Mid-infrared dual-comb spectroscopy with quantum cascade lasers

Since the introduction of the first commercial instruments in the late 1960s and 1970s, Fourier transform infrared (FTIR) spectroscopy has been synonymous with infrared spectroscopy.¹ FTIR spectrometers have been adapted for use in many challenging applications and have paved the way for widespread application of infrared spectroscopy in industrial and laboratory settings. The broadband polychromatic nature of the light emitted by a thermal blackbody-like emitter (globar) used in most FTIRs has provided access to a range of wavelengths from 1 to 20 µm and beyond. Unfortunately, the large physical dimensions of the emitter have made it virtually impossible to collimate the beam over longer propagation distances. The latter is a prerequisite for high-spectral resolution gas phase spectroscopy at reduced pressures or remote sensing scenarios. Another significant limitation stems from the low power available per spectral band, which implies long integration times to obtain a satisfactory signal-to-noise ratio (SNR).

Some of these limitations have been overcome with the development of tunable diode laser absorption spectroscopy (TDLAS) enabled by rapid progress in semiconductor laser technology. High-spectral purity continuous wave (cw) emitters of spatially and temporally coherent light have shown to be compatible with optical cavities and multi-pass cells to reach unprecedented detection limits of simple gas molecules even in extraterrestrial conditions.² For infrared spectroscopy³ and microscopy⁴ of samples in condensed phase, external cavity lasers have been used to extend the optical tuning range of semiconductor lasers from several cm⁻¹ to 100 cm⁻¹ (tens of GHz to 3 THz, respectively). Mechanical scanning of wavelength provided by an external grating acting as a mode selection mechanism implies that sub-ms time resolution is only achievable for a single wavelength at a time and imposes additional technical difficulties, such as mode-hops⁵ (rapid wavelength changes) and beam-steering effects⁶ (see also Table II).

For many decades, a significant gap has remained between the broadband coverage offered by thermal sources and single-frequency semiconductor lasers. This gap was filled in the mid-infrared region by the quantum cascade laser (QCL) frequency comb—a source that operates as a broadband emitter of hundreds of lines equivalent to an array of discrete phase-locked lasers. This class of sources is critically important for probing more complex molecular absorbers with broad absorption features in the hundreds of GHz range or even multi-compound mixtures.

Besides continuous wave light sources, much attention has been devoted to optically pumped pulsed mode-locked lasers. This is because they naturally merge the idea of laser emission (which is conventionally narrowband) with the broadband spectral coverage of thermal sources. Enabled by saturable absorption,^7,8 mode-locked lasers produce optical pulses in the fs and ps range (10⁻¹⁵–10⁻¹² s) at 0.7–2.4 µm of wavelength with high peak intensities and a well-defined repetition rate. Their short pulse duration implies that they emit a polychromatic spectrum (compare time-bandwidth product), and the well-defined pulse repetition rate yields equidistantly spaced discrete lines in the frequency domain. Not only do they offer broadband spectral coverage, but they also unlock various nonlinear frequency conversion schemes, like difference and sum frequency generation, to access other spectroscopy-relevant spectral regions. One of the major breakthroughs after the development of the first mode-locked lasers came with the invention of the f-to-2f self-referencing technique.⁹ It enabled one to define the frequency of each optical line by two easily accessible microwave frequencies: an offset frequency f₀ and repetition rate f_rep. By locking them to primary frequency standards, an optical frequency comb (OFC) was obtained. Such a source was analogous to an optical frequency ruler, where each line was located at a frequency f_n such that f_n = nf_rep + f₀. In other words, equidistant lines were offset from a harmonic grid by a common frequency. The development of the OFC technique was recognized by the Nobel Prize committee in 2005¹⁰ and has obviously further propelled advances on the source side. Efforts undertaken by numerous research groups have rendered ultra-broadband frequency combs with multi-octave coverage in spectral regions ranging from the extreme ultraviolet (UV)¹¹ to the terahertz (THz).¹² 

Many techniques have been developed to perform spectroscopy using optical frequency combs to probe samples in all aggregate states. Thorough reviews can be found in Refs. 13 and 14. Briefly, the most straightforward way to apply an OFC for spectroscopy has been to use it as a broadband illumination source instead of the globar in FTIR. This has already provided advantages such as much higher radiance leading to an SNR improvement,¹⁵ while the modal phase coherence of frequency combs was exploited to achieve spectral resolution orders of magnitude below the path-limit of the FTIR spectrometer.^16,17 Other techniques to resolve comb modes were also developed, including grating-based virtually imaged phase array (VIPA) spectrometers¹⁸ that disperse comb lines on a CCD camera¹⁹ or comb-cavity Vernier spectrometers with line-filtering properties.²⁰ Arguably, the most celebrated scheme has employed the multiheterodyne frequency configuration of dual-comb spectroscopy, where a second frequency comb replaces mechanically scanned interferometers or dispersive elements, enabling high-resolution broadband spectroscopy over short time scales. On the other hand, the need for a pair of mutually coherent mode-locked lasers with frequency conversion stages to access spectroscopy-relevant longer-wavelength regions (3–15 µm) has initially restricted this technique to highly specialized laboratories. This changed with the discovery of frequency comb emission from electrically pumped quantum cascade lasers in 2012,²¹ which will be discussed in the next section.

Quantum cascade lasers (QCLs), first demonstrated in 1994,²² have expanded the realm of applications of infrared spectroscopy due to their simplicity of operation and high-power, narrow-linewidth emission in the important fingerprint spectral region. These merits are particularly well suited for gas phase spectroscopy and spectroscopic sensing of small molecules with narrow absorption lines.²³ Since 1994,²² QCLs have undergone rapid technological development. Together with interband cascade lasers (ICLs),²⁴ they are now the dominant semiconductor light source in the mid-infrared region. QCLs are based on inherently fast intersubband transitions as opposed to interband transitions.²⁵ The gain recovery of mid-infrared QCLs is in the order of 1 ps,²⁶ which is two orders of magnitude faster than conventional interband lasers and much shorter than the cavity round-trip time of typical QCLs (∼100 ps for a 4.5 mm long cavity). Such fast gain dynamics effectively preclude energy storage over consecutive cavity round trips, which is a prerequisite for mode locking using saturable absorption employed in traditional OFCs emitting optical pulses.²⁷ The QCL, in contrast, favors continuous wave operation due to the fast gain medium, which acts as a fast saturable gain.¹⁶ Broadband polychromatic emission in the form of multimode emission without any modal phase relationship could provide some advantages over globars for FTIR spectroscopy, but can one do better? Can true frequency comb emission be obtained from a QCL at all?

An answer to this question was provided in 2012, when room-temperature frequency comb emission from a mid-infrared QCL (7 µm of center wavelength) with a ∼7.5 GHz repetition rate was observed and characterized.²¹ However, unlike all prior mode-locked comb sources with amplitude modulation (AM), the electric field waveform generated by the QCL had a drastically different temporal profile: It was frequency modulated (FM) with a quasi-constant intensity. This controversial discovery has sparked a heated debate in the ultrafast optics community. Until recently, it was assumed that modal equidistance and existence of a (carrier envelope, CEO) offset frequency—prerequisites for an OFC—can originate only from pulse formation mechanisms. It was rather unclear what mechanisms can produce an OFC inside a single-section Fabry–Pérot semiconductor laser device. For the interested reader, the locking mechanism in QCL devices and the developed methods to observe the presence of phase-locked frequency comb emission are reviewed in the following section. The application-focused reader may skip to the next section and consider FM comb emission from QCLs proven.

