2D show an apparatus and survey ringdown spectra collected with MRCI, in embodiments.

[0012] FIGs. 3A–3D illustrate a cavity swept lock, in embodiments.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0013] FIGs. 4A–4D illustrate a data-processing workflow for MRCI, in embodiments.

[0014] FIG. 5 is a flowchart of a method for MRCI, in embodiments.

[0015] FIG. 6 is a functional diagram of a modulated ringdown comb interferometer, in embodiments.

[0016] FIG. 7 is a functional diagram of a modulated ringdown comb interferometer, in embodiments. DETAILED DESCRIPTION Introduction

[0017] Achieving the most sensitive multi-species trace gas sensing is critical for many applications, including precise determination of complex molecular structures, real-time reaction kinetics, atmospheric sensing, and breath-based medical diagnostics. An enhancement cavity formed by a pair of high-reflectivity optical mirrors is a powerful tool for this goal as it enables light to circulate through the same sample, which significantly enhances the absorption path length for sensing the weakest absorption signals. However, laser light cannot be coupled through a cavity unless the laser frequency is precisely matched to a cavity resonance, i.e., the optical frequency accumulates a cavity roundtrip phase delay equal to integer multiples of 2π (see FIG. 1A). Multiple cavity resonances are established with varying spectral gaps between each other caused by intracavity dispersion arising from the refractive index of loaded gas samples and the mirror reflectivities.

[0018] Cavity enhancement with broadband spectral coverage for multi-species detection uses an optical frequency comb, a type of broadband laser that consists of multiple narrow-linewidth optical lines (i.e., comb lines, comb modes, or optical-comb frequency components) emitted at strictly evenly spaced frequencies. All comb lines can sequentially and deterministically couple into the cavity if each of them is swept across a cavity resonance at sufficient modulation depth (see FIG. 1B). Normally measured with a grating monochromator for experiments conducted in the near-infrared, the ringdown time ^^^characterizing the temporal width of the cavity transmitted burst indicates the number of roundtrips laser light can sustain within the cavity. This forms the basics of cavity ringdown spectroscopy for ultra- sensitive detection of intracavity absorption from the loaded samples. Yet, such a normal scheme is challenging to extend to the mid-infrared due to the fact that current mid-infrared photodetection technologies lack detectivity and speed.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0019] The present embodiments implement modulated ringdown comb interferometry (MRCI), a variant of ringdown interferometry. Advantageously, MRCI achieves unrestricted comb-cavity enhancement across an unprecedented spectral range in the mid- infrared. In some embodiments, MRCI uses a Michelson interferometer (see Michelson interferometer 110 in FIG. 1B), commonly believed to be restricted to measuring laser fields with static intensity in time, as a spectrometer for measuring the frequency comb’s intensity dynamics generated from the transient comb-cavity coupling scheme. The temporal dynamics are measured in the Fourier frequency domain, where reflection from a moving mirror in an arm of the Michelson interferometer 110 offsets the Fourier harmonic frequencies decomposing the periodically generated intensity dynamics. As a result, each comb line is measured at anisolated radio frequency (RF) of ^^^ + ^^, where ^ is an integer representing the Fourierharmonic order,is the cavity-length modulation frequency, and ^^is the Doppler frequency shift. Because the comb lines have the same modulation rate ^^, but different Doppler shifts ^^, the intensity dynamics can be observed in parallel for different comb lines free from spectral overlap. Broadband ringdown spectroscopy down to comb-mode resolution is thus realized.

[0020] In MRCI, the transient comb-cavity coupling scheme brings general applicability to sensing real-world samples containing unknown, complex, and dynamic molecular contents. With dispersion immunity, scalable enhancement for both cavity finesse and spectral coverage can be achieved. The present embodiments achieve high products of finesse and optical spectral coverage exceeding 107cm-1, highlighting a new capability to navigate through intense absorption features from molecules including H2O, HDO, CH4, CO2, and N2O to detect the lowest concentration species (hydrocarbons, alcohols, nitric oxide, aldehyde, ketone, and others) at sensitivities down to a few parts-per-trillion. In some embodiments, the optical frequency combs is free-running, obviating the need for high bandwidth feedback loops. This simplification makes possible highly automated and efficient data collection for large-scale measurements. Modulated Ringdown Comb Interferometry

[0021] In some of the present embodiments, MRCI encompasses two ingredients. The first is a high-finesse cavity swept-locked to the incident optical frequency comb to permit its periodic transmission through the cavity once per cavity length sweep. The second is an asynchronous, passively scanning Michelson interferometer to read out the periodic cavity transmission bursts through balanced detection.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0022] FIG. 1C shows how the recorded interferogram, as a function of time, allows high-spectral resolution determination of the ringdown time spectrum. Panel (a): consider a single transmission intensity burst generated from the ^thcavity resonance. Panel (b): related by a Fourier transformation ℱ, the shape of the spectral envelope uniquely determines the ringdown time. A shorter ringdown time corresponds to a flatter spectral envelope. Panel (c): the same transmission bursts repeat with a periodicityPanel (d): the original spectral information is now transformed into a set of Fourier components spaced by ^^. Panel (e): record the same periodic transmission bursts through balanced detection from the Michelson interferometer 110 of FIG. 1B. Panel (f): all Fourier components are frequency-displaced with a corresponding Doppler shift that is proportional to the ^thcavity-resonance wavenumber and the velocity of the delay stage. The spectral information for this cavity resonance is thus engineered into an “RF spectral comb” having a uniform spacing ofand a spectral intensity distribution determined by the cavity ringdown time. Multiple RF spectral combs generated from optical-comb components transmitted through different cavity resonances share the same spacing but with unique frequency offsets (see FIG. 1D). Determination of the spectral intensity distributions for multiple sets of RF spectral combs thus realizes broadband simultaneous ringdown measurements. For more details, see the section below titled “Methodology.”

[0023] The dispersion immunity brought by MRCI allows scalable expansion of spectral coverage up to the full coating range of high-finesse cavities. In the experimental demonstration described below, two high-finesse cavities with intersecting optical axes were used to probe the same gas sample (see FIG. 3A). One cavity permits spectroscopic data collection over the 1,850–2,230 cm-1wavenumber range (near 5-µm in wavelength; finesse peaks at 23,000; ringdown time peaks at 17 µs), the other cavity at the 2,700–3,330 cm-1range (near 3-µm in wavelength; finesse peaks at 14,000; ringdown time peaks at 8 µs). Data collection for either cavity is highly automated (see the section below titled “Methodology”). The comb source used for the 5-µm cavity is a synchronously pumped, singly resonant optical parametric oscillator (OPO) using a type-I phase-matched zinc germanium phosphide crystal. Demonstration in Spectroscopic Sensing

[0024] As an experimental demonstration, MRCI was performed on samples of exhaled breath collected from nasal and oral respiratory airways and ambient air. With a new record-level finesse and broadband coverage, subtle differences in these complex gas samples can now be accurately quantified for a comprehensive list of trace molecular species. Notably,PATENT Attorney Docket No. UOCO.P2093WO / 00637650 nitric oxide, the species with administrative approval for asthma monitoring, can now be robustly measured on top of the strongly saturated water absorption background and down to its extremely low concentration of ~20 ppb.

[0025] Sample data collected with two seconds of acquisition time and with the cavity filled with an exhaled breath sample using MRCI are presented in FIG. 2B. Trend lines highlight the two sets of Fourier components generated at two different optical frequencies. Insets show the fitted ringdown values. A faster drop in spectral intensities with the increase of Fourier harmonics order corresponds to a longer cavity ringdown time. Measured for the same breath sample, full-coverage ringdown spectrum determined from the 5-µm cavity is presented in FIG. 2C, and from the 3-µm cavity is presented in FIG. 2D. Comparing the ringdown data measured with cavity loaded with breath against that measured with empty cavity (held at base pressure below 3 mTorr), the reduction in the ringdown time as a function of optical frequency yields a plethora of molecular absorption features generated uniquely from the loaded breath molecules inside the cavity. Ringdown data measured for the empty 5-µm cavity correspond to water absorption signals at below 2,000 cm-1, arising from residual water molecules present inside the cavity. The ringdown spectrum measured for the empty 3-µm cavity was found to exhibit large but slowly varying reflectivity oscillating at a period of about 300 cm-1wavenumbers, a feature likely due to the mirror coating design. For both cavities, neither the intracavity dispersion from the non-flat mirror spectral response nor the strong molecular absorptions prohibit the utilization of the entire spectral range of the high-reflectivity coatings for ultra-sensitive absorption spectroscopy.

[0026] Molecular cross-section data for twenty species is fitted to the experimental data. Strongly elevated concentrations of multiple species were identified in nasal breath compared with oral breath. The current study focuses primarily on reporting the new capabilities for characterizing highly dispersive gas samples. Nevertheless, this powerful capability may be used to establish a broader utility to observe simultaneously a rich variety of biological conditions through non-invasive measurements of breath.

[0027] Normalized to the acquisition time of 100 s, molecular concentration sensitivities into the parts-per-trillion level (below 1 ppb) are found for fifteen out of the twenty species. The best sensitivity at 8 ppt is obtained for N15NO and15NNO. The minimum detectable absorption at one second per spectral element is 3×10-10cm-1Hz-1 / 2for the 5-µm cavity and 6×10-10cm-1Hz-1 / 2for the 3-µm cavity. DiscussionPATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0028] In the absence of intracavity dispersion, traditional techniques that use interferometry readout of static laser fields generated from tight comb-cavity coupling (i.e., concurrent matching frequencies of different comb lines to different cavity resonances) can be employed. Detection sensitivity down to the quantum shot noise limit can be achieved in the near-infrared leveraging the maturity of photodetectors. MRCI is invented as a new spectroscopy technique for the necessity to enable practical measurement of gas samples of arbitrary dispersion profile. For measurements of exhaled breath and ambient air in the mid- infrared molecular fingerprint region, displacement of cavity resonance frequencies occurs in a range amounting to > 100 times of cavity resonance width even for modest cavity finesse at 1,000. Tight comb-cavity coupling must be replaced with transient coupling schemes and ringdown detection is leveraged as a sensitive technique for probing intracavity absorption.

