Terahertz Frequency Metrology for Spectroscopic Applications: a Review

We provide an overview on terahertz (THz) frequency metrology, starting from the nowadays available continuous wave THz sources, discussing their main features such as tunability, spectral purity, and frequency referencing to the primary frequency standards. A comparison on the achieved results in high-precision molecular spectroscopy is given and discussed, and finally, a special emphasis poses on the future developments of this upcoming field.

tunability ranges, being based on the emission of a finite set of discrete molecular transitions. Extended tunability has been achieved with the tunable far-infrared spectrometer (TuFIR) hybrid approach discussed in Sect. 2. 3.
In an optically pumped FIR laser, the molecular gas is excited from its ground vibrational state to a higher vibrational state by an external laser source. Population inversion is then obtained between two rotational levels within the excited vibrational state. The pump source must be tuned so that its wavelength closely coincides with an absorption line in the molecular gas. Typically, the molecular absorption bands lie in the 10 μm region (0.1 eV). All infrared emitting lasers are potential candidates for FIR laser pumping, provided that the FIR laser medium absorbs the pump wavelength. Carbon dioxide (CO 2 ) lasers are powerful emitters of coherent radiation at a large number of discrete wavelengths around 10 μm. Since typical molecular absorption bands lie within the CO 2 laser spectrum, FIR laser pumps are almost exclusively CO 2 lasers. The energy difference between two rotational levels within one vibrational state is typically 0.01 eV, which is the energy of FIR photons. Stimulated Raman emission is also a contributor to the richness of FIR laser lines, especially at high pump power levels. However, even at low pump powers, laser emissions that cannot be assigned to a specific transition in the FIR laser medium are attributed to stimulated Raman. A vast number of polyatomic molecules have been found to lase when optically pumped by infrared lasers. Heavy molecules preferentially produce longer wavelengths. For example, formic acid, HCOOH, lases at longer wavelengths than the lighter methyl alcohol, CH 3 OH. The heavier CF 2 Br, pumped by a CO 2 , laser yields the long FIR laser wavelength of 2140 μm (140 GHz). Despite the abundance of observed FIR laser lines, only a limited number of them are powerful enough to be used in practical applications. Usually, a FIR laser molecule only produces a few powerful laser lines. Due the complexity of its FIR molecular spectrum, and the consequent richness of FIR laser lines, methyl alcohol has become a favorite FIR laser medium. Normal (non-isotopic) CH 3 OH alone is responsible for more than 300 FIR laser lines, ranging from 30 to 1200 μm. In methyl alcohol, the overlap between the strongly absorbing CO-stretch band and the CO 2 laser spectrum is excellent; the CH 3 OH molecule shows fairly large permanent electric dipole moments both parallel and orthogonal to the axis of symmetry. Exchanging H by D in the CH 3 OH, thereby obtaining the deuterated species CH 3 OD, CH 2 DOH, CHD 2 OH, CD 2 OH, and CD 3 OD, does not shift the CO-stretch band appreciably; the absorption still falls within the CO 2 laser spectrum. On the other hand, different FIR laser lines do appear since the rotational and vibrational levels are affected. It is also possible to exchange the ordinary 12 C carbon atom by the 13 C isotope.
In general, using an FIR laser involves bulky instrumentation, sophisticated operation, and is intrinsically limited by the active media transition frequencies. Consequently, availability of a new generation of compact, reliable THz sources is the key for the development of the largely underdeveloped THz range. In this framework and THz-emitting Quantum Cascade Lasers (QCLs) are proving to be good candidates to fill this gap. For a dimension comparison between QCLs and molecular lasers, see Fig. 1. The first report on QCLs operating at THz frequencies is relatively recent [8] and, despite the cryogenic operation temperatures (<199 K) [9], THz QCLs have attracted considerable attention thanks to the high output power (>100 mW), spectral purity, stability, compactness, and reliability, and have now a realistic chance of making a deep impact on technological applications. Among the features that a THz QCL has to show to become a useful radiation source for many applications is a reliable, repeatable, and tunable single-frequency emission. Many quantum-design related approaches, technological solutions, and/or optical configurations have been indeed recently tested to tune the emission frequency of THz QC sources over quite a large bandwidth [10].