The first challenge in studying FM comb emission properties came with the lack of suitable diagnostic tools. The celebrated intensity autocorrelation (IAC) measurement that relies on a quadratic intensity response of a nonlinear crystal or two-photon absorption in a nonlinear photodetector is not suitable for continuous-wave-like sources. Its interferometric realization (fringe-resolved IAC, FRIAC) showed a peak-to-background ratio of 8:3, which signified the lack of pulses but nothing more. Both implementations failed to provide informative quantities from a comb characterization perspective like the phase profile. Understanding the nature of FM phase locking required a different approach.

Before 2012, semiconductor lasers had already been known to exhibit the phenomenon referred to as “passive FM locking”²⁸ or “single-section passive mode locking.”²⁹ Its signature was the appearance of a strong microwave intermode beat note at the repetition frequency, which results from a coherent superposition of beating signals between all pairs of longitudinal modes in the emitted spectrum. It is important to note, however, that the appearance of a strong beat note does not prove comb operation. Its narrow width and noise-pedestal-free shape only indicate that there is at least one pair of lines exhibiting some degree of mutual phase relationship. Other lines may not be locked and thus not contribute to the beat note at all. The coherence between these lines was instead studied using a linear interferometric technique known as intermode beat spectroscopy,²¹ which was further adopted in a coherent, phase-sensitive adaptation referred to as Shifted Wave Interference Fourier Transform Spectroscopy (SWIFTS).^30,31 It is analogous to performing Fourier transform spectroscopy of the microwave intermode beat note with a suitable quadrature demodulator. Not only does the technique enable one to study the modal phase relationship but also allows for quantifying the degree of coherence between all comb lines. It can also serve for reconstruction of the emitted waveform’s characteristics like instantaneous intensity or frequency in both, AM and FM frequency combs.³² 

Before the emergence of these interferometric techniques, experimental evidence for the FM properties was provided by observing the amplitude of the intermode beat when inserting a wavelength-dependent absorber into the optical beam.³³ This absorber acted as a frequency-to- amplitude (FM to AM) converter or simply an FM discriminator. As expected, the intermode beat intensity increased by almost two orders of magnitude (16 dB).

The postulated mechanism for FM phase locking in QCLs was four-wave mixing (FWM), which being a χ⁽³⁾ process exists in many two-level systems^34–36 and ensures line equidistance. Without its presence, the frequency-dependent line spacing would be governed by the intracavity group velocity dispersion (GVD), which lacks the line equidistance property of a frequency comb [Fig. 1(a)]. Note that unlike in the case of saturable absorption, modal phase synchronization (locking) does not imply all phases being equal. The term “phase locking” is used in a wider sense: It means establishing an arbitrary fixed phase relationship that lasts indefinitely. Further characterization of QCL combs using SWIFTS revealed that the natural emission pattern follows a simple linear chirp,^37,38 see Fig. 1(b). A linearly swept instantaneous frequency corresponds to a parabolic modal phase profile, which is what one expects from a source with a saturable gain (inverse saturable absorber)—lasing at a constant frequency cannot be sustained. Interestingly, the modal phases are not random. Instead, the modal phase difference is splayed over a range of −π to π, which implies that modes lying symmetrically from the center oscillate in counterphase.³⁹ This also means that AM is strongly suppressed while the laser periodically sweeps the range of emitted frequencies without the need for any microwave generators. However, when the self-frequency sweep reaches the turnaround point (end of the scanned range), amplitude pulsation occurs and results in mixed FM/AM operation. In fact, most QCL comb devices emit waveforms with pronounced deviations from a simple linear chirp.³⁷ The instantaneous intensity oscillates in these devices too, which is why an intermode beat note can be observed. A pure FM emitter would not possess it at all.

Following the characterization experiments, many researchers have attempted to model and understand the striking phenomenon of FM locking.^41–45 It is now agreed that the requirements for an FM comb to form are spatial hole burning (SHB) to enable multimode operation in tandem with FWM (originating from the Kerr nonlinearity) responsible for modal spacing equalization and a suitable range of group velocity dispersion (GVD).^46–50 Sometimes, the natural GVD is unfavorable for a comb to form (e.g., too high); therefore, several approaches to engineer it have been proposed, like the use of dispersive facet coatings,⁴⁶ waveguide engineering,^49–51 or the incorporation of a Gires–Tournois interferometer.⁵² Unlike in the case of passive optical microcavities, FWM by itself is insufficiently strong in Fabry–Pérot semiconductor lasers to produce sidebands in the spectrum.⁴³ A recent review by the authors of the work of Silvestri et al. greatly summarizes the modeling and characterization efforts of the QCL comb community.⁵³ 

From a mathematical perspective, an elegant theory to describe FM combs using the nonlinear Schrödinger equation (NLSE) has been developed by Burghoff.⁵⁴ In the presence of a saturable absorber, as in conventional mode-locked lasers, a stable solution to the NLSE is the celebrated optical soliton, which propagates without changing its shape and is inherently localized. A train of optical solitons generated in ring cavities corresponds to an AM comb. In contrast, in the presence of fast saturable gain, GVD, and Kerr nonlinearity, a delocalized extended wave referred to as an extendon is generated as a solution to the NLSE. It extends over a full linear cavity round trip with a piecewise quadratic phase profile and depends on the phase potential of the NLSE instead of intensity. Such a wave repeats itself with consecutive round trips, thus giving rise to an FM comb.

For this tutorial, it is important to explicitly define what a semiconductor laser OFC is. The ultrafast optics community (focused on traditional AM mode-locked lasers) requires active stabilization of f₀ and f_rep for a source to meet the OFC criterion. Unfortunately, unlike f_rep, f₀ is difficult to access and typically requires one to utilize the f-to-2f offset measurement technique.⁹ It explicitly requires the presence of optical pulses to facilitate spectral broadening to an octave and doubling the frequency of the red part of the spectrum via nonlinear processes.

In conventional mode-locked lasers (that emit optical pulses), the offset frequency f₀ results from a phase slip (carrier-envelope phase, CEP) between the carrier wave and the pulse envelope peak.⁹ However, it is not obvious how one can define it for FM sources if pulses do not exist, and the waveform extends over the full round-trip period. It turns out that FM modulation can be seen as generation of equally spaced sidebands around a specific carrier frequency f_c, which can take an arbitrary value. This carrier frequency is aligned with one of the comb lines so that it still follows the comb formula f_c = nf_rep + f_0. In the time domain, this means that the produced E-field waveform with maximal chirp exhibits a certain phase increment between points spaced in time by 1/f_rep (exactly one period). This manifests itself as a common offset frequency in the spectrum just like in AM combs. The E-field waveform requires multiple round trips to reproduce itself, for instance k would be required for an offset frequency of f₀ = f_rep/k. This is the same condition as for OFC emitting pulses.