[0029] The traditional ringdown detection scheme based on grating monochromators is not practical in the mid-infrared primarily due to the slow integration time (≥ 10 μs) of detector arrays to resolve ringdown dynamics. Sensing intracavity absorption is instead based on measuring the cavity-transmitted power rather than ringdown. Highly dispersive virtually imaged phase arrays etalons are required to achieve sufficient spectral resolution (< 1 GHz) and the instantaneous spectral coverage is compromised to well below 100 cm-1, limited by the size of the detector arrays. High optical power loss and strong detection noise floor impose further technological restrictions to achieve the highest possible detection sensitivity. Recent reports on using interferometry for implementing ringdown detection in the near-infrared commonly necessitate the use of tight comb-cavity coupling: High coherence control must be achieved either for ringdown events generation or for the interferometry scan. While spectral resolution below 1 GHz can be achieved, spectral coverages are compromised to < 50 cm-1and finesse- coverage products are more than 60 times below that available with standard grating monochromators. The tradeoff in spectral coverage, cavity finesse, and spectral resolution of previous techniques prevent the development of more extreme sensing capability. This longstanding bottleneck is resolved by MRCI, which makes it possible for extensive compound determination for real-world gas samples which oftentimes are highly dispersive and contain unknown chemical compounds.

[0030] Yet, for the technique to be practically usable in real-world, large-scale settings, equal emphasis must be placed on robustness and simplicity. In MRCI, the occurrence frequency of cavity transmission bursts and the Doppler frequency ^^must have theirfrequency-mixed signals (^^^ + ^^, ^ is an integer; see FIG. 1D) accurately determined freefrom spectral overlap for broadband ringdown detection. The ^^is real-time perturbed byPATENT Attorney Docket No. UOCO.P2093WO / 00637650 comb-cavity frequency detuning, cavity length piezo hysteresis, and intracavity dispersion (see FIGS. 3A–3D), whileis real-time perturbed by mechanical stage moving jitter. Accurate detection of frequency-mixed signals can be achieved directly through data analysis of the collected interferogram. Two pivotal data reconstruction techniques, pulse paring and demodulation, are introduced and detailed in the section titled “Methodology” and FIGS. 4A– 4D. Spectroscopy data collection can thus be executed despite the lack of mutual coherence between ringdown generation and interferometry scan. MRCI thus do not require comb frequencies to be measured and stabilized to cavity resonances at high servo bandwidth. Furthermore, cavity swept lock permits the entire instantaneous comb bandwidth to be coupled through the cavity despite presence of strong intracavity dispersion, and it enables a large dynamic range for servo and robust sustentation against incidental mechanical jittering. These make MRCI more practical and robust for out-of-laboratory operation and at much-improved data collection efficiency.

[0031] Finally, extending comb cavity ringdown to over 1,000 cm-1mid-infrared spectral coverage is a new opportunity to simultaneously detect chemical compounds of various sizes. Molecular absorption spectra are robustly determined against a constant baseline determined by the reflectivity of the cavity mirrors. This provides exceptional versatility to the technique for demanding applications of measuring absorption features of arbitrary spectral widths. As evident from the presented results, rovibrational quantum state-resolved absorption signals from small molecules with a characteristic spectral width of ~0.1–1.0 GHz (e.g., nitric oxide) and state-unresolved signals from medium-size molecules at ~10–100 cm-1(e.g., acetone and methanol) are all robustly measured. This unique and outstanding capability paves the way to detect viruses and proteins that are potentially present in breath, where absorption features span over 1,000 cm-1. Direct quantification of viral loads in exhaled breath can allow differentiation of viral response from host response in diagnostic scenarios such as detection of SARS-CoV-2 infection to yield extra mechanistic insights. MRCI may also be used for fundamental molecular spectroscopic studies. Applications

[0032] Highly-dispersive gas samples can now be extensively measured. MRCI resolves important issues that have prevented promising trace-gas sensing tools from further development, thereby providing a robust prospect for a wide-range deployment for real-world applications. MRCI enables: (1) weakest, strongest, sharpest, and broadest absorption features to be robustly measured; (2) scalable improvement to both spectral coverage and cavity finessePATENT Attorney Docket No. UOCO.P2093WO / 00637650 without sacrificing spectral resolution; (3) comb sources operating under free-running condition and no requirement of high-bandwidth servo control; and (4) robust, automatable, and high-efficiency data collection.

[0033] New practical possibilities are opened for the creation of large-scale breathomics databases for investigations of multiple health-related symptoms in parallel, as well as monitoring campaigns over multiple gaseous emissions for climate change from energy production, vehicles, soil microbes and vegetation. High-quality comb-based massive datasets generated for the “odor” of gas samples will be integrated with machine learning analysis tools to facilitate more thorough investigations for science and new applications for industry. Free from chemical derivatization;, instrument calibration, and absorptions external to the enclosed cavity regions, MRCI promises enhanced accuracy, reproducibility, and reduced labor for large-scale detection of chemical compounds of various sizes and of different isotopologue, isomer, and isobar origins. MRCI can be broadly employed, requiring minimum to no modification of existing experimental setups, and can stimulate high-throughput mirror coating characterization and molecular database construction particularly for weakly absorbing large- size molecules. Comb sources produced with integrated photonics may leverage the compatibility of MRCI with free-running combs to facilitate the construction of portable devices. When laser sources are constructed with simplicity and are robust to maintain working condition, dual-comb readout may prove to be advantageous over Michelson interferometry in practicality for implementing MRCI.

[0034] Lastly, MRCI is a massively multiplexed information extraction methodology, where channels of information encoded into the field intensity dynamics of frequency comb lines can be determined in parallel from its frequency domain equivalent carriers. The method can be generalized to measuring periodic dynamics of various kinds, for example chemical kinetics. To illustrate, assume a chemical reaction is triggered in a repetitive manner with a repetition period commensurate with the reaction time scale. The chemical reaction kinetics will be multiplied with the cavity ringdown dynamics in real time and their product can be determined via MRCI. Pre-measured empty cavity ringdown dynamics allows the chemical reaction kinetics to be separated out. While the interferometry scan can take potentially a much longer time than a single-shot reaction event, this process can be regarded as signal averaging for better signal-to-noise ratio in addition to ensuring sufficient spectral resolution. Methodology

[0035] Cavity Swept Lock and Intracavity DispersionPATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0036] ^^^^, ^^^^, and FSR: The shape of the cavity transmission bursts is dependent on the values of comb repetition rate (^^^^), comb offset frequency (^^^^), and cavity free spectral range (FSR). A general scheme for implementing a cavity swept lock is presented in FIGS. 3A–3D. This scheme is used to fully stabilize all relevant frequencies (^^^^, ^^^^, and FSR) for our data collected at comb-mode resolution. For all data shown in the main text, where data are all collected at a modest instrument resolution ≥ 800 MHz, only the servo utilizing the odd-order Fourier components is engaged for locking the ^^^^-FSR detuning. This is because comb and cavity absolute frequencies change less than the magnitude of instrument resolution over the acquisition time of interferogram. Further engaging the rest of the servos is thus not necessary.

[0037] Intracavity Dispersion: Molecular absorption and mirror spectral response result in displacement of cavity resonance frequency to mismatch from comb lines. To quantify,consider a cavity of length ^^^^, intracavity refractive index ^ = 1 + ∆^, and phase delay perreflection from a cavity mirror surface ∆ . The resonant condition requires the roundtrip total phase delay to be integer multiples of 2^: 2!^^^^ + 2∆ = 2"^, (1)equivalently, $% ⋅ 2^^^^ + ∆^ ⋅ $% ⋅ 2^^^^ + ∆ / ^ = ", (2)where ! = 2^'1 + ∆^($% is wave vector, $% is wavenumber, and " is an integer mode index.From cavity finesse ) defined as the ratio of cavity FSR to cavity resonance width, comb walksoff from cavity resonances when the molecular dispersion term ∆^ ⋅ $% ⋅ 2^^^^ or the mirrordispersion term ∆ / ^ in Eqn.2 exceeds that corresponding to changing " by 1 / ). To illustrate, consider the measurement of an ambient air sample over 2,700–3,330 cm-1. Dispersion isestimated at ∆^ ≥ 4 × 10. / and ∆ ≥ 50°. Take $% = 3,000 cm-1, ^^^^ = 50 cm. Moleculardispersion contributes ∆^ ⋅ $% ⋅ 2^^^^ = 0.12. Displacement in cavity resonance frequencies frommolecular dispersion alone thus amounts to 12% of the cavity free spectral range and 120 times the cavity resonance width, if finesse ) = 1,000. A transient coupling scheme should be employed to replace tight comb-cavity coupling.

[0038] Modulated Ringdown Comb Interferometry (MRCI)

[0039] Theoretical Model: Consider a high-finesse cavity made of two identical cavity mirrors is coupled with a passively-stable frequency comb. The cavity length isPATENT Attorney Docket No. UOCO.P2093WO / 00637650 modulated by a sinusoidal wave at rate ^^centered at the cavity length of 01. The mean cavity free spectral range is matched to the comb spacing. The cavity resonance frequency modulation depth is smaller than the comb spacing but is large enough to permit the instantaneous comb bandwidth to resonant couple through the cavity twice per modulation period: once when the cavity length is swept up, the other down. For simplicity, imagine for now the incident comb is shut off during the cavity length down sweep (more discussions later). Transmission burstsfrom all cavity resonances thus have the same time periodicityand emitted allfrom the cavity up sweep. The electric field intensity components 346^^^,^'5(3 in time 5 from the ^-th cavity resonance can be decomposed into a Fourier serieswhere ^8^is the ^-th Fourier component from the ^-th resonance, and < denotes imaginary unit.