Design and fabrication of such long wavelength QC lasers have been challenging: the emission, at energies lower than the optical phonon one, makes e-e scattering a non-negligible energy relaxation channel; the huge free-carrier absorption becomes the dominant factor of the waveguide losses; anomalies in semiconductor dielectric constant further complicate laser action and must be taken into account for designing optical waveguides that overcome the high losses and the practical limitations induced by prohibitive layer thickness; the non-radiative lifetimes (~0.5 ps) of intersubband transitions become very short, and carriers tend to enhance the leakage channels. Also, it is important to stress that, at terahertz frequencies, the spontaneous emission lifetime is typically much longer (microseconds) compared to the nonradiative lifetimes (picoseconds), so that radiative relaxation does not play any role in transport below threshold.
At this time, three active region types have successfully been used in terahertz QCLs: chirped superlattice, bound-to-continuum, and resonant phonon. In addition, a combined scheme of bound-to-continuum and resonant phonon, namely interlaced photon-phonon cascade, has been also developed. The first report on a THz QCL [11] exploited a careful design of the active region, based on a chirped superlattice and an asymmetric low-loss waveguide, and emission at 4.4 THz was achieved.
The gain spectrum of THz QCLs can be engineered by quantum design to be broadband over a range even larger than 1 THz, or, alternatively, to be electrically tunable over a 240-GHz bandwidth, exploiting the Stark shift of the lasing transition.
For reliable and repeatable single-mode tuning, QCL emitters based on external cavities can be realized using either a movable mirror to control the cavity length or a rotating grating for wavelength-dependent feedback. First-order and second-order DFB THz QCLs also provide stable single-mode emission, via temperature/current tuning, though usually on a more limited range, of only a few gigahertz. However, if coupled to an external microcavity, wider tuning ranges (≈20 GHz) can be achieved by mechanically changing the size of the external Fig. 1 Comparison between the bulky instrumentation of a molecular FIR laser, whose cavity length is in the order of 2 m, and a THz Quantum Cascade Laser, whose dimensions are in the millimeter range microcavity. As an alternative, the lateral size of the lasing mode can be manipulated in narrow (~λ/3) ridge, double-metal THz DFBs via electromechanical components, to tune the laser emission widely and continuously on a bandwidth of over 330 GHz [12].
Another surprising characteristic of THz QCLs is their spectral purity. In fact, frequencyand phase-stabilized, high-power, and reliable solid-state terahertz sources can find application in a number of fields, spanning from far-IR astronomy [13] and high-precision molecular gas spectroscopy [14] to high-resolution coherent imaging and telecommunications [15,16]. In this context, knowledge of the intrinsic linewidth, ultimately related to the uncertainty principle of quantum mechanics, plays a key role, determining the maximum achievable spectral resolution and coherence length in a free-running laser. Recently, the spectral purity of a THz QCL has been investigated via the measurement of its frequency-noise power spectral density (FNPSD), providing an experimental evaluation and a theoretical assessment of its intrinsic LW. Intensity measurements were performed to retrieve information in the frequency domain by converting the laser frequency fluctuations into detectable intensity (amplitude) variations [17]. These measurements led to a measured FWHM δυ = 90 ± 30 Hz [18]. This result has been later independently confirmed [19].