It is important to note that chirping an optical pulse from a comb source even to a maximum is routinely performed in dispersive Fourier transform (DFT)⁵⁵ and does not alter the source comb properties. It only converts the pulsed AM waveform into a dispersed FM signal captured by an oscilloscope. Recent works have shown the ability to do the opposite: compress chirped FM waveforms natively emitted by QCL⁵⁶ and diode laser combs⁵⁷ to AM waveforms (with optical pulses), where the aforementioned CEO definition could be easily applied. This means that no matter the emission pattern, the critical requirement for an OFC is just waveform periodicity, and not its shape, which translates into modal equidistance and phase coherence, and it is these two that distinguish an OFC from a multimode source.

Line equidistance in FM combs can be characterized through SWIFTS (which assesses it down to a level defined by the acquisition time on the order of seconds) or via multiheterodyne beating.²¹ In the latter case, it has been proven down to a mHz level using frequency counters. To ensure immunity to environmental perturbations, f_rep that governs the line spacing can be locked using the microwave injection locking technique, where a sinusoidal microwave signal is injected into the device.^39,58 Locking the offset frequency f₀ is possible via modulating the injection current⁵⁹ or optical injection locking.⁶⁰ Unfortunately, such schemes rely on interlocking between two sources (i.e., a true OFC operating in the relevant spectral region⁶¹ is locked to the QCL). This is how full stabilization of QCL comb frequencies (via interlocking) has been obtained;⁶¹ yet, in most practical applications it is not strictly required—locking one degree of freedom at a time suffices. For the FM comb community, a sufficient condition for a device to be called an OFC is simply full line equidistance and high phase coherence rather than stabilization of all lines to a primary frequency standard.

A plausible option for self-referencing to directly measure f₀ and stabilize it to a frequency standard without an auxiliary source or f-to-2f interferometry is the one-pulse-delay interferometer,⁶² which requires an acousto-optic modulator and a stabilized interferometer. However, to date it has not been implemented for QCL comb systems, except for a digital realization for offset frequency retrieval in FTIR experiments.⁶³ 

Apart from the initially discovered FM output and thanks to the rich laser dynamics present in QCLs, other comb laser regimes have been identified. Harmonic frequency combs⁶⁴ with sub-THz repetition frequencies as well as Kerr solitons from a ring resonator featuring a temporal width of ∼3 ps⁶⁵ have been observed. Still, in this tutorial we will focus on QCLs operating in the fundamental FM regime with inherent advantages for spectroscopy applications. When compared with harmonic combs, the fundamental mode-spacing is denser, resulting in more sample points on the frequency axis per fixed spectral range. The FM nature also minimizes amplitude modulation in these devices as opposed to the high-intensity, short pulses of Kerr solitons. This is very advantageous for two reasons. First, this reduces the amplitude of the center burst in a dual-comb spectrometer and distributes the power more evenly over the entire interferogram, therefore effectively increasing the dynamic range of the detection system. Second, it reduces the risk of nonvoluntary nonlinear effects in the sample path.

Shortly after the demonstration of comb operation of mid-infrared QCLs, their THz counterparts also revealed OFC characteristics^30,66 with many similarities to their mid-infrared siblings. To date, THz-QCL gas phase measurements of dynamically varying mixtures,⁶⁷ hyperspectral imaging of solid samples,⁶⁸ and etalon spectroscopy using the DCS approach have been shown.⁶⁹ Given the more mature technology and possibility to generate portable and compact spectrometers, this tutorial will focus on dual-comb applications of QCLs operating in the mid-infrared region. One can remain optimistic that recent advances in THz QCLs operating up to 250 K using a thermoelectrical cooler⁷⁰ will also lead to cryogenic free dual-comb spectrometers based on THz QCLs, also considering the QCL’s high-speed self-detection capabilities.⁷¹ 

With QCL dual-comb spectroscopy available for users of diverse backgrounds, this tutorial paper intends to discuss relevant aspects of this emerging technique as well as its current applications for a broad audience. Following the introductions to QCL frequency combs in this section, we review how these laser sources can be employed for infrared spectroscopy, focusing on dual-comb spectroscopy, and compare them to other frequency comb sources in this spectral region. The distinct features of QCL frequency combs are intimately related to the properties of QCL dual-comb spectroscopy and the fields of application of this technique. The final chapters provide an overview of current applications of QCL dual-comb spectroscopy as well as an outlook on future developments.

Spectroscopy with QCL frequency comb can be performed using the various techniques of frequency comb spectroscopy discussed above and reviewed in Refs. 13 and 14. Examples include Vernier filtering using an optical cavity⁷² and Fourier transform frequency comb spectroscopy with sub-nominal resolution.⁶³ The majority of research with QCL frequency combs, including applications reviewed in this manuscript, has, however, been conducted using dual-comb spectroscopy due to its synergy with the characteristics of these comb sources. In the following, we introduce dual-comb spectroscopy followed by a comparative discussion of QCL frequency combs with other FC sources and the implications of the distinct differences for spectroscopic applications.

Dual-comb spectroscopy (DCS) leverages the frequency comb properties of a pair of sources to perform broadband optical measurements without resorting to any moving parts or dispersive elements.^73,74 In DCS, one establishes a link between the optical and microwave domain through optical heterodyning between a pair of sources that exhibit mutual spectral overlap. Interaction between the two sources produces an interferogram like that in FTIR spectroscopy which, however, results from the E-field cross-correlation⁷⁵ rather than autocorrelation. Since this technique is phase-sensitive, it provides both absorption and dispersion information. Unfortunately, it puts sharp constraints on the phase noise (frequency stability and jitter) properties of the QCL comb source. Therefore, a phase-incoherent multimode source will not work here and will not offer the advantages of DCS discussed below.

Figures 2(a) and 2(b) show schematic beam paths of a dual-comb experiment in the so-called amplitude and phase-sensitive configuration, also known as symmetric and asymmetric,⁷⁶ respectively. In the amplitude-sensitive configuration, both lasers are attenuated by the sample, which increases the sensitivity. In the phase-sensitive configuration, only the sample comb passes through the sample and is combined with the local oscillator comb thereafter. In this configuration, both the attenuation and the phase shift induced by the sample are measurable.

From the first inequality in (3), one may conclude that, by increasing the difference in repetition frequency between the two frequency combs, τ and hence the time resolution between consecutive spectra can be decreased to arbitrary values. This, however, neglects the fact that any one mode of the sample frequency comb interferes with any mode of the local oscillator comb. Equation (2) is only an approximation assuming that f_S,n − f_LO,k≠n is larger than the detector bandwidth for all n, k, i.e., interference of any mode with the “second to next” neighboring local oscillator mode yields beat note frequencies outside the detector bandwidth. As can be readily shown, this condition cannot be fulfilled if $Δfrep>frep2N$⁠, where N is the number of modes emitted by the comb, since, in this case, the beat note frequency of at least one mode with the “second to next” neighboring mode f_S,n − f_LO,n+1 is smaller than f_s,n − f_LO,n. By expressing N as a function of the spectral coverage f_span of the frequency comb source, $N=fspanfrep$⁠, one arrives at the important observation that the time resolution achievable in a dual-comb spectroscopy experiment scales with $frep−2$ and f_span [compare (3)]. It is further worth noting that the limit expressed in (3) relies on the strict phase relation between comb modes. In phase-incoherent multimode devices, the beat notes would appear broadened or even unresolved, with no phase relation between the individual lines, yielding crosstalk between neighboring heterodyne lines.