[0040] The Michelson interferometer 110 of FIG. 1B introduces an amplitude modulation to the field intensity profile that is specific to the optical frequency. The laser- intensity-versus-time readout by the photodetectors PD1 and PD2 in the Michelsoninterferometer 110 is expressed as, where ^^ = !^G is theDoppler frequency, !^is the optical wavenumber, and G is the optical path length difference scanning speed assumed to be a constant (more discussions later). The signal carriers arespectrally detected at RF frequencies over |^^^ ± ^^| and the non-cosine modulated terms at|^|^^. They are spectrally isolated if ^^for all ^ are non-equal to integer multiples of ^^ / 2. Acquisition time per interferogram should be sufficiently long to resolve different RF frequency components. Spectral intensities measured by the non-cosine modulated terms can be used to determine detectors’ relative response function to implement digital-autobalancing. The balanced intensity output I'5( after canceling out the non-cosine modulated terms is given byThe expression of 34^^^,^'5(36for a single transmission burst (i.e., cavity resonance swept through a comb line for once) has been previously derived in the art. Omitting commonproportionality and assuming 2 ≫ ^LM,^, ^8^ can be calculated from:PATENT Attorney Docket No. UOCO.P2093WO / 00637650where ^^^,^ = ^ / '−2 ln X^( is the cavity ringdown time, ^ =the cavity round trip time,Y the speed of light, X^ = Z6^ expA−'[^ / 2( ⋅ 201F the round-trip field loss, Z^ the mirrorreflection coefficient,the intracavity absorption coefficient, and erfc the error function. The Λ^'5Q(is given bywhere G^^^^is the cavity length swept velocity when the ^thresonance is scanned to the comb line.

[0041] Eqn.5 is used to fit the experimental data for |^8^| at the same ^ over different ^ to find ^^^,^. For molecular spectroscopy with the goal to determine [^, one compares theringdown times measured with the cavity loaded≠ 0) and unloaded= 0) with asample. From dependence of ^ on X^one finds

[0042] Experimental Implementation: MRCI was implemented with the incident comb power passively staying on, a high-finesse cavity swept-locked to the incident comb lines, and a passively-scanning Michelson interferometer. Theand ^^are not perfectly their sources of incoherence are detailed below along with data-analysis strategies. These strategies are the key to removing the experimental needs to ensure perfect coherence inand that can require dramatically increased experimental complexity and robustness. A detailed workflow of the entire data processing workflow is presented in FIGs. 4A–4D and can be used as a reference for the rest of this section.

[0043] (1) The ^^: Due to the presence of intracavity dispersion and piezo hysteresis, comb lines on cavity resonance at an earlier time during the cavity length up sweep are on resonance at a later time during the cavity down sweep. While two transmission bursts are emitted from each resonance per cavity length round trip modulation period 2, intensity component 34^^^,^'5(36generally repeats at periodicity 2 rather than 2 / 2: the base pattern in repetition is that from a cavity up and subsequent down sweeps. For Eqn. 5 to be directly applicable, the transmission bursts are processed from the up sweep separately from the downPATENT Attorney Docket No. UOCO.P2093WO / 00637650 sweep. A logical TTL function phase-synchronized with the cavity modulation signals is digitally generated in the data post-processing to pick up transmission bursts solely from the up (or down) sweep via multiplication to the raw interferogram data. This is equivalent to having the incident comb source shut off during the down (or up) sweep. The two sets of interferograms, one containing only the up-sweep bursts and the other only the down-sweep bursts, are processed separately by Eqn. 5. On a separate note, for dispersion spectroscopy, the interferogram can be processed without separating the up and down bursts. In this case, Eqn. 5 would be modified to account for an emission time delay between adjacent up and down bursts from the expected time separation at 2 / 2.

[0044] (2) The ^^: Conventional delay stage introduces considerable mechanical jitters causing ^^to fluctuate in time. The jittering ofis small compared with the value of but must be properly accounted for in the data processing for applications at high spectral resolution. Eqn. 4 should be modified into a more general form by replacing ^^5 with !^aHere, a is the optical path length difference scanned. In the experiment, the real-time mapping from a to 5 can be measured with a wavelength-stable continuous-wave (CW) laser co- propagating with the comb light into the Michelson interferometer 110. With both interferogram and space variable a sampled by the same time array, explicit dependence of interferogram on time can be removed using demodulation: first multiply the interferogramI'5, a( with digitally-generated exp'−<^Q^^5( sampled over the same time array, thenbandpass centered at the Doppler frequencies b^^c. This results in

[0045] The demodulated output is a complex function of a single variable a and can be non-uniformly Fourier transformed to find the spectrum of |^8h^| vs. !^. The value of |^8h^| for all ^ can thus be determined. The demodulation is repeated at different ^Qspanning from negative to positive integers: The|^8h^|and|^.8h^|are independently measured by two signalcarriers at ^Q^^but probe the same physical quantity |^8h^| = |^.8h^|(see Eqn. 5). They can be averaged together to improve the signal-to-noise ratio.

[0046] To summarize, the raw interferogram is first linearly decomposed into two interferograms: one contains bursts exclusively from the up sweep, the other down. The twoPATENT Attorney Docket No. UOCO.P2093WO / 00637650 interferograms are respectively processed to find |^8^| containing non-negative ^ using the demodulation technique. The two sets of |^8^| are averaged together and Eqn. 5 is applied to find the ringdown time ^LM,^.

[0047] Apparatus

[0048] Frequency Combs: The comb used for the data collection in the 2,700−3,330 cm-1spectral range is a singly resonant optical parametric oscillator (OPO) built from a periodically poled lithium niobate crystal and synchronously pumped by an ytterbium comb that has a 137-MHz repetition rate, is centered at 1064 nm, and has a pulse duration of 100 fs. The instantaneous bandwidth is 180 cm-1. Spectral tuning is performed by translating the crystal location. More details about this OPO can be found in F. Adler et al, “Phase- stabilized, 1.5 W frequency comb at 2.8–4.8 μm,” Optics Letters 34, pp. 1330–1332 (2009).

[0049] The comb used for the 1,850−2,230 cm-1spectral range is a singly-resonant OPO using a zinc-germanium-phosphide (ZGP) crystal synchronously pumped by a thulium comb that has a 110-MHz repetition rate, is centered at 1,960 nm, and has a pulse duration of 1,000 fs. This comb source is developed to overcome the difficulty of oxide materials in generating sufficient power per comb tooth at ≥ 4.8 µm wavelengths necessary for cavity- enhanced spectroscopy. The ZGP-based OPO was designed using the same strategy previously described in K. Iwakuni et al., “Phase-stabilized 100 mW frequency comb near 10 μm,” Appl Phys B 124, 128 (2018). Both the spatial walk-off from birefringence and the temporal walk- off from group velocity mismatch are considered to design the threshold pump power sufficiently small compared to the maximum available pump power. A ring-cavity geometry, as opposed to a linear geometry, is adopted to avoid round-trip signal absorption loss from one additional pass through the crystal. The cavity is unpurged but sealed for passive frequency stability. The OPO had an instantaneous bandwidth of 60 cm-1and is broadly tunable in the range of 1,850−2,230 cm-1simply by translating the OPO cavity length to manipulate the group delay dispersion. Limited power below 1,850 cm-1is due to water absorption and above 2,230 cm-1due to carbon dioxide absorption. When the idler is tuned to 2,040 cm-1(4.9 µm), powers per comb tooth up to 13 µW is measured. This is more than a factor of ten higher than all comb sources ever reported near the 5-µm wavelength.

[0050] Data Acquisition: Both cavities have their mean free spectral range matched to twice the frequency spacing of the incident combs. The 5-µm cavity is 68 cm in length and modulated at 13 kHz. The 3-µm cavity is 55 cm in length and modulated at 18 kHz. The modulation rates are set sufficiently low to enable sufficient time for each ringdown event toPATENT Attorney Docket No. UOCO.P2093WO / 00637650 eventually decay down to the detector noise floor. Modulation rate is higher for the 3-µm cavity due to smaller cavity ringdown times overall. Servo error signals generated from the photodetectors in the Michelson interferometer are summed and demodulated at the third harmonics of the modulation frequencies for feedback stabilization of the cavity lengths to the comb line frequencies. For details, see the section on cavity swept lock. The OPO spectral center and the interferometer’s optical path length difference (OPD) scanning speed are feedforward adjusted automatically. The feedforward map ensures every interferogram scan is executed such that the Doppler frequencies stay about the same and are sufficiently detuned from integer multiples of half the cavity length modulation rates to avoid spectral overlapping issues. For each scan, the OPD scanning speed is fixed at no faster than 200 mm / s. A total of three data channels low passed at 1 MHz are collected simultaneously at a 2-MSPS sampling rate and 16-bit resolution: interference fringes for the comb light using two photodetectors, and for the CW laser with one photodetector. No photodetectors require cooling by liquid nitrogen. Data collection for the entire 5-µm spectral range (1,850–2,230 cm-1) is fully automated with servo loops robustly sustained. For the 3-µm range (2,700–3,330 cm-1), however, the actuator used for adjusting the crystal location introduces considerable mechanical jitter that occasionally disengages the servo. Breath and air data are processed at 800-MHz instrument resolution and the acquisition time per interferogram is about two seconds. Full survey spectrum for the 5-µm spectral range at 380 cm-1coverage is collected with ~500 interferograms, while ~1,400 interferograms for the 3-µm range at 630 cm-1coverage. Empty cavity survey data are collected at 2-GHz instrument resolution. For all experimental data, the demodulation is performed for the 5-µm data up to the 20thFourier harmonics and the 3-µm data up to the 15thharmonics. This utilized signal carriers measured with high signal-to-noise ratio up to 300-kHz electronic bandwidth. Fourier-harmonics spectra collected at different spectroscopic regions are summed before determining the full coverage ringdown spectrum. Data analysis is performed using a 32-core 3975WX CPU, used for accelerating ringdown extraction, and an RTX A6000 GPU, used for accelerating demodulation and Fourier transforms.