Experiments on frequency and phase stabilization of THz QCLs have involved right from the beginning some secondary frequency standards. In 2005, phase-lock of a 3-THz QCL to the 3.1059368 THz line of a methanol gas laser was demonstrated [20] and, in 2009, frequency stabilization of a single-mode THz QCL against the 2.409293 THz line of a CH 2 DOH gas laser was obtained [21]. In both cases, a narrowing of the beat signal was observed (65 and 3 kHz, respectively). The first THz QCLs locked to microwave-driven harmonically generated THz sources were demonstrated at 1.5 THz [22], 2.7 THz [23] (phase locks), and 2.3 THz [24] (frequency lock). Despite the high complexity, the high electrical power consumption, and the very low efficiency with a few pW radiation powers, such kinds of sources provide a very narrow and absolute-frequency reference, suitable for mixing with the THz QCL in a sensitive detector such as a hot-electron bolometer. Frequency-locking of a THz QCL to a molecular reference was demonstrated in a first-derivative, direct absorption spectroscopy configuration, without any significant narrowing of the QCL emission [25][26][27]. Phase-locking to an optical FCS (OFCS) was achieved only recently [28] by locking a 2.5-THz QCL to the nth harmonic of the repetition rate of a mode-locked erbium-doped fiber laser. By implementing a technique first reported by Löffler et al. [29], the comb generated by the mode-locked laser was mixed in a non-linear crystal with a CW THz QCL, thus generating THz sidebands around the near-IR carrier. The beating between the original comb and its shifted replicas provides the signal for closing the phase-lock loop. A similar scheme, but using a photoconductive antenna instead of electrooptic detection, was presented 1 year later [30].
Despite the clear advantage of room temperature detection of the beat-note, the latter approaches are based on low-efficiency up-conversion processes, and they inherently require a CW THz power in the milliwatt range. This can represent a limitation in many cases, when enough output power is not available or when it is needed for other simultaneous applications (i.e., spectroscopy). One possible solution has been provided by moving to a THz detection, where the beat-note can be acquired by a square-law THz detector with much higher efficiency, therefore involving only a small fraction of the overall emitted QCL power. Of course, this detection involves a free-standing and air-propagating THz FCS.
Coherent THz pulses are largely employed in time-domain spectroscopy, and even though this review focuses on CW sources, a focus on THz comb is needed for their importance in narrowing and referencing THz QCLs. In fact, the intrinsic comb nature of pulsed THz sources used in time-domain spectroscopy has been recently demonstrated [31,32], and direct use of such sources as frequency Bruler^for a THz QCL has ever been reported in 2012, by phaselocking a single-frequency CW QCL emitting at 2.5 THz onto a single tooth of a THz FCS [33]. This allowed for not only the direct knowledge of the QCL absolute frequency, but also a narrowing of its emission linewidth down to around 130 Hz could be observed.
In [33], the THz comb generation is realized by optical rectification, in Cherenkov configuration [34], of a femtosecond mode-locked Ti/Sa laser in a single-mode waveguide. The waveguide is fabricated on a MgO-doped LiNbO 3 crystal plate [35]. The generated radiation is a train of coherent THz pulses, carrying a very large spectral content (from few GHz up to more than 6 THz). Since the pulses are identical, the comb-like spectrum of the infinite train has a perfectly zero offset, and a spacing corresponding to the repetition rate of the pump laser, which was stabilized against a Rb-GPS (Global Positioning System) disciplined 10 MHz quartz oscillator. Stability in the megahertz range was obtained for the repetition rate, and this ensures stability of each tooth of the THz comb at the 100 Hz level, depending on the order of the tooth.

Frequency Up-conversion of Microwaves in Solid-State Diodes
Coming from the electronics frequency range, planar Schottky diodes based on frequency multiplier chains pumped at W-band with 100-150 mW have reached 1.9 THz in 2004 [36][37][38] with sufficient power to be used as a LO for HEB mixers [39,40]. These chains produce from several to tens of microwatts at room temperature but, as reported in an earlier work [41], they greatly improve upon cooling at 120-150 K and in 2005, a 2 × 3 × 3 multiplier chain reached a record 100 μW at 1.665 THz [42]. Moreover, they are electronically and continuously tunable over 10-15% of their bandwidth, and they are intrinsically easy to be referenced to primary frequency standards, in order to precisely know their emission frequency.
In the 1990s, frequency multipliers based on whisker-contacted Schottky diodes played an important role in the development of heterodyne receivers for radio astronomy as they appeared to be the only available solution for the LOs of space-borne submillimeter-wave heterodyne instruments. However, in 1992, this already mature technology was still unable to pass the 1 THz milestone [43]. Progress toward the terahertz region was reported by Crowe and Rüdiger Zimmerman [44] just before Zimmerman reached 1.135 THz in 1998 with an allsolid-state source that produced 40 μW of output power [45].