Among the various types of frequency comb sources, QCL frequency combs feature a distinct set of properties: (i) high repetition frequency f_rep of typically 10 GHz, (ii) high optical power, typically between 50 mW and 1 W, (iii) direct emission in the mid-infrared between 5 and 10 µm, covering the fingerprint range, and (iv) emission from chip-scale, direct current driven devices. The relevance of these features for mid-infrared dual-comb spectroscopy shall be discussed.

The significance of f_rep for the time resolution in dual-comb spectroscopy was clarified in Eq. (3). To stress this aspect, it is worth considering typical values for f_rep. Typical values for mode-locked lasers are on the order of 50–250 MHz, and typical spectral coverage f_span is 100 to 400 cm⁻¹.⁷⁷ Entering in Eq. (3) yields a maximum time resolution τ of ∼1 ms. In contrast, due to scaling with f_rep⁻², typical values for QCL frequency combs yield τ ∼ 40 ns. In practice, technical limitations set by the detector bandwidth currently limit the time resolution to typically 1 µs. High f_rep and low τ comparable to QCL frequency combs can be achieved with micro-resonator based frequency combs, but the optical power of these sources is much lower.⁷⁸ 

Besides the time resolution, f_rep governs the spectral sampling of infrared spectra. Optimal values of spectral sampling depend on the width of the spectral features of the sample to be resolved.⁷⁹ For small molecules in the gas phase, the width of isolated rovibrational absorption lines typically falls in the range between 0.2 and 0.005 cm⁻¹, depending on the temperature and pressure. Hence, the native sampling of QCL frequency combs given by f_rep ∼ 0.3 cm⁻¹ is too sparse for many applications, which can, however, be overcome by spectral interleaving (compare the section titled “High-resolution molecular spectroscopy”).

In liquids and solids, absorption bands have a typical width of 10s of cm⁻¹. Hence, f_rep ∼ 10 GHz = 0.3 cm⁻¹ is sufficient to resolve the spectrum well above the typically chosen spectral resolution of 2–8 cm⁻¹ in FTIR spectroscopy. In fact, for broad absorption bands, concentrating the optical power available from the laser source in a smaller number of modes can be beneficial: With typically 200–300 modes and an optical power of 50 mW–1 W,⁴⁰ the optical power per mode is close to 1 mW. This is approximately four orders of magnitude higher than typical values for conventional mode-locked fiber combs in the near-infrared spectral range⁸⁰ and five orders of magnitude higher than difference frequency generated combs at wavelengths > 5 µm.⁷⁷ The high power per mode is important when the measured sample absorbs a large portion of the light,^81,82 as is the case, for example, in aqueous solutions. In this case, and/or at short integration times (high time resolution, few repetitions), the detection noise floor will readily exceed the amplitude of individual sub-µW comb modes, while mW level modes will show a much higher signal-to-noise ratio.

The availability of direct mid-infrared emission in the mid-infrared from chip-scale sources greatly simplifies the optical and electronic setup of spectrometers, avoiding nonlinear frequency conversion. Therefore, the complete optical assembly for generating and combining two mid-infrared frequency combs can be as compact as 13 × 11 × 5 cm³,⁸³ and the same drive electronics used for conventional QCLs can be used without radio frequency drive electronics. Being direct current driven, thermoelectrically cooled, and millimeter-sized, QCL comb devices are simple in operation and lend themselves to compact system integration,⁸⁴ leading to the introduction of commercial tabletop spectrometers in 2018 using the dual frequency comb technique.⁸⁵ 

To make best use of QCLs for dual-comb spectroscopy and achieve high-quality spectra with high signal-to-noise ratio, some practical aspects and technological challenges must be taken into consideration that shall be discussed in this section.

First, the currently predominantly employed free-running operation of QCLs implies that the signal and local oscillator comb exhibit uncorrelated fluctuations in f_rep and f₀, yielding jitter on Δf_rep and Δf₀ on the order of ∼1 kHz and ∼3 MHz, respectively, on ms time scales. This causes broadening of the heterodyne beat notes and, together with amplitude noise on the lasers themselves, multi-percent level noise on their amplitudes and phases. This effect, however, can be effectively mitigated by employing a reference beam path in which both the local oscillator and signal comb bypass the sample. The so obtained reference multiheterodyne spectrum from a second detector is then used to normalize the amplitude and phase of each beat note, which allows reaching the noise floor reported below. Alternatively to a reference beam path, it is possible to employ digital phase correction to compensate for the beat note broadening effect.^86,87 Two global time-varying frequencies are extracted from a dual-comb signal and applied in counterphase to all lines. It is worth noting that this approach is only possible due to the strict phase relation between the modes of QCL frequency combs and cannot be employed to phase-incoherent multimode devices.

After applying the techniques described above, the noise floor at short time scales is limited by the number of photons, i.e., optical power, available per spectral element. Besides the emitted optical power, the linear range of the detector often limits the optical power per detected spectral element. Mercury cadmium telluride (HgCdTe) detectors currently employed in QCL dual-comb spectroscopy saturate at power levels below 1 mW. Generally speaking, single-frequency laser spectrometers achieve the highest SNR at short time scales, as all power is concentrated in one spectral element, while broadband rapid-scan FTIR instruments achieve large spectral coverage at the expense of low SNR.⁸⁸ With N ∼ 200–300 spectral elements, QCL frequency combs operate on a middle ground, with sufficient bandwidth to cover vibrational absorption bands in condensed phase and high SNR on short time scales. This is illustrated by an Allan deviation analysis⁸⁹ shown in Fig. 3. It represents, as a function of integration time, the standard deviation σ of the absorbance $α=−log10T=−log10I(t)I0$⁠, with I(t) and I₀ the intensity as a function of time and at t = 0, respectively. Three datasets are presented, related to practical data-handling aspects of the setup used: a “Short Term” measurement, where a continuous stream of interferograms is evaluated at 4 µs time resolution, and two “Long-Term” measurements, where a stream of acquisitions of 4 ms duration is evaluated. The long-term measurement, labeled as “Buffered,” was taken by acquiring consecutive measurements as fast as possible, so a data buffer fills up after ∼20 s, prohibiting the acquisition of new data. For the data labeled “Long Term,” the acquisition frequency is chosen such that data processing keeps up with data acquisition, so measurements at this frequency can continue for hours and days, if necessary. As the standard deviation varies with the power of the heterodyne beat note, the best beat note and 75th percentile beat note are shown for each dataset. Note that, for long-term measurements, the integration time is only 8% (“Long Term”) and 30% (“Buffered”) of the measurement time plotted on the abscissa of Fig. 3. For a single heterodyne beat note, the standard deviation reaches about 10⁻³ at 10 µs and about 10⁻⁴ at 1 ms integration time. For longer integration times, the standard deviation remains at ∼10⁻⁴ for measurement times up to 1000 s (about 16 and a half minutes), limited by slow instrument drifts. Further improvement of the signal quality can be achieved by spectral averaging of multiple beat notes or averaging of multiple sample measurements. Note that this noise floor is achieved without spectral averaging. The additional gain from spectral averaging to a resolution sufficient for the majority of condensed phase samples is apparent in the spectra shown in Figs. 3(b) and 3(c). As a reference, state-of-the-art FTIR instruments achieve 10⁻⁵ absorbance noise on second time scales due to the exceptionally stable emission characteristics of thermal emitters. It is further worth noting that an absorbance noise floor of 10⁻⁵–10⁻⁶ can be achieved with QCL dual-comb spectroscopy in repeated time-resolved measurements, for example, in experiments triggered by a light pulse, where slow drifts are effectively eliminated due to the background taken immediately before each trigger event (compare section dual-comb spectroscopy in biophysics).⁹⁰ 

To achieve the signal-to-noise ratio shown in Fig. 3, some noise sources commonly encountered in dual-comb spectroscopy must be minimized.