[0051] Molecular Line Fitting

[0052] The HITRAN database (2020 edition) is used to extract concentrations totaling twenty molecular species. Methanol (214.2 K, 102.7 Torr) and acetone (209.8 K, 109.6 Torr) use cross sectional data directly measured at specific temperature and pressure due to lack of line intensity data. For the other eighteen species, cross-sectional data are calculated from the line intensity data using Voigt profiles that were evaluated at the experimental conditions of 293.15 K and 100 Torr. Pressure broadening by air is considered while self-pressure broadeningPATENT Attorney Docket No. UOCO.P2093WO / 00637650 and pressure-shift of the line frequency centers were ignored. Molecular-line fitting is performed bearing in mind that:

[0053] (i) Not all species that are detected may be fitted, which could partly be attributed to lack of cross-sectional data. For exhaled breath, more than 1,000 species have been reported and over 40% are hydrocarbons that can be spectroscopically detected in the wavelength range of 3–4 µm. Unfitted species that may be present in the experimental data are generally large molecules exhibiting state-unresolved ultra-broadband absorption features.

[0054] (ii) Strongly absorbing species, such as water and carbon dioxide, produced saturated absorption features. Errors in the prediction of absorption lineshape at far-off line centers prohibits strongly absorbing species to have their concentrations determined over their saturated absorption regions. Further, other species that are weakly absorbing could also be impacted when their line centers spectrally overlap with the saturated absorption features.

[0055] The following fitting strategy was developed: a sliding window was used to select one 10-cm-1spectral range at a time for molecular line fitting. The window is step incremented by 0.5 cm-1to uniformly sample the whole coverage (1,010 cm-1). Least squares fitting is performed with residuals given by the difference in the logarithm of absorption spectra. After fitting all windows separately, a slowly varying baseline is introduced for the whole coverage to vertically offset the simulated data to better match the experimental data. Here, the baseline is used to mimic the absorption features from unknown species and correct mis- predicted absorption lineshape from the fitted species, both assumed to have a bandwidth broader than the window width at 10 cm-1. With its introduction, the fittable species can be determined at minimized influence from these broadband features. The spectrum is re-fitted with the simulated data added to the baseline in each window to compare it to the experimental data. After all windows are fitted, the baseline is updated again. The whole process is iterated five times for convergence. Fitted concentrations from the final iteration run are used for determining molecular concentrations. For each species, the concentration is determined from windows where it can be reliably fitted. For the two saturated species CO2 and H2O, windows where their max absorption coefficient exceeds 10-4cm-1are discarded from statistical evaluations. For the other eighteen species, only windows where the maximum absorption cross sections are within 70 % of their strongest absorption cross section in the whole coverage are considered. The cutoff percentage is chosen sufficiently low so that each species has its concentration evaluated generally from more than 30 windows for convergence. For each window, the averaging weight is given by its maximum absorption cross section. The weighted mean and error are determined for each species over its considered windows.PATENT Attorney Docket No. UOCO.P2093WO / 00637650 Figure Legends

[0056] FIGs. 1A–1D: Modulated Ringdown Comb Interferometry (MRCI)

[0057] In FIG. 1A, an optical cavity formed by a pair of high-reflectivity mirrors enables significant enhancement in light-molecule absorption path length. The finesse and free spectral range (FSR) are two quantities characterizing the frequency width and spacing, respectively, of the cavity resonances formed from the field boundary conditions. Their typical values are given. A larger cavity finesse leads to narrower cavity resonances but provides stronger enhancement to the absorption path length. A continuous-wave laser precisely matched to a cavity resonance in optical frequency will be coupled through the cavity. Dispersion originating from the loaded gas and mirror spectral response can displace the frequency of the cavity resonances.

[0058] In FIG. 1B, a high-finesse cavity, periodically modulated in cavity length, transmits an incident mid-infrared comb once per cavity length sweep. Within a single sweep, laser fields from different cavity resonances are emitted sequentially. Traditionally, the periodic cavity transmission bursts are measured with a photodetector array following grating separation. Fitting to the temporal envelope yields ringdown time that ultra-sensitively probes the intracavity absorption. In the experimental demonstration described herein, instead, a passively scanning Michelson interferometer was used for spectroscopy readout.

[0059] FIG. 1C illustrates physical interpretations of the recorded interferogram. Shown are the intensity subcomponents versus time generated from a single cavity resonance. Through the addition of time periodicity and readout by interferometry, the spectral information generated by the same cavity resonance is engineered into an “RF spectral comb” having (i) RF frequency components uniformly spaced by the cavity-length modulation frequency, (ii) an RF frequency offset given by the Doppler frequency, and (iii) a spectral envelope determined by the ringdown time.

[0060] In FIG. 1D, the Fourier transform of the recorded interferogram encompasses a series of RF spectral combs spectrally isolated from each other for simultaneous broadband high-resolution ringdown measurements. For simplicity in the discussion, the emission rate of cavity transmission bursts and Doppler frequency are assumed to be constants. This is not a mandatory requirement in experimental realizations: their lack of mutual coherence can be robustly tackled with data reconstruction techniques. See the section titled “Methodology” for more details.

[0061] FIGs.2A–2D: Apparatus and Survey Ringdown Spectra Collected with MRCI In FIG.2A, an apparatus includes two sets of high-finesse cavities that are used for probing thePATENT Attorney Docket No. UOCO.P2093WO / 00637650 same gas sample. One cavity operates at a 5-μm wavelength range (finesse ~23,000) to produce a spectral output between 1,850–2,230 cm-1. The other cavity operates at a 3-μm wavelength range (finesse ~14,000) to produce a spectral output between 2,700–3,330 cm-1.

[0062] FIG. 2B shows sample spectral data processed from the recorded interferogram data collected at a two-second acquisition time and an 800-MHz instrument resolution with the cavities loaded with a breath sample. Trend lines highlight the Fourier components from different cavity resonances. The decay in spectral intensities with increasing Fourier-harmonic order is fitted to an analytically derived formula to determine the ringdown time. Insets show fitted ringdown times, where circles are experimental data and solid lines are the fitted curves.

[0063] FIGs.2C and 2D show survey ringdown spectra measured for the same breath samples employing the two cavities. Data show the ringdown data measured for cavity held at base pressure (below 3 mTorr). The decrease in ringdown times in data measured for breath from that cavity at base pressure gives molecular absorption spectrum measured from the loaded breath sample.

[0064] FIGS. 3A–3D: Cavity Swept Lock on Comb-Cavity Frequency Detuning

[0065] In FIGs. 3A–3D, both intracavity dispersion from molecules and mirrors and piezo hysteresis associated with the cavity-length modulation are neglected for clarity. FIG.3A is a servo schematic for a cavity swept lock. Error signals generated from demodulating the observed cavity transmission bursts at the even and odd harmonics of the cavity length modulation frequency ^^, are used for ensuring comb frequencies (^^^^and ^^^^) and cavity free spectral range (FSR) can all be precisely stabilized to each other. Through further locking the ^^^^to an external frequency reference, the absolute frequencies for ^^^^, ^^^^, and FSR can be fixed. To explain why the transmission bursts can be demodulated at even or odd harmonics of ^^for respectively locking either the ^^^^-FSR frequency detuning or the absolute frequency drift in ^^^^, consider the three cases shown in FIGs. 3B, 3C, and 3D. Here, the triangle waves represent the cavity-length periodic sweep. Different curves represent transmission bursts generated from three different comb lines. The thick curve is the sum of the three comb signals and represents the time-dependent signal measured by a photodiode placed at the cavity transmission side. For each case, the intensity spectrum of the Fourier transform of the black curve time signal is shown to the right. Comparing FIG. 3B with FIG. 3C, when the ^^^^-FSR frequency detuning is non-zero, the black curve observed from one consecutive cavity up and down sweeps form a base pattern repeating at rate ^^. This leads to non-zero amplitudes atPATENT Attorney Docket No. UOCO.P2093WO / 00637650 odd harmonics of ^^. Comparing FIG. 3B with FIG. 3D, the slow drift in ^^^^may result in comb lines coming on cavity resonances sequentially. The black curve becomes wider in shape and its Fourier decomposition has less intense high-frequency components. Thus, the even harmonics decrease in intensity (excluding the zeroth order, which will not change).

[0066] FIGs. 4A–4D: A Data-Processing Workflow for MRCI

[0067] A step-by-step workflow for MRCI includes processing of the raw interferogram data (i.e., the sequence of interferograms) to obtain broadband cavity ringdown values. See the section titled “Methodology” for more details. The element labeled “PD” is a photodetector. Additional Embodiments

[0068] FIG. 5 is a flowchart of a method 500 for modulated ringdown comb interferometry, in accordance with some of the present embodiments. The method 500 may be performed by any of the modulated ringdown comb interferometers described herein (e.g., see FIG.2A). The method 500 includes steps 504, 506, 508, 510, and 512. In certain embodiments, the method 500 further includes step 502, step 514, step 516, or a combination thereof. The method 500 may include additional steps not shown in FIG. 5.

[0069] In the step 504 of the method 500, an optical frequency comb is coupled into an optical cavity forming a plurality of resonances. In one example of the step 504, a mid- infrared frequency comb is coupled into the high-finesse optical cavity of FIG. 1B. In some embodiments, the optical cavity is a Fabry-Perot cavity having first and second cavity mirrors that face each other to form the plurality of cavity resonances. In this case, the optical frequency comb may be coupled into the Fabry-Perot cavity via transmission through one of the first and second cavity mirrors. The optical cavity may have a finesse up to several thousand, or more. In particular, the optical cavity may have a product of finesse and optical spectral coverage that is 107cm-1or more. The “optical spectral coverage” is the range of wavenumbers over which the cavity finesse is large enough for the optical cavity to be useful for high-resolution frequency-comb spectroscopy. A large optical spectral coverage and finesse may be obtained by constructing the optical cavity with broadband high-reflectivity mirrors.

[0070] In the step 506 of the method 500, the optical frequency comb or optical cavity is modulated, at a modulation frequency, such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts. In some embodiments, the optical path length of the optical cavity is modulated to vary the resonant frequenciesPATENT Attorney Docket No. UOCO.P2093WO / 00637650 relative to fixed optical frequencies of the comb lines. The optical path length may be modulated by varying the physical path length of the optical cavity. For example, the physical distance between two cavity mirrors forming a Fabry-Perot cavity may be varied by actuating a piezoelectric transducer or motorized translation stage that spatially moves one of the cavity mirrors with respect to the other. Regardless of whether the optical frequency comb or the Fabry-Perot cavity is modulated, the Fabry-Perot cavity, in response to the modulation, transmits the sequence of transmission bursts through one of the first and second cavity mirrors.