At submillimeter wavelengths and until the year 2000, whisker-contacted diodes outperformed Schottky planar diodes introduced in the mid-1980s by Cronin and Law [46] at the University of Bath, UK, and shortly later by Bishop and Mattauch [47] at the University of Virginia (UVa), USA. However, at millimeter-wavelength Schottky planar discrete diodes started to give better performance due to the use of multiple anodes in balanced configurations. Erickson's balanced doublers, proposed and demonstrated in [48][49][50], have become a standard topology for frequency multiplication due to their good performance. Gallium arsenide (GaAs)-based heterostructure barrier varactors (HBVs) were introduced by Kollberg and Rydberg at the University of Chalmers [51], Sweden, as alternate diodes, and this technology took a significant turn in the late 1990s when Lippens and Mélique at IEMN, France, introduced indium phosphide (InP)-based multiple barrier devices [52]. The results obtained with a 250-GHz waveguide tripler (11% efficiency and 9.5 mW of output power) demonstrated that HBV technology was a serious challenger to the classic and simpler Schottky technology.
However, despite further efforts by IEMN, Chalmers, and UVa, HBV multipliers did not reach the level of performance of Schottky multipliers. Another technique to build devices that exhibit internal symmetries was explored by Krach [53]. It gave a conversion efficiency of 22% for a 230-GHz planar diode tripler.
In the mid-1990s, the release of powerful commercial three-dimensional (3D) field-solvers (Ansoft HFSS) and non-linear circuit simulators (HP-now-Agilent MDS-now-ADS) transformed the way frequency multipliers were designed and built. These codes greatly increased the accuracy and the speed of the calculations necessary to optimize frequency multipliers. Erickson and Tiovunen pioneered the design of a four-anode balanced doubler at 170 GHz, entirely with HFSS and MDS [54]. Within the years 1995-2000, it became clear that discrete planar diodes were limited in frequency due to their size and the difficulty to connect them to the circuit with sufficient precision. Their integration on a circuit featuring several matching elements and providing precise connections to the waveguide block was necessary. The precision of the machining of the waveguide blocks plays a fundamental role in the working of THz frequency multipliers. For instance, JPL 1.9 THz tripler chips are inserted in a channel which width and depth are respectively 38 and 12 μm. The required precision for the alignment of the chip in the channel or the alignment of the two halves of the block is 2 to 3 μm.
These sources have recently been demonstrated and characterized for THz frequencies up to 2.75 THz [55][56][57][58], with output powers in the 1-14 μW range, as can be seen from Fig. 2. This source is based on a 91.8-to 101.8-GHz synthesizer followed by a power amplifier and three cascaded frequency triplers. It demonstrates, for the first time, that purely electronic solid-state sources can generate a useful amount of power in a region of the electromagnetic spectrum where lasers (solid state or gas) were previously the only available coherent sources.