Optical feedback must be avoided. Optical feedback refers to light that is fed back into a laser through reflection or scattering. Optical feedback disturbs the stable emission of QCL frequency combs and induces rapid fluctuations in the emitted spectrum. Hence, any transmitted optical element in the beam path must be tilted. Optical isolation typically required to prevent stray light from entering the laser can be achieved using neutral density filters or Faraday rotators.

Interference fringes are a common source of noise in all spectrometers. They arise from interference between fractions of the beam reaching the detector through different paths, typically due to reflections from plane parallel surfaces (etalon). Fringes are observed if the difference in optical path length of the alternate beam paths is smaller than the reciprocal resolution of the spectrometer, while fringes of smaller periodicity are effectively averaged out. As the linewidth of QCL frequency combs on the μs to ms time scale is on the order of 3 × 10⁻⁵ cm⁻¹, high-frequency fringes may show up in a spectrum, even though the spectral sampling is much larger, e.g., 0.3 cm⁻¹ (compare section entitled “High-resolution molecular spectroscopy”). Hence, care must be taken to avoid both short and long etalons.

Finally, averaging is an essential step to achieve better signal-to-noise ratio and also to deal with the very high data rate produced by QCL dual-comb spectroscopy. Typically, detector signals are acquired for acquisition lengths of 100 µs to tens of ms. The data of an acquisition are split up into slices of a given processing time of typically 1 µs–16 µs, yielding up to 1000 transmission and phase spectra per ms. In short-term measurements, the averaged slices before an external trigger are typically used as a background for transmission spectra. The resulting spectra are so-called "difference spectra", comparing post-trigger transmission to pre-trigger transmission. For static samples or samples with ms to s dynamics, it is convenient to average over all spectra obtained from an acquisition and repeat acquisitions with a given repetition frequency of typically 1–50 Hz. In this long-term mode, one or multiple acquisitions taken prior to the experiment serve as a background.

Table I summarizes the main fields of application of QCL dual-comb spectroscopy with the relevant information accessible from vibrational spectra, commonly used spectroscopic techniques other than QCL dual-comb spectroscopy, and common experimental challenges. More details and additional references for each field of application are provided in the following paragraphs.

To understand the strengths and weaknesses of QCL DCS for a given application, the technique must be compared with other spectroscopic techniques, as listed in Table I. For an overview, the diverse alternative techniques are sorted in groups of similar characteristics. Table II lists these groups alongside their common advantages and disadvantages. The authors stress that this list and the exemplified techniques are not exhaustive. More details are available in the review articles referenced in Table I, and additional discussion is provided in the following paragraphs. Some of the characteristics given in Table II shall be briefly discussed here.

Spectral coverage describes the spectral range accessible with a given instrument, either in a single-shot experiment or by wavelength tuning. While large spectral coverage is always desirable, it also implies that the available optical power is distributed over a larger number N of spectral elements. Hence, the power per resolved spectral element is reduced by a factor of N, yielding a SNR that is lower by a factor of $N$ compared to a single wavelength measurement of same power. Moreover, in rapid-scan FTIR measurements, highest SNR is achieved by reducing the spectral range by spectral filters due to improved exploitation of the detector dynamic range, as demonstrated in Ref. 123.

This scaling law holds true for a photon limited noise floor typically encountered at short integration times. For longer integration times, typically seconds, coherent artifacts and intensity noise of the light source are commonly the dominant sources of noise. The most common coherent artifacts are optical fringes¹²¹ in the frequency domain and speckles⁸¹ in the spatial domain. Intensity noise is typically higher for laser sources than for thermal emitters. Additional effects like fluctuations in the emitted polarization orientation as well as optical feedback may cause additional noise on the power received at the detector element.

The beam quality of the probing beam dictates the optical design of the spectrometer and the sample interface. Thermal light sources emit their radiation from an extended surface area of at least several mm², which is effectively limited in commercial spectrometers by a variable circular aperture in the beam path. To guide the beam along extended beam paths necessary for increased spectral resolution, the aperture must be limited to <1 mm, reducing the optical throughput and SNR.¹¹⁰ In contrast, laser sources typically emit close to Gaussian beams, which greatly simplifies the definition of optical beam paths and allows coupling the beam to single-mode fibers and optical cavities, among others.

Systems of biological relevance, such as proteins and other oligo- and polypeptides, as well as nucleic acids, have been extensively studied by (time-resolved) infrared spectroscopy. Infrared spectroscopy can reveal information about the structure and function of these molecular systems in isolation as well as in biologically relevant environments.¹⁰¹ Perhaps the most studied spectral region is the Amide I region, around 1600–1800 cm⁻¹, due to its prominence in the spectra of polypeptides.^101,124 The Amide II and Amide III bands (1450–1600 and 1300–1450 cm⁻¹) are less frequently studied.^125,126 Protonation and de-protonation and changes in redox state of side chains can be followed by infrared difference spectroscopy.⁹¹ If site-specific information is desired, site-specific infrared probes such as CO, CN, N₃, and NO groups and noncanonical amino acids can be used to investigate a specific target within the biomolecule.¹²⁷ A major advantage of infrared spectroscopy is its ability to determine dynamics over time scales ranging from femtoseconds to hours, all of which are relevant for biophysics.

Infrared spectroscopy of biological molecules faces several challenges. First, strong absorption by water and the protein backbone limits the cell thickness to ∼8 µm in the amide band region (∼22 cm⁻¹ M⁻¹ absorption coefficient for the bending mode at 1645 cm⁻¹) and requires temperature stabilization to avoid changes in the background absorption spectrum. To mitigate this, studies are often conducted in deuterated conditions, which alters the sample properties.^128,129 High protein concentrations are typically used to achieve sufficient sensitivity,¹³⁰ yielding viscous and foaming liquids that form bubbles in the liquid cell.¹³¹ Additionally, not all biological processes are cyclic, thus necessitating methods that capture information in a single experiment or within a few repetitions. Sometimes, they require complete sample replacement for every repetition.¹³² 

Rapid-scan FTIR spectroscopy is applicable to noncyclic experiments and achieves a time resolution of ∼10 ms,^133,134 with 10 µs demonstrated.⁸⁸ However, the spectral resolution is comparably low (16 or 32 cm⁻¹ are common), and the SNR is limited by the low available power per spectral element. Step-scan FTIR achieves microsecond to nanosecond time resolution, but it requires tens of thousands to hundreds of thousands of repetitions per measurement.⁹⁵ Finally, ultrashort pulsed lasers with high repetition rate enable single-shot spectroscopy with 10 µs time resolution and fs to ns time resolution in cyclic experiments.^135,136 The required setups are, however, complex and expensive and currently out of reach of nonexpert operators.