[0071] Alternatively or additionally, the step 506 may be performed by driving an intracavity electro-optic modulator to vary the path-averaged refractive index of the optical cavity. In this case, the optical path length of the optical cavity is changed without physically modulating the position of a cavity mirror. However, an intracavity modulator will introduce loss that reduces the cavity finesse. The intracavity modulator may also reduce optical spectral coverage, depending on the transmission of the electro-optic material it uses.

[0072] In other embodiments, the step 506 is performed by modulating the optical frequency comb to vary the optical frequencies of the comb lines relative to fixed resonant frequencies of the cavity resonances. For example, one or both of the offset frequency of the optical frequency comb and the repetition rate of the optical frequency comb may be modulated.

[0073] In one example of the step 506, the high-finesse cavity of FIG. 1B is periodically modulated in cavity length such that the incident mid-infrared frequency comb is transmitted through the optical length once per cavity-length sweep. In some embodiments, the optical-cavity length is modulated using a cavity swept lock, such as the implementation described above in the section titled “Cavity Swept Lock and Intracavity Dispersion.”

[0074] In the step 508 of the method 500, the sequence of transmission bursts is combined with a multichromatic reference beam to generate a combined optical beam. The step 508 may be performed with an optical beam combiner (e.g., a beamsplitter cube or plate beamsplitter). In one example of the step 508, the Michelson interferometer 110 of FIG. 1B includes a beamsplitter (BS2) that combines the Doppler-shifted optical beam from the scanning arm of the interferometer with a portion of the sequence of transmission bursts that propagates through the unscanned arm of the Michelson interferometer.

[0075] In the step 510 of the method 500, the combined beam is detected with a photodetector to generate a sequence of interferograms. The sequence of interferograms has an intensity spectrum comprising a plurality of radio-frequency (RF) combs. Each of the plurality of RF combs has a plurality of RF components that are uniformly spaced by the modulation frequency. The plurality of RF combs having a respective plurality of RF-comb offsetPATENT Attorney Docket No. UOCO.P2093WO / 00637650 frequencies that are unique. In one example of the step 510, the photodetector PD1 in FIG. 1B detects the combined beam formed from the beam combiner BS2 of the Michelson interferometer 110. FIG. 1D shows an example of the sequence of interferograms and an example of the RF combs.

[0076] In the step 512 of the method 500, the plurality of RF components of each RF comb are fit to obtain a respective one of a plurality of best-fit decay times. This plurality of best-fit decay times collectively forms a ringdown spectrum. In one example of the step 512, FIGS. 2B–2D show examples of ringdown spectra obtained via signal processing of the photodetector signals obtained via the Michelson interferometer of FIG.2A. In another example of the step 512, the two insets of FIG. 2B show how the amplitudes of the Fourier harmonics of two different RF combs (corresponding to two different optical frequencies) are fit to obtain two respective best-fit ringdown times. Other best-fit parameters may be ignored or used for other purposes. A different type of analytical function may be used for the fitting.

[0077] In some embodiments, the method 500 further includes the step 502, in which the gas sample is loaded into the optical cavity. In this case, the steps 504, 506, 508, 510 occur while the gas sample is located in the optical cavity. The resulting ringdown spectrum obtained by the step 512 contains spectroscopic information about one or more species of the gas sample. In one example of the step 502, FIG.2A shows how a gas is loaded into the two optical cavities used for the experimental demonstration described above. For clarity, the gas sample is shown in FIG. 2A as a collection of ball-and-stick molecules.

[0078] In some embodiments, the method 500 further includes the step 514, in which an absorption spectrum of the gas sample is calculated based at least in part on the ringdown spectrum. In one example of the step 514, the steps 504, 506, 508, 510, and 512 are repeated while the optical cavity is empty. In this case, the step 512 produces a baseline ringdown spectrum. The ring-down times of the baseline ringdown spectrum are inverted to obtain an inverted baseline ringdown spectrum. Similarly, the ring-down times of the ringdown spectrum obtained with the gas sample in the optical cavity are inverted to obtain an inverted ringdown spectrum. The inverted baseline ringdown spectrum is then subtracted from the inverted ringdown spectrum to generate the absorption spectrum.

[0079] In other embodiments, the method 500 further includes the step 516, in which the absorption spectrum is processed to determine a concentration of at least one atomic or molecular species of the gas sample. In one example of the step 516, concentrations are extracted from an absorption spectrum using molecular line fitting. Details about molecular line fitting are described in the section above titled “Molecular Line Fitting.”PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0080] In some embodiments, the method 500 further includes a step in which the multichromatic reference beam is generated from the sequence of transmission bursts by (i) splitting the sequence of transmission bursts into a first optical beam and a second optical beam and (ii) scanning an optical delay to transform the first optical beam into a Doppler-shifted optical beam. In these embodiments, the step 510 is performed by combining the Doppler- shifted optical beam with the second optical beam. In one example of these embodiments, the Michelson interferometer 110 of FIG. 1B has a beamsplitter BS1, a scanning arm with the optical delay, an unscanning arm (i.e., an arm that does not introduce any Doppler shift), and a beam combiner BS2. The beamsplitter BS1 splits the sequence of transmission bursts between the two arms. The beam combiner BS2 combines the two arms to output one or two combined beams (which are then detected with the photodetectors PD1 and PD2 in FIG. 1B).

[0081] In some of the embodiments of the method 500 that generate a Doppler-shifted optical beam, the method 500 may further includes the step of linearly moving a delay stage at a constant velocity. A retroreflecting mirror is mounted on the delay stage, as shown in FIG. 2A. This linear motion is repeated in a back-and-forth manner to produce motion represented by a triangle wave. Examples of this triangle wave are shown in FIGS. 3A–3D. However, the delay stage may be moved in other ways (e.g., non-linearly) without departing from the scope hereof.

[0082] In some embodiments, the multichromatic reference beam is a second optical frequency comb that is different from the first optical frequency comb (i.e., the optical frequency comb that is coupled into the optical cavity). For clarity, this first optical frequency comb is also referred to herein as the “signal comb” while the second optical frequency comb is also referred to herein as the “reference comb.” The signal comb lines of the signal comb form a one-to-correspondence with the reference comb lines of the reference comb. Each signal comb line mixes with its corresponding reference comb line during photodetection to produce the fundamental component of a corresponding RF comb. Specifically, for each signal comb line, the corresponding reference comb line is the comb line of the reference comb that is closest in frequency to that of the signal comb line. The resulting heterodyne beat signal is therefore the lowest-frequency RF beat note between the signal comb line and the entire reference comb.

[0083] To ensure that the RF combs have unique RF-comb offset frequencies, and therefore that each RF component of each RF comb has a unique RF frequency, the reference comb may have a uniform spacing (i.e., repetition rate) that is different from that of the signal comb. The difference between the repetition rate of the signal comb and the repetition rate of the reference comb is referred to herein as the repetition-rate difference frequency. The RF- comb offset frequencies are uniformly spaced by the repetition-rate difference frequency.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0084] In some embodiments, the method 500 further includes the step of generating the signal and reference combs. The signal and reference combs may be generated from two separate frequency-comb sources (e.g., mode-locked femtosecond lasers) that operate independently of each other. Alternatively, the signal and reference combs may be generated from a dual-comb source that generates two pulse trains (i.e., optical frequency combs) from the same laser cavity. Typically, the two pulse trains generated by a dual-comb source have slightly different repetition rates.

[0085] In some of the embodiments that use two different optical frequency combs (e.g., signal and reference combs), one or both of the frequency combs may be controlled such that their offset frequencies are the same. For example, the method 500 may further include the steps of (i) phase-locking a first offset frequency of the first optical frequency comb to an RF reference frequency and (ii) phase-locking a second offset frequency of the second optical frequency comb to the RF reference frequency. Alternatively, the method 500 may include the step of phase-locking the first offset frequency to the second offset frequency.

[0086] In other embodiments that use two different optical frequency combs, one or both of the frequency combs may be controlled such that their repetition rates are different. For example, the method 500 may further includes the steps of (i) phase-locking a first repetition rate of the first optical frequency comb to a first RF reference frequency and (ii) phase-locking a second repetition rate of the second optical frequency comb to a second RF reference frequency that is different from the first RF reference frequency. Alternatively, the method 500 may include the step of phase-locking the first repetition rate of the second repetition rate.

[0087] FIG. 6 is a functional diagram of a modulated ringdown comb interferometer 600, in accordance with some of the present embodiments. The interferometer 600 implements the method 500 of FIG. 5. The interferometer 600 includes an optical cavity 606, an input coupler 604, a modulator 612, a beam combiner 616, a photodetector 630, and a signal- processing circuit 640. However, the interferometer 600 may include additional components that are not shown in FIG. 5, as needed to perform the method 500 and implement any other functionality described herein.

[0088] The optical cavity 606 forms a plurality of cavity resonances that have different resonant frequencies. The input coupler 604 couples an optical frequency comb 602 into the optical cavity 606. In the example shown in FIG. 6, the optical cavity is a Fabry-Perot cavity having a first cavity mirror 608 and a second cavity mirror 610 that face each other to form the cavity resonances. In this case, the input coupler 604 couples the optical frequency comb 602 into the Fabry-Perot cavity via transmission through the first cavity mirror 608. AlsoPATENT Attorney Docket No. UOCO.P2093WO / 00637650 in FIG.6, the input coupler 604 is a steering mirror. However, the input coupler 604 may include additional or alternative components (e.g., lenses, mirrors, etc.) for mode-matching the optical frequency comb 602 to the cavity resonances.

[0089] The modulator 612 modulates, at a modulation frequency, the optical frequency comb 602 or the optical cavity 606 such that the optical cavity 606, in response to each of a plurality of comb lines of the optical frequency comb 602 being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts 614. In the example of FIG. 6, the Fabry-Perot cavity, in response to the modulation, transmits the sequence of transmission bursts 614 through the second cavity mirror 610.