Frequency Down-conversion of Visible/Infrared Light in Non-linear Media and Photoconductive Antennas
Historically, frequency conversion by non-linear generation has been a key tool for synthesizing radiation, often continuously tunable over wide intervals, in regions where lasers did not directly emit. Non-linear difference frequency generation has been conveniently used for many years to generate cw radiation throughout the infrared region, till the millimetric range, where microwave sources had been available as coherent sources before the advent of the laser. In the 1980s, two metrological-grade techniques, based on difference frequency generation by non-linear conversion, were developed. The first of these is generation of microwave sidebands on the strongest FIR laser lines in Schottky diodes [59][60][61][62]. It produces tunable FIR radiation up to about 3 THz (100/cm −1 ) with 5 kHz linewidths and roughly 500 kHz accuracy. Thanks to the pioneering work of K.M. Evenson and co-workers, a significant improvement in accuracy and tunability range were obtained with the tunable far-infrared spectrometer (TuFIR) [63]. It is based on a synthesis of FIR radiation starting from infrared and microwave radiation mixed onto a non-linear device, namely the metal-insulator-metal (MIM) diode. The MIM diode consists of an electrochemically sharpened tungsten whisker (25-pm diameter and 3 to 7 mm long) contacting a metal base. The metal base has a naturally occurring thin-oxide insulating layer. Both nickel and cobalt have been used as base materials, but cobalt is generally more consistent in the production of third-order FIR radiation. In a later version [64], the TuFIR spectrometer produces radiation through non-linear mixing of three radiations (two from CO 2 lasers and one from a microwave source; see Fig. 3) in a third-order MIM diode [65,66]. The MIM diode generates microwave sidebands on the CO 2 difference frequency. The standard frequency coverage of such a source is from 0.3 to 6 THz; the lower limit is set by the bolometer and the upper limit by the largest difference frequency between the two CO 2 lasers. The MIM diode was for many years a key device for frequency metrology, and it is interesting to compare its properties with present-day photoconductive (PC) switches for photomixing observation.
CW THz generation by photomixing is based on the THz-periodic generation of electrons and holes in semiconductors by absorption of two interfering laser beams of frequencies υ 1 and υ 2 , for which υ THz = υ 2 − υ 1 . The interference results in intensity modulation of the laser beams with the frequency υ THz and, hence, the THz-periodic carrier generation. The emission of THz radiation originates either directly from the (phase-coherent) acceleration of the Fig. 3 Schematic view of a typical TuFIR spectrometer. Reprinted with permission from [66], copyright 1998 individual photogenerated carriers or from the collective current of these carriers, generated within a very small area and being fed into an antenna.
Low temperature (LT) GaAs is the prevailing PC antenna material for this technique because of its high mobility and short lifetime. A photomixer is a compact, solid-state device. The tuning range can be exceptionally broad provided that a high-quality, tunable, dual-frequency laser system is available. The primary disadvantage of this method is that the output power is relatively low compared with other techniques of CW THz generation. Its optical-to-THz conversion efficiency is 10 −6 -10 −5 , and the typical output power is in the microwatt range. Because photomixing requires continuous optical excitation, the maximum THz output power is limited by the low thermal conductivity of LT-GaAs (~15 W/mK). The damage threshold of a 1-μm LT-GaAs layer with biased electrodes is less than 10 5 W/cm 2 of optical excitation [67].
A typical photomixer includes an antenna structure of metal on a LT-GaAs layer grown on a SI-GaAs substrate. A silicon hyper-hemispherical lens is attached to the back side of the substrate. A commonly used antenna structure for photomixing is the logarithmic uniplanar spiral antenna (log-spiral antenna) with interdigitated electrode fingers [68]. It has the advantage of a broad tuning range: its radiation pattern, impedance, and polarization remain virtually unchanged below 1 THz. The carrier lifetime of LT-GaAs and the RC time constant of a typical log-spiral photomixer are a few hundred femtoseconds; thus, frequency roll-off begins around 1 THz. The generated radiation power is proportional to ω −4 at higher frequencies. At the expense of a broad tuning range, the log-spiral antenna has low output power due to the relatively low radiation resistance. The radiation power can be enhanced by using resonant antenna structures [69]. The radiation resistance of a dipole antenna on a GaAs substrate peaks around L/λ R = 0.3 [70], which corresponds to the resonant frequency ν R ≈ 1.8 THz for the 50-μm dipole. The measured output power is maximized near 1.2 THz and extends throughout the broad spectral range from 0.5 to 2 THz.
Higher output powers are achievable using more sophisticated antenna designs, such as dual dipole antennas. The advantages over simpler antenna designs include a more symmetric radiation pattern and a higher radiation resistance [71]. In this design, the electrode capacitance is canceled out by the inductive tuning when the length of the transmission line is adjusted to the resonant frequency. Thus, the radiation resistance is determined mainly by the carrier lifetime. The output spectra of the dual-dipole antennas demonstrate significantly higher powers near their resonant frequencies when compared to the output from the log-spiral antenna [71]. The maximum output powers of the dual-dipole antennas are 3, 2, 0.8, and 0.3 μW at 0.9, 1.0, 1.6, and 2.7 THz, respectively. The peak power declines with increasing frequency as ω −2 , while the log-spiral output is proportional to ω −4 .