QCL frequency combs are a valuable complement to these techniques. As outlined above, the combination of the GHz repetition rate, 50–100 cm⁻¹ coverage, and >50 mW optical power results in spectra covering the typical width of the amide and other absorption bands at 1 µs time resolution and 10⁻³ to 10⁻⁴ transmission noise at 1 ms integration time. The substantial reduction in necessary experimental repetitions was first demonstrated for bacteriorhodopsin, a light-driven proton pump. The authors of the work of Klocke et al. studied the isomerization step of the chromophore retinal in the protein via characteristic changes in C–C stretching and C–C–H rocking modes in the 1230–1200 cm⁻¹ range.⁸² The study of Schubert et al. targeted the C=N stretching mode of the retinal Schiff base and conformational changes of the protein in the 1680–1620 cm⁻¹ region as well as the protonation of an aspartic acid in the 1770–1720 cm⁻¹ range.⁹⁰ Exemplary spectra and time-traces recorded of bacteriorhodopsin after photoexcitation at 532 nm are given in Fig. 4.⁹⁰ After 500 repetitions recorded in 250 s, spectra with a noise floor of 15 µOD (0.000 007 transmission noise) were obtained. This allowed retrieving the kinetics of the photocycle of bacteriorhodopsin observed previously in 20 000 repetitions via step-scan FTIR.⁸² The reduced measurement time is critical to lower the consumption of precious biological samples. Another advantage stems from the fact that the liquid cell thickness was increased by a factor of four compared to step-scan FTIR measurements, which provided increased sensitivity and improved sample handling.⁵ 

The capability of single-shot time-resolved measurements was also exploited to study the hydrolysis kinetics of GTP by GTPase Gα_i via phosphate stretching bands in the 1280–1210 cm⁻¹ region.¹³⁷ Here, the reaction was initiated using photoremovable protective groups¹³² (“caged compounds”) on the guanosine triphosphate (GTP) substrate that released GTP upon UV excitation. Due to the irreversible nature of photolysis, the small number of necessary repetitions in QCL dual-comb spectroscopy significantly reduces the sample consumption in these experiments.

Combustion systems involve high pressures and temperatures, resulting in fast reaction kinetics with multiple species involved. Reaction kinetics at well-defined conditions are commonly studied in 0D model systems such as shock tubes, where a gas is compressed by a shock wave. Infrared spectroscopy is used to retrieve species concentrations, temperature, and pressure at the necessary temporal resolution of ∼10 µs to 1 ms.^102,138,139 Experimental challenges beyond the time resolution arise from an abrupt jump in temperature and pressure and mechanical vibrations upon impact of the shock wave, resulting in beam deflection and offsets in the recorded transmission. With few exceptions,¹⁴⁰ shock tube experiments typically require cleaning of the apparatus and exchange of the burst diaphragm after each shot and consume large amounts of sample gas. This results in a very limited number of repetitions per workday, if any. Hence, the target species, temperature and pressure must be measured with sufficient sensitivity with a very limited integration time of typically <1 ms. Further challenges are the spectral resolution necessary to resolve absorption lines of gaseous species and thermal emission from the hot gas that may saturate the detector.

Direct absorption spectroscopy and frequency modulation spectroscopy based on narrowly tunable laser sources are common techniques for retrieving the concentration of few small molecular species.^102,103,141 While frequency comb spectroscopy with mode-locked lasers has been applied to other areas of combustion diagnostics,^142–144 the achievable time resolution has been insufficient for shock tube experiments.

QCL dual-comb spectroscopy offers a significantly larger spectral coverage than diode lasers that may include absorption lines of multiple target species and can further cover broad absorption bands of heavier molecules. The coverage of many rovibrational transitions further allows determination of temperature, and enables improved baseline removal, including broadband background absorbers, in particular soot. Depending on the gas temperature and pressure, the spectral sampling of typically 0.3 cm⁻¹ of QCL dual-comb spectroscopy can be insufficient to capture the absorption line shape of a target gas, and some absorption lines in the covered spectral range may not be captured. Although a denser spectral sampling is often desirable, spectroscopic models can also be fitted to undersampled spectra, allowing accurate retrieval of concentrations and temperature.

QCL dual-comb spectroscopy was first validated against two individual distributed-feedback laser diagnostics in a shock tube experiment oxidizing a fuel (propyne) into water¹⁴⁵ as shown in Fig. 5(a). The speciation results of both laser diagnostics are shown in panel (b) and are compared to the USC Mech. II kinetic mechanism numerical model with good agreement. Simultaneous quantification of four hydrocarbon species in nonreactive shocks has been shown in the work of Zhang et al.¹⁴⁶ 

Simultaneous retrieval of gas temperature was demonstrated based on the spectrum of methane in Ref. 147. The measured absorption is shown in Fig. 6(a) as blue triangles along with simulated spectra. Time-dependent simulated and measured temperatures are shown in panel (b). QCL dual-comb spectroscopy was further used to determine chemical rate coefficients for the decomposition of trioxane with formation of formaldehyde.¹⁴⁸ 

The term “stand-off” refers to measurements during which the operator is at a safe distance from the typically solid state and potentially harmful sample to be studied. Vibrational spectroscopy offers the unique capability to gather spectroscopic information from a distance that can identify a substance with sufficient selectivity. Both infrared backscattering imaging spectroscopy^114,149 and Raman imaging spectroscopy^115,150 have been investigated for this purpose.

Common challenges in stand-off detection include the low power of backscattered light, the necessity to transmit through ambient air and adherence to eye safety limits. The latter, in combination with small Raman cross sections, strongly limits the sensitivity of Raman spectroscopic stand-off detection. Eye safety limits in the infrared region are approximately two orders of magnitude higher than in the visible region, allowing the use of infrared lasers exceeding 100 mW output power. With the exception of fog and haze, transmission through ambient air is higher in the visible spectral range.¹⁵¹ Nevertheless, atmospheric windows coinciding with the mid-infrared fingerprint region can be exploited. To identify substances reliably, high signal-to-noise ratio and spectral bandwidth are required. The latter is determined by the laser source. The former is often limited by the power of the backscattered light and coherent artifacts in the form of speckles.¹⁵² 

QCLs are appealing for stand-off detection due to their compact form-factor and high optical power in excess of 100 mW. External cavity and broadband Fabry–Pérot QCLs were employed to raster-scan across a surface while typically collecting the backscattered light on a focal plane array detector.^152–155 Note that direct (specular) reflection is avoided as specular reflection spectra are less sensitive to trace residues and show little resemblance to transmission reference spectra.¹⁵⁶ Diffusely backscattered spectra may also differ from transmission spectra and are affected by surface material, coverage, and grain size of a powder. Suitable databases and/or spectral modeling must hence be employed.¹⁵² 