[0090] Also in FIG.6, the modulator 612 modulates the position of the second mirror 610 to vary the resonant frequencies of the cavity resonances relative to fixed optical frequencies of the comb lines of the optical frequency comb 602. Alternatively, the modulator 612 may drive a frequency-comb source (not shown in FIG. 6) to vary an offset frequency or repetition rate of the optical frequency comb 602. In this case, the optical frequencies of the comb lines are swept relative to fixed resonant frequencies of the optical cavity 606.

[0091] The beam combiner 616 combines the sequence of transmission bursts 614 and a multichromatic reference beam 618 to generate a combined optical beam 620. The photodetector 630 detects the combined optical beam 620 to generate a sequence of interferograms 632. The sequence of interferograms 632 has an intensity / amplitude spectrum comprising a plurality of RF combs. Each of the plurality of RF combs has a plurality of RF components that are uniformly spaced by the modulation frequency. The plurality of RF combs have a respective plurality of RF-comb offset frequencies that are unique.

[0092] The signal-processing circuit 640 fits the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times. The plurality of best- fit decay times form a ringdown spectrum 642. The ringdown spectrum 642 may be further processed into an absorption spectrum, which in turn may be processed to determine the concentration of one or more species of a gas sample 650 confined inside the optical cavity 606 (i.e., in a region of space located between the cavity mirrors 608 and 610).

[0093] In some embodiments, the modulated ringdown comb interferometer 600 includes additional components to generate the multichromatic reference beam 618 from the sequence of transmission bursts 614. These additional components include (i) an input beamsplitter that splits the sequence of transmission bursts 614 into a first optical beam and a second optical beam and (ii) a scannable optical delay that, when actuated, transforms the first optical beam into a Doppler-shifted optical beam. In these embodiments, the beam combinerPATENT Attorney Docket No. UOCO.P2093WO / 00637650 616 combines the Doppler-shifted optical beam and the second optical beam to generate the combined optical beam 620. The input beamsplitter, scannable optical delay, and beam combiner collectively form a Michelson interferometer with a scanning arm (see the Michelson interferometer 110 of FIG.1B). The photodetector 630 detects one output port of the Michelson interferometer. However, the interferometer 600 may further include a second photodetector to detect the second output port of the Michelson interferometer. In this case, the signal-processing circuit 640 processes both photodetector signals to determine the ringdown spectrum 642.

[0094] FIG. 7 is a functional diagram of a modulated ringdown comb interferometer 700, in accordance with some of the present embodiments. The interferometer 700 is similar to the modulated ringdown comb interferometer 600 of FIG. 6 but further includes a second optical frequency comb 718 that serves as the multichromatic reference beam 618 of FIG. 6. For clarity, the optical frequency comb 602 is also referred to herein as the “signal comb” while the second optical frequency comb 718 is also referred to herein as the “reference comb.” The signal comb lines of the signal comb form a one-to-correspondence with the reference comb lines of the reference comb. Each signal comb line mixes with its corresponding reference comb line during photodetection to produce the fundamental component of a corresponding RF comb. Specifically, for each signal comb line, the corresponding reference comb line is the comb line of the reference comb that is closest in frequency to that of the signal comb line. The resulting heterodyne beat signal is therefore the lowest-frequency RF beat note between the signal comb line and the entire reference comb.

[0095] To ensure that the RF combs have unique RF-comb offset frequencies, and therefore that each RF component of each RF comb has a unique RF frequency, the reference comb may have a uniform spacing (i.e., repetition rate) that is different from that of the signal comb. The difference between the repetition rate of the signal comb and the repetition rate of the reference comb is referred to herein as the repetition-rate difference frequency. The RF- comb offset frequencies are uniformly spaced by the repetition-rate difference frequency.

[0096] In some embodiments, the modulated ringdown comb interferometer 700 further includes a first frequency-comb source 704 the generates the optical frequency comb 602. The interferometer 700 may further include a second frequency-comb source 706 that generates the second optical frequency comb 718. The frequency combs 602 and 718 may be generated from two separate frequency-comb sources (e.g., mode-locked femtosecond lasers) that operate independently of each other. Alternatively, the frequency combs 602 and 718 may be generated from a dual-comb source that generates two pulse trains from the same laser cavity.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0097] In other embodiments, the modulated ringdown comb interferometer 700 further includes one or more phase-locked loops that stabilize one or both of the optical frequency combs 602 and 718. In one embodiment, the interferometer 700 includes a first phase-locked-loop circuit that controls the first frequency-comb source 704 to phase-lock the offset frequency (^^^^) of the optical frequency comb 602 to an RF reference frequency. The interferometer 700 also includes a second phase-locked-loop circuit that controls the second frequency-comb source 706 to phase-lock the offset frequency of the second optical frequency comb 718 to the same RF reference frequency used with the first phase-locked circuit.

[0098] In another embodiment, the modulated ringdown comb interferometer 700 includes a phase-locked-loop circuit that controls the first frequency-comb source 704 to phase- lock the offset frequency of the optical frequency comb 602 to the offset frequency of the second optical frequency comb 718.

[0099] In another embodiment, the modulated ringdown comb interferometer 700 includes a first phase-locked-loop circuit that controls the first frequency-comb source 704 to phase-lock the repetition rate of the optical frequency comb 602 to a first RF reference frequency. The interferometer 700 also includes a second phase-locked-loop circuit that controls the second frequency-comb source 706 to phase-lock the repetition rate of the second optical frequency comb 718 to a second RF reference frequency that is different from the first RF reference frequency. The difference between the first and second RF reference frequencies is the repetition-rate difference frequency described above.

[0100] In another embodiment, the interferometer 700 includes a phase-locked-loop circuit that controls the first frequency-comb source 704 to offset-lock the repetition rate of the optical frequency comb 602 from the repetition rate of the second optical frequency comb 718. The frequency offset used for this offset lock sets the repetition-rate difference frequency. Combinations of Features

[0101] Features described above as well as those claimed below may be combined in various ways without departing from the scope hereof. The following examples illustrate possible, non-limiting combinations of features and embodiments described above. It should be clear that other changes and modifications may be made to the present embodiments without departing from the spirit and scope of this invention:

[0102] (A1) A method for modulated ringdown comb interferometry includes coupling an optical frequency comb into an optical cavity forming a plurality of cavity resonances. The method also includes modulating, at a modulation frequency, the opticalPATENT Attorney Docket No. UOCO.P2093WO / 00637650 frequency comb or the optical cavity such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts. The method also includes combining the sequence of transmission bursts with a multichromatic reference beam to generate a combined optical beam. The method also includes detecting the combined optical beam with a photodetector to generate a sequence of interferograms. The sequence of interferograms has an intensity spectrum comprising a plurality of radio-frequency (RF) combs. Each of the plurality of RF combs has a plurality of RF components that are uniformly spaced by the modulation frequency. The plurality of RF combs have a respective plurality of RF-comb offset frequencies that are unique. The method also includes fitting the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times. The plurality of best-fit decay times form a ringdown spectrum.

[0103] (A2) In the method denoted (A1), the method further includes loading a gas sample into the optical cavity. The steps of coupling, modulating, combining, and detecting occur while the gas sample is in the optical cavity.

[0104] (A3) In the method denoted (A2), the method further includes calculating an absorption spectrum of the gas sample based at least in part on the ringdown spectrum.

[0105] (A4) In the method denoted (A3), the step of calculating the absorption spectrum includes repeating, while the optical cavity is empty, the steps of coupling, modulating, combining, detecting, and fitting to obtain a baseline ringdown spectrum. The step of calculating further includes inverting each of a plurality of best-fit decay times of the baseline ringdown spectrum to obtain an inverted baseline ringdown spectrum and inverting each of the plurality of best-fit decay times of the ringdown spectrum to obtain an inverted ringdown spectrum. The step of calculating further includes subtracting the inverted baseline ringdown spectrum from the inverted ringdown spectrum.

[0106] (A5) In the method denoted (A3) or (A4), the method further includes processing the absorption spectrum to determine a concentration of at least one atomic or molecular species of the gas sample.

[0107] (A6) In any of the methods denoted (A1) to (A5), the optical frequency comb has a first optical-comb offset frequency and a first optical-comb repetition rate. The multichromatic reference beam includes a second optical frequency comb having a second optical-comb offset frequency and a second optical-comb repetition rate. The second optical- comb offset frequency is equal to the first optical-comb offset frequency. The second optical- comb repetition rate is different from the first optical-comb repetition rate by a repetition-ratePATENT Attorney Docket No. UOCO.P2093WO / 00637650 difference frequency. The plurality of RF-comb offset frequencies are uniformly spaced by the repetition-rate difference frequency.

[0108] (A7) In any of the methods denoted (A1) to (A6), the step of modulating includes modulating one or both of an offset frequency of the optical frequency comb and a repetition rate of the optical frequency comb.

[0109] (A8) In any of the methods denoted (A1) to (A7), the step of modulating includes modulating a length of the optical cavity.

[0110] (A9) In the method denoted (A8), the step of modulating the length of the optical cavity includes implementing a cavity swept lock.

[0111] (A10) In any of the methods denoted (A1) to (A9), the optical cavity has a product of cavity finesse and optical spectral coverage that is 107cm-1or more.

[0112] (A11) In any of the methods denoted (A1) to (A10), the method further includes generating the multichromatic reference beam from the sequence of transmission bursts by (i) splitting the sequence of transmission bursts into a first optical beam and a second optical beam and (ii) scanning an optical delay to transform the first optical beam into a Doppler-shifted optical beam. The step of combining includes combining the Doppler-shifted optical beam and the second optical beam to generate the combined optical beam.

[0113] (A12) In the method denoted (A11), the step of scanning includes linearly moving a delay stage at a constant velocity.

[0114] (A13) In any of the methods denoted (A1) to (A12), the method further includes generating the optical frequency comb with a first frequency-comb source. The method further includes generating a second optical frequency comb with a second frequency-comb source. The multichromatic reference beam includes the second optical frequency comb.

[0115] (A14) In the method denoted (A13), the step of generating the optical frequency comb and the step of generating the second optical frequency comb include generating the optical frequency and the second optical frequency comb with a dual-comb source.

[0116] (A15) In the method denoted (A13) or (A14), the method further includes phase-locking a first offset frequency of the optical frequency comb to an RF reference frequency. The method further includes phase-locking a second offset frequency of the second optical frequency comb to the RF reference frequency.