QCL frequency combs have been used for stand-off detection of explosives.^81,157 Savitski and co-workers used a pair of QCL frequency combs covering the spectral range from 1200 to 1285 cm⁻¹ to detect the backscattering spectrum collected via a reflective telescope (see Fig. 7).⁸¹ As focal plane arrays are not fast enough to resolve the multiheterodyne dual-comb spectrum, the backscattered light was focused on a single point detector. A hyperspectral image covering 18 × 18 cm² was obtained by scanning the beam via a galvo-mirror pair across the target placed 3 m from the source. In 2 s measurement time, a hyperspectral image with 400 pixels was obtained that allowed the detection of 30 µg/cm² of cyclotrimethylene trinitramine (RDX)⁸¹ (see Fig. 7). Further improvements in sensitivity can be expected from various methods of speckle mitigation.¹⁵⁸ 

The main drawbacks of QCL frequency combs for stand-off detection are their limited spectral range and their susceptibility to optical feedback. To increase the spectral coverage, simultaneous operation of co-aligned laser modules has recently been demonstrated.¹⁵⁹ As illustrated in Fig. 8(a), a polarizer was used as the beam combining element and the polarization of one laser module was rotated by 90° with respect to the other module. Hence, one beam is transmitted, while the other is reflected at the polarizer, yielding maximum optical power of both beams on the sample. The orthogonal polarization also avoids interference beat notes between spectrally overlapping comb lines of lasers of different modules. To disentangle the multiheterodyne spectra of the two modules, they were separated along the frequency axis by slightly tuning the laser temperatures and currents [see Fig. 8(b)]. The two spectra can then be processed without interference between them by applying a digital low-pass and high-pass filter, respectively.

High-resolution vibrational spectroscopic data play a central role in various fields by providing essential information, including spectroscopic constants,¹⁶⁰ line shape parameters,¹⁶¹ line strengths,¹¹² and molecular number densities for research in physical chemistry, atmospheric science, astrophysics, and astrochemistry. Such data are applied for atmospheric remote sensing,^162,163 investigation of exoplanetary atmospheres,¹⁶⁴ and molecules in the circumstellar and interstellar medium¹⁶⁵ and numerous fields of fundamental research.¹⁶⁶ 

Besides the obvious technological challenge of achieving the required spectral resolution, ensuring accuracy for the assigned frequency axis represents a challenge that requires establishing a link between primary or secondary frequency standards and the spectroscopic data. Advanced techniques, in particular experiments in pulsed molecular jets, require simultaneously high spectral resolution and time resolution to study transient species of sub-ms lifetime.¹⁶⁶ High spectral resolution naturally yields increased susceptibility to optical fringing, which must be met by careful optical design. Ideally, all of the above should be tackled with a spectral coverage of 100s of cm⁻¹.

High-resolution FTIR spectroscopy¹⁶⁷ has been the method of choice for broadband measurements at a spectral resolution on the order of 10⁻²–10⁻³ cm⁻¹. Spectrometers based on tunable lasers trade spectral coverage for improved spectral resolution <10⁻⁴ cm⁻¹ and reduced measurement time and instrument cost and also allow measurements of transient species.¹²² Today, frequency comb spectroscopy is established as a prime tool to obtain high accuracy spectroscopic data.^13,168,169 The spectral properties of frequency combs exhibit an extraordinary overlap with the requirements of high-resolution molecular spectroscopy. A sample’s infrared spectrum is probed at 100s to 100 000s of equidistant points spanning 10s to 100s of cm⁻¹. The frequency axis accuracy of all datapoints is achieved by controlling or measuring only two parameters: the offset frequency f₀ and repetition frequency f_rep.

Before further discussion of high-resolution frequency comb spectroscopy, a brief definition of spectral resolution is needed. Griffiths¹⁷⁰ offers two definitions for spectral resolution, one as “the minimum separation of two infinitely sharp lines of equal intensity that allows the presence of two lines to be seen in the measured spectrum,” and the second as “the full width at half height (FWHH) of an infinitely sharp isolated line measured by a spectrometer.” Applying the first definition, one may consider the repetition frequency f_rep of a frequency comb to determine the resolution of a dual-comb spectrometer. However, since the first definition insinuates that any separation of two lines larger than the resolution renders observable lines, this first definition appears ill-suited for the situation at hand since the presence of two lines also may not be seen for separations larger than f_rep, e.g., 2.7 × f_rep. The second definition can be applied by assuming that the frequency of the infinitely sharp line coincides with a mode of the frequency comb. In this case, the FWHH, which will be referred to as full width at half maximum (FWHM) from here on, of the measured line would be equivalent to the linewidth (FWHM) of the comb mode on the time scale of the measurement.

It is hence customary for frequency comb spectroscopy to discriminate between spectral resolution and point spacing, i.e., the distance in cm⁻¹ or Hz between adjacent points in the measured spectrum. While the former is limited by frequency drifts and phase noise on different time scales, the latter can be reduced by combining spectra recorded consecutively with tuned f₀ or f_rep, i.e., by spectral interleaving.^111,171 A spectrum obtained from q interleaved measurements spanning an overall tuning range of f_rep hence has a reduced point spacing of f_rep/q. QCL frequency combs are tunable by both, current and temperature, where the first is typically preferred because it is much faster.

The applications presented below employ free-running QCL frequency combs. Because they are not referenced—neither to each other nor to a stable frequency reference—the frequencies of their teeth are not known a priori and must be calibrated by measurements of f₀ and f_rep. One may argue that, since the f₀ and f_rep are also free-running and hence fluctuate, an exact relation between all comb modes does not exist as it constantly changes, defying the advantages of frequency combs for high-resolution spectroscopy. However, as shown below, the fluctuation of f₀ and f_rep is sufficiently small such that the advantages of frequency combs are preserved for most applications. The following examples showcase different approaches to obtaining a MHz-level accurate frequency axis for interleaved QCL dual-comb spectroscopy. It is important to note that QCL frequency combs can be fully stabilized by injection locking and locking of f₀ (compare the section titled “Broader-sense definition of an optical frequency comb”).^61,172

An approach to high-resolution molecular spectroscopy employing two QCLs in a dual-comb configuration named rapid-sweep mode was demonstrated by a group at Empa¹⁷³ in 2020. Here, a triangular modulation with a period of 120 ms was applied to the bias current of both lasers to tune the frequencies across the gap between adjacent comb modes. By tuning both lasers simultaneously, the multiheterodyne interference spectrum remains within the detection bandwidth throughout the measurement. By dividing the acquired interferograms into short slices, about 30 000 spectrally offset spectra were obtained. Since the modulation ramps were programmed with an amplitude that resulted in a maximum shift of ∼f_rep, the point spacing after interleaving was reduced to f_rep/30 000 ≈ 300 kHz. Figure 9 shows the absorption and dispersion spectra of methane gas at 107 hPa in the range 1170 to −1225 cm⁻¹ (left) and an isolated absorption line with a Voigt fit and residuals (right). The determination of the frequency axis required calibration of $f0t$ and $frept$⁠, whereby several absorption lines with known center frequencies were measured. Instead of calibration, the authors of the work of Komagata et al. measured $frept$ directly from the voltage modulation on the laser electrodes.¹⁷⁴ Furthermore, $f0t$ was measured by recording the beating of one comb mode with a distributed-feedback QCL that was locked to an absorption line of N₂O, achieving an overall frequency uncertainty of 600 kHz.

An alternative tuning procedure was described in the work of Lepère et al.¹⁷⁵ Instead of continuously tuning both lasers, in the so-called step-sweep mode, the interferogram is sampled only while the lasers are under constant current. Then, one laser is “stepped” while the second is kept fixed. The frequency shift of the first laser relative to the second is measurable as a shift in beat note frequencies. In a second step, the second laser is also stepped to prevent the beat notes from progressively wandering outside the detection bandwidth. In this way, the changes in f₀ and f_rep for every step are directly measured with respect to the initial step. Calibration of f₀ and f_rep at the first step based on known absorption lines hence yields a fully calibrated wavenumber axis. The agreement of the frequency axis was found to be within 4 MHz of the reference measurement of N₂O¹⁷⁶ (see Fig. 10).

The authors of the work of Agner et al. performed interleaved measurements in step–sweep mode in a pulsed supersonic beam (see Fig. 11).¹⁷⁷ Pulsed supersonic jet expansions and molecular beams prepare molecules in the gas phase at low temperatures, which is essential to reduce complexity and congestion in rovibrational spectra of large molecules and to unambiguously assign quantum numbers to observed absorption lines.¹⁶⁶ Absorption lines with a full width at half maximum of 30 MHz were resolved with a temporal resolution of 4 µs.¹⁷⁷ 

During the first ten years of QCL frequency combs, they have been employed in diverse fields of academic research, primarily using dual-comb spectroscopy. The application space of QCL frequency combs reviewed in this tutorial, leaning more toward samples in liquid and solid states, is distinctly different from that of more established frequency comb sources that have been used with remarkable success mostly for high-resolution and precision spectroscopy of gases.¹⁴ As discussed in this tutorial, this is a direct consequence of the distinctly different properties of QCL frequency combs compared to these sources, in particular the higher repetition frequency of typically 10 GHz and higher optical output power in the 5–10 µm spectral range of up to 1 W.

The high repetition frequency of QCL frequency combs enables sub-µs time resolution in dual-comb measurements, which has been exploited in the context of biophysical processes and combustion diagnostics. Beyond that, probing rapid molecular kinetics in the infrared region has significant potential in spectroelectrochemical studies,^179,180 for stopped-flow measurements,¹⁸¹ and for analyzing polymerization kinetics.¹⁸² 

In the future, fast measurements with sub-10⁻³ transmission noise recorded in under 1 ms may be utilized for high-throughput applications, involving the rapid analysis of numerous samples, as seen in hyperspectral imaging, waste sorting,¹⁸³ and quality control for incoming and outgoing products. Future applications may further exploit the polarized nature of QCL emission in polarimetry¹⁸⁴ and vibrational circular dichroism experiments.¹⁸⁵ 

The high repetition frequency of QCL frequency combs and their typically free-running operation (without active stabilization of f₀ or f_rep) are generally disadvantages in gas phase measurements with high spectral resolution. Nevertheless, measurements employing spectral interleaving demonstrated agreement with reference values of measured line positions and linewidths to within 5 MHz (1.7 × 10⁻⁴ cm⁻¹).¹⁷⁵ By referencing f_rep and f₀ to frequency standards, a combined uncertainty of the frequency axis below 1 MHz was demonstrated.¹⁷⁴ With these figures of merits, QCL combs are well suited, among others, for accurate line parameter studies.¹⁷⁵ Intriguingly, μs time resolution enabled by the high repetition frequency was shown to be an important feature for high-resolution measurements of transient species in pulsed supersonic expansions.¹⁷⁷ In the future, even higher frequency accuracy could be achieved with fully stabilized QCL frequency combs.¹⁷² 

Quantum cascade detectors becoming commercially available are an interesting new development for QCL DCS. Although they suffer from a lower responsivity than the currently employed HgCdTe photodetectors, their bandwidth reaches 10s of GHz. This allows taking full advantage of the maximum bandwidth of DCS set by f_rep [see Eq. (3)], enabling a time resolution of a few 100s of ns. Additionally, this would allow performing interleaved measurements in step-sweep mode without stepping the local oscillator frequency, thus improving the frequency accuracy of the technique.

Another promising development is the recent demonstration of thermoelectrically cooled THz QCLs.¹⁸⁶ If cryogen-free, broadband THz frequency combs become available, the application space exploiting high temporal and spectral resolution of QCL dual-comb spectroscopy would be extended to the realm of THz spectroscopy.

Finally, QCL dual-comb spectroscopy holds the promise of massive miniaturization. With dual-comb spectroscopy from two QCL combs on a single chip shown,¹⁸⁷ and on-chip integration of quantum cascade laser and detector shown,¹⁸⁸ it is only a matter of time until complete on-chip integration of a QCL dual-comb spectrometer is demonstrated. Such a device may find use in airborne or space applications, mobile or handheld spectrometers, or miniaturized optical sensors.

In concluding remarks, it may be worth mentioning that in addition to DCS, QCL combs can also extend the wavelength coverage of spectroscopic techniques tailored to the multi-GHz line spacing of other comb platforms in the mid-infrared-like interband cascade lasers¹⁸⁹ or quantum well diode laser frequency combs.¹⁹⁰ Such techniques lift the challenging requirement for a fast mid- or longwave-infrared photodetector. For instance, QCL combs can be employed for mode-resolved Vernier spectroscopy, which would offer ms acquisition time of broadband spectra with a cavity-enhanced interaction length.⁷² Even short, cm-sized optical cavities can serve for this purpose and provide tens of meters of effective path length. Another potentially interesting niche is MHz-resolution Fourier transform spectroscopy (FTS) obtainable at millimeter-sized displacements of the interferometer’s moving arm.⁶³ This technique is a chip-scale adaptation of sub-nominal resolution FTS,¹⁶ which offers resolution enhancement factors relative to the nominal (up to 1000× or more) for sources with narrow optical linewidths and multi-GHz repetition rates.

L. A. Sterczewski acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 101027721.

J.H. acknowledges funding from the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie Grant Agreement No. 101032761.

L.S. acknowledges funding from Horizon 2020 Framework Program (Grant No. 101027721) and J.H. acknowledges funding from Horizon 2020 Framework Program (Grant No. 101032761).

J.H., R.H., Ma.G., Mi.G., A.H. and M.M. are current and past members of IRsweep AG, now part of Sensirion AG, which commercializes QCL dual-comb spectrometers. Their function may be perceived as a conflict of interest by some readers. IRsweep and Sensirion hold up the highest ethical standards in their scientific work. The authors do not financially participate in the commercial success of the company. L.S. declares no conflict of interest.

Jakob Hayden: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal). Markus Geiser: Writing – original draft (equal); Writing – review & editing (equal). Michele Gianella: Writing – original draft (equal); Writing – review & editing (equal). Raphael Horvath: Writing – original draft (equal); Writing – review & editing (equal). Andreas Hugi: Writing – original draft (equal); Writing – review & editing (equal). Lukasz Sterczewski: Conceptualization (equal); Validation (equal); Writing – review & editing (equal). Markus Mangold: Conceptualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

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