[0117] (A16) In the method denoted (A13) or (A14), the method further includes phase-locking a first offset frequency of the optical frequency comb to a second offset frequency of the second optical frequency comb.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0118] (A17) In any of the methods denoted (A13) to (A16), the method further includes phase-locking a first repetition rate of the optical frequency comb to a first RF reference frequency. The method further includes phase-locking a second repetition rate of the second optical frequency comb to second RF reference frequency that is different from the first RF reference frequency.

[0119] (A18) In any of the methods denoted (A13) to (A16), the method further includes phase-locking a first repetition rate of the optical frequency comb to a second repetition rate of the second optical frequency comb such that the first repetition rate is different from the first reference frequency.

[0120] (A19) In any of the methods denoted (A1) to (A18), the step of fitting includes processing the plurality of RF components of each RF comb to obtain a sequence of amplitudes of the plurality of RF components. The step of fitting further includes fitting the sequence of amplitudes to an analytical function to obtain the respective one of the plurality of best-fit decay times.

[0121] (A20) In any of the methods denoted (A1) to (A19), the optical cavity is a Fabry-Perot cavity having first and second cavity mirrors that face each other to form the plurality of cavity resonances. Furthermore, the step of coupling includes coupling the optical frequency comb into the Fabry-Perot cavity via transmission through the first cavity mirror. The Fabry-Perot cavity, in response to the step of modulating, transmits the sequence of transmission bursts through the second cavity mirror.

[0122] (B1) A modulated ringdown comb interferometer includes an optical cavity forming a plurality of cavity resonances and an input coupler configured to couple an optical frequency comb into the optical cavity. The modulated ringdown comb interferometer further includes a modulator configured to modulate, at a modulation frequency, the optical frequency comb or the optical cavity such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts. The modulated ringdown comb interferometer further includes a beam combiner configured to combine the sequence of transmission bursts and a multichromatic reference beam to generate a combined optical beam. The modulated ringdown comb interferometer further includes a photodetector configured to detect the combined optical beam to generate a sequence of interferograms. The sequence of interferograms has an intensity spectrum comprising a plurality of radio-frequency (RF) combs. Each of the plurality of RF combs has a plurality of RF components that are uniformly spaced by the modulation frequency. The plurality of RF combs has a respective plurality of RF-combPATENT Attorney Docket No. UOCO.P2093WO / 00637650 offset frequencies that are unique. The modulated ringdown comb interferometer further includes a signal-processing circuit configured to fit the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times. The plurality of best-fit decay times forming a ringdown spectrum.

[0123] (B2) In the modulated ringdown comb interferometer denoted (B1), the modulated ringdown comb interferometer further includes a gas source configured to load a gas sample into the optical cavity.

[0124] (B3) In the modulated ringdown comb interferometer denoted (B2), the signal- processing circuit is further configured to calculate an absorption spectrum of the gas sample based at least in part on the ringdown spectrum.

[0125] (B4) In the modulated ringdown comb interferometer denoted (B3), the signal- processing circuit is further configured to process the absorption spectrum to determine a concentration of at least one atomic or molecular species of the gas sample.

[0126] (B5) In any of the modulated ringdown comb interferometers denoted (B1) to (B4), the optical frequency comb has a first optical-comb offset frequency and a first optical- comb repetition rate. Furthermore, the multichromatic reference beam includes a second optical frequency comb having a second optical-comb offset frequency and a second optical-comb repetition rate. The second optical-comb offset frequency is equal to the first optical-comb offset frequency. The second optical-comb repetition rate is different from the first optical- comb repetition rate by a repetition-rate difference frequency. The plurality of RF-comb offset frequencies are uniformly spaced by the repetition-rate difference frequency.

[0127] (B6) In any of the modulated ringdown comb interferometers denoted (B1) to (B5), the modulator is configured to modulate one or both of an offset frequency of the optical frequency comb and a repetition rate of the optical frequency comb.

[0128] (B7) In any of the modulated ringdown comb interferometers denoted (B1) to (B6), the modulator includes a piezoelectric transducer configured to modulate a length of the optical cavity.

[0129] (B8) In the modulated ringdown comb interferometer denoted (B7), the modulated ringdown comb interferometer denoted further includes modulator circuitry configured to control the optical cavity to implement a cavity swept lock.

[0130] (B9) In any of the modulated ringdown comb interferometers denoted (B1) to (B8), the optical cavity has a product of cavity finesse and optical spectral coverage that is 107cm-1or more.PATENT Attorney Docket No. UOCO.P2093WO / 00637650

[0131] (B10) In any of the modulated ringdown comb interferometers denoted (B1) to (B9), the modulated ringdown comb interferometer further includes an input beamsplitter configured to split the sequence of transmission bursts into a first optical beam and a second optical beam. The modulated ringdown comb interferometer further includes a scannable optical delay configured to transform the first optical beam into a Doppler-shifted beam. The beam combiner is configured to combine the Doppler-shifted beam and the second optical beam to generate the combined optical beam.

[0132] (B11) In the modulated ringdown comb interferometer denoted (B10), the optical delay includes a delay stage and a delay-stage controller configured to linear move the delay stage at a constant velocity.

[0133] (B12) In any of the modulated ringdown comb interferometers denoted (B1) to (B11), the modulated ringdown comb interferometer further includes a first frequency-comb source configured to generate the optical frequency comb. The modulated ringdown comb interferometer further includes a second frequency-comb source configured to generate a second optical frequency comb. The multichromatic reference beam includes the second optical frequency comb.

[0134] (B13) In the modulated ringdown comb interferometer denoted (B12), the modulated ringdown comb interferometer further includes a dual-comb source that includes the first frequency-comb source and the second frequency-comb source.

[0135] (B14) In the modulated ringdown comb interferometer denoted (B12) or (B13), the modulated ringdown comb interferometer further includes a first phase-locked loop configured to phase-lock a first offset frequency of the optical frequency comb to an RF reference frequency. The modulated ringdown comb interferometer further includes a second phase-locked loop configured to phase-lock a second offset frequency of the second optical frequency comb to the RF reference frequency.

[0136] (B15) In the modulated ringdown comb interferometer denoted (B12) or (B13), the modulated ringdown comb interferometer further includes a phase-locked loop configured to phase-lock a first offset frequency of the optical frequency comb to a second offset frequency of the second optical frequency comb.

[0137] (B16) In any of the modulated ringdown comb interferometers denoted (B12) to (B15), the modulated ringdown comb interferometer further includes a first phase-locked loop configured to phase-lock a first repetition rate of the optical frequency comb to a first RF reference frequency. The modulated ringdown comb interferometer further includes a second phase-locked loop configured to phase-lock a second repetition rate of the second opticalPATENT Attorney Docket No. UOCO.P2093WO / 00637650 frequency comb to a second RF reference frequency. The first RF reference frequency is different from the first RF reference frequency.

[0138] (B17) In any of the modulated ringdown comb interferometers denoted (B12) to (B15), the modulated ringdown comb interferometer further includes a phase-locked loop configured to phase-lock a first repetition rate of the optical frequency comb to a second repetition rate of the second optical frequency comb. The first repetition rate is different from the second repetition rate.

[0139] (B18) In any of the modulated ringdown comb interferometers denoted (B1) to (B17), the signal-processing circuit is configured to process the plurality of RF components of each RF comb to obtain a sequence of amplitudes of the plurality of RF components. The signal-processing circuit is further configured to fit the sequence of amplitudes to an analytical function to obtain the respective one of the plurality of best-fit decay times.

[0140] (B19) In any of the modulated ringdown comb interferometers denoted (B1) to (B18), the optical cavity is a Fabry-Perot cavity having first and second cavity mirrors that face each other to form the plurality of cavity resonances. The input coupler is configured to couple the optical frequency comb into the Fabry-Perot cavity via transmission through the first cavity mirror. The Fabry-Perot cavity, in response to the modulator modulating the optical frequency comb or the optical cavity, transmits the sequence of transmission bursts through the second cavity mirror.

[0141] (B20) In any of the modulated ringdown comb interferometers denoted (B1) to (B19), the signal-processing circuit includes one or more processing units and a memory in communication with the one or more processing units. The memory stores machine-readable instructions that, when executed by the one or more processing units, control the signal- processing circuit to fit the plurality of RF components of each RF comb to obtain the respective one of the plurality of best-fit decay times.

[0142] (B21) In any of the modulated ringdown comb interferometers denoted (B1) to (B20), the signal-processing circuit includes a field-programmable gate array (FPGA), a programmable logic device (PLD), or a combination thereof.

[0143] Changes may be made in the above methods and systems without departing from the scope hereof. It should thus be noted that the matter contained in the above description or shown in the accompanying drawings should be interpreted as illustrative and not in a limiting sense. The following claims are intended to cover all generic and specific features described herein, as well as all statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.

Claims

PATENT Attorney Docket No. UOCO.P2093WO / 00637650 CLAIMS What is claimed is:

1. A method for modulated ringdown comb interferometry, comprising: coupling an optical frequency comb into an optical cavity forming a plurality of cavity resonances; modulating, at a modulation frequency, the optical frequency comb or the optical cavity such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts; combining the sequence of transmission bursts and a multichromatic reference beam to generate a combined optical beam; detecting the combined optical beam with a photodetector to generate a sequence of interferograms, the sequence of interferograms having an intensity spectrum comprising a plurality of radio-frequency (RF) combs, each of the plurality of RF combs comprising a plurality of RF components that are uniformly spaced by the modulation frequency, the plurality of RF combs having a respective plurality of RF-comb offset frequencies that are unique; and fitting the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times, the plurality of best-fit decay times forming a ringdown spectrum.

2. The method of claim 1, further comprising loading a gas sample into the optical cavity; wherein said coupling, said modulating, said combining, and said detecting occur while the gas sample is in the optical cavity.

3. The method of claim 2, further comprising calculating an absorption spectrum of the gas sample based at least in part on the ringdown spectrum.

4. The method of claim 3, wherein said calculating the absorption spectrum comprises:PATENT Attorney Docket No. UOCO.P2093WO / 00637650 repeating, while the optical cavity is empty, said coupling, said modulating, said combining, said detecting, and said fitting to obtain a baseline ringdown spectrum; inverting each of a plurality of best-fit decay times of the baseline ringdown spectrum to obtain an inverted baseline ringdown spectrum; inverting each of the plurality of best-fit decay times of the ringdown spectrum to obtain an inverted ringdown spectrum; and subtracting the inverted baseline ringdown spectrum from the inverted ringdown spectrum.

5. The method of claim 3, further comprising processing the absorption spectrum to determine a concentration of at least one atomic or molecular species of the gas sample.

6. The method of claim 1, wherein: the optical frequency comb has a first optical-comb offset frequency and a first optical-comb repetition rate; the multichromatic reference beam comprises a second optical frequency comb having a second optical-comb offset frequency and a second optical-comb repetition rate, the second optical-comb offset frequency being equal to the first optical- comb offset frequency, the second optical-comb repetition rate being different from the first optical-comb repetition rate by a repetition-rate difference frequency; and the plurality of RF-comb offset frequencies are uniformly spaced by the repetition-rate difference frequency.

7. The method of claim 1, wherein said modulating comprises modulating one or both of an offset frequency of the optical frequency comb and a repetition rate of the optical frequency comb.

8. The method of claim 1, wherein said modulating comprises modulating a length of the optical cavity.PATENT Attorney Docket No. UOCO.P2093WO / 00637650 9. The method of claim 8, wherein said modulating the length of the optical cavity comprises implementing a cavity swept lock.

10. The method of claim 1, the optical cavity having a product of cavity finesse and optical spectral coverage that is 107cm-1or more.

11. The method of claim 1, further comprising generating the multichromatic reference beam from the sequence of transmission bursts by: splitting the sequence of transmission bursts into a first optical beam and a second optical beam; and scanning an optical delay to transform the first optical beam into a Doppler- shifted optical beam; wherein said combining comprises combining the Doppler-shifted optical beam and the second optical beam to generate the combined optical beam.

12. The method of claim 11, wherein said scanning comprises linearly moving a delay stage at a constant velocity.

13. The method of claim 1, further comprising: generating the optical frequency comb with a first frequency-comb source; and generating a second optical frequency comb with a second frequency-comb source, the multichromatic reference beam comprising the second optical frequency comb.

14. The method of claim 13, wherein said generating the optical frequency comb and said generating the second optical frequency comb comprises generating the optical frequency comb and the second optical frequency comb with a dual-comb source.

15. The method of claim 13, further comprising: phase-locking a first offset frequency of the optical frequency comb to an RF reference frequency; and phase-locking a second offset frequency of the second optical frequency comb to the RF reference frequency.PATENT Attorney Docket No. UOCO.P2093WO / 00637650 16. The method of claim 13, further comprising phase-locking a first offset frequency of the optical frequency comb to a second offset frequency of the second optical frequency comb.

17. The method of claim 13, further comprising: phase-locking a first repetition rate of the optical frequency comb to a first RF reference frequency; and phase-locking a second repetition rate of the second optical frequency comb to a second RF reference frequency that is different from the first RF reference frequency.

18. The method of claim 13, further comprising phase-locking a first repetition rate of the optical frequency comb to a second repetition rate of the second optical frequency comb such that the first repetition rate is different from the first reference frequency.

19. The method of claim 1, wherein said fitting comprises: processing the plurality of RF components of each RF comb to obtain a sequence of amplitudes of the plurality of RF components; and fitting the sequence of amplitudes to an analytical function to obtain the respective one of the plurality of best-fit decay times.

20. The method of claim 1, wherein: the optical cavity is a Fabry-Perot cavity having first and second cavity mirrors that face each other to form the plurality of cavity resonances; said coupling comprises coupling the optical frequency comb into the Fabry-Perot cavity via transmission through the first cavity mirror; and the Fabry-Perot cavity, in response to said modulating, transmits the sequence of transmission bursts through the second cavity mirror.

21. A modulated ringdown comb interferometer, comprising: an optical cavity forming a plurality of cavity resonances; an input coupler configured to couple an optical frequency comb into the optical cavity;PATENT Attorney Docket No. UOCO.P2093WO / 00637650 a modulator configured to modulate, at a modulation frequency, the optical frequency comb or the optical cavity such that the optical cavity, in response to each of a plurality of comb lines of the optical frequency comb being swept across any one of the plurality of cavity resonances, transmits one of a sequence of transmission bursts; a beam combiner configured to combine the sequence of transmission bursts and a multichromatic reference beam to generate a combined optical beam; a photodetector configured to detect the combined optical beam to generate a sequence of interferograms, the sequence of interferograms having an intensity spectrum comprising a plurality of radio-frequency (RF) combs, each of the plurality of RF combs comprising a plurality of RF components that are uniformly spaced by the modulation frequency, the plurality of RF combs having a respective plurality of RF-comb offset frequencies that are unique; and a signal-processing circuit configured to fit the plurality of RF components of each RF comb to obtain a respective one of a plurality of best-fit decay times, the plurality of best-fit decay times forming a ringdown spectrum.

22. The modulated ringdown comb interferometer of claim 21, further comprising a gas source configured to load a gas sample into the optical cavity.

23. The modulated ringdown comb interferometer of claim 22, the signal-processing circuit being further configured to calculate an absorption spectrum of the gas sample based at least in part on the ringdown spectrum.

24. The modulated ringdown comb interferometer of claim 23, the signal-processing circuit being further configured to process the absorption spectrum to determine a concentration of at least one atomic or molecular species of the gas sample.

25. The modulated ringdown comb interferometer of claim 21, wherein: the optical frequency comb has a first optical-comb offset frequency and a first optical-comb repetition rate; the multichromatic reference beam comprises a second optical frequency comb having a second optical-comb offset frequency and a second optical-comb repetition rate, the second optical-comb offset frequency being equal to the first optical-PATENT Attorney Docket No. UOCO.P2093WO / 00637650 comb offset frequency, the second optical-comb repetition rate being different from the first optical-comb repetition rate by a repetition-rate difference frequency; and the plurality of RF-comb offset frequencies are uniformly spaced by the repetition-rate difference frequency.

26. The modulated ringdown comb interferometer of claim 21, the modulator being configured to modulate one or both of an offset frequency of the optical frequency comb and a repetition rate of the optical frequency comb.

27. The modulated ringdown comb interferometer of claim 21, the modulator comprising a piezoelectric transducer configured to modulate a length of the optical cavity.

28. The modulated ringdown comb interferometer of claim 27, further comprising modulator circuitry configured to control the piezoelectric transducer to implement a cavity swept lock.

29. The modulated ringdown comb interferometer of claim 21, the optical cavity having a product of cavity finesse and optical spectral coverage that is 107cm-1or more.

30. The modulated ringdown comb interferometer of claim 21, further comprising: an input beamsplitter configured to split the sequence of transmission bursts into a first optical beam and a second optical beam; and a scannable optical delay configured to transform the first optical beam into a Doppler-shifted optical beam; wherein the beam combiner is configured to combine the Doppler-shifted optical beam and the second optical beam to generate the combined optical beam.

31. The modulated ringdown comb interferometer of claim 30, the optical delay comprising a delay stage and a delay-stage controller configured to linear move the delay stage at a constant velocity.

32. The modulated ringdown comb interferometer of claim 21, further comprising: a first frequency-comb source configured to generate the optical frequency comb; andPATENT Attorney Docket No. UOCO.P2093WO / 00637650 a second frequency-comb source configured to generate a second optical frequency comb, the multichromatic reference beam comprising the second optical frequency comb.

33. The modulated ringdown comb interferometer of claim 32, further comprising a dual- comb source that includes the first frequency-comb source and the second frequency- comb source.

34. The modulated ringdown comb interferometer of claim 32, further comprising: a first phase-locked loop configured to phase-lock a first offset frequency of the optical frequency comb to an RF reference frequency; and a second phase-locked loop configured to phase-lock a second offset frequency of the second optical frequency comb to the RF reference frequency.

35. The modulated ringdown comb interferometer of claim 32, further comprising a phase-locked loop configured to phase-lock a first offset frequency of the optical frequency comb to a second offset frequency of the second optical frequency comb.

36. The modulated ringdown comb interferometer of claim 32, further comprising: a first phase-locked loop configured to phase-lock a first repetition rate of the optical frequency comb to a first RF reference frequency; and a second phase-locked loop configured to phase-lock a second repetition rate of the second optical frequency comb to a second RF reference frequency that is different from the first RF reference frequency.

37. The modulated ringdown comb interferometer of claim 32, further comprising a phase-locked loop configured to phase-lock a first repetition rate of the optical frequency comb to a second repetition rate of the second optical frequency comb such that the first repetition rate is different from the second repetition rate.

38. The modulated ringdown comb interferometer of claim 21, the signal-processing circuit being configured to: process the plurality of RF components of each RF comb to obtain a sequence of amplitudes of the plurality of RF components; andPATENT Attorney Docket No. UOCO.P2093WO / 00637650 fit the sequence of amplitudes to an analytical function to obtain the respective one of the plurality of best-fit decay times.

39. The modulated ringdown comb interferometer of claim 21, wherein: the optical cavity is a Fabry-Perot cavity having first and second cavity mirrors that face each other to form the plurality of cavity resonances; the input coupler is configured to couple the optical frequency comb into the Fabry- Perot cavity via transmission through the first cavity mirror; and the Fabry-Perot cavity, in response to the modulator modulating the optical frequency comb or the optical cavity, transmits the sequence of transmission bursts through the second cavity mirror.

40. The modulated ringdown comb interferometer of claim 21, the signal-processing circuit comprising: one or more processing units; and a memory in communication with the one or more processing units; the memory storing machine-readable instructions that, when executed by the one or more processing units, control the signal-processing circuit to fit the plurality of RF components of each RF comb to obtain the respective one of the plurality of best-fit decay times.

41. The modulated ringdown comb interferometer of claim 21, the signal-processing circuit comprising a field-programmable gate array (FPGA), a programmable logic device (PLD), or a combination thereof.

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