In order to avoid the limitation due to the low threshold for thermal damage of LT-GaAs, as well as to increase photocurrent, the optical excitation area can be made larger and can be illuminated with an extended beam from a high-power laser. If the dimensions of the illumination area are comparable to or larger than the wavelength of radiation, the optical excitation should be phase-matched with the THz radiation [72]. Another approach to enhance photocurrent is to replace LT-GaAs substrates with semiconductor heterostructures such as p-in photodiodes. A unitraveling-carrier photodiode (UTC-PD) contains a collection layer of InP which takes advantage of the exceptionally high-electron mobility in the material and of the optical excitation at the optical communication wavelength, 1.55 μm [73].
In order to overcome the limitations set by the finite carrier lifetime of semiconductor materials that induce a pronounced roll-off at higher frequencies, limiting the available THz power beyond 1 THz to a few microwatts, the approach of difference frequency generation (DFG) in non-linear crystals can be followed. DFG is a second-order non-linear optical process; thus, it requires a non-centrosymmetric crystal. Like photomixing, the optical source for DFG are two narrowband laser beams with slightly different frequencies υ THz = υ 2 − υ 1 . In these DGF operations, phase matching is crucial for efficient THz generation, and therefore, various phase matching methodologies have been developed, such as birefringence and quasiphase matching, using periodically poled crystals or Cherenkov emission schemes.
Regarding the materials used, various crystals have been explored for THz DFG, starting with simple bulk configurations in inorganic materials, such as lithium niobate (LiNbO 3 ) [74,75], lithium tantalate (LiTaO 3 ) [75,76], zinc telluride (ZnTe) [77,78], indium phosphide (InP) [79], gallium arsenide (GaAs) [80], gallium selenide (GaSe) [81,82], cadmium telluride (CdTe) [83], cadmium zinc telluride (CdZnTe) [84], and gallium phosphide (GaP) [85]. The main limitation of these sources relies on the low power achieved, due to the low efficiency of the second-order phenomenon implied in the generation. In order to improve the overall generation efficiency, different experimental setups have been tested. One favorable option is to enclose the generation crystal into an optical cavity for the visible/near IR radiation. These setups can achieve output powers of a few milliwatts at 1.9 THz with intracavity power as high as 500 W [86]. The generated power scales with the product of the intensities of the two pump beams; as a consequence, the emitted THz power can be also enhanced by confining the THz radiation in a small space. To this purpose, ridged [87] and implanted (Garcia Lopez et al., submitted for publication) waveguides have been realized, achieving efficiencies of 10 −9 in the THz generation.
Another important kind of source that relies on χ 2 processes is the optical parametric oscillator (OPO), which has recently been successfully used for THz generation. For example, CW THz output tunable between 1.3 and 1.7 THz was produced in a cascaded parametric process, where the resonantly enhanced near infrared Bsignal^wave of a primary 1030-nm pumped periodically poled lithium niobate (PPLN) OPO served as pump for the secondary OPO (using the same PPLN crystal and the same optical cavity) with the backward THz wave as the Bidler^ [95]. In another setup, based on a pump enhancement cavity with a finesse of 500 at the 1030 nm pump wavelength, the pump radiation was directly converted to THz, via an OPO process using PPLN, with tunability from 1.2 to 2.9 THz [96]. In both cases, the output power was limited to few microwatts due to strong THz absorption in lithium niobate.
Intracavity THz generation using a doubly resonant optical parametric oscillator (DR-OPO) for CW operation was first proposed and implemented in [97], where a quasi-phase-matched (QPM) OP-GaAs crystal was placed inside the cavity of an OPO, designed for type II parametric interaction (signal and idler orthogonally polarized) to allow narrow linewidths when operating near degeneracy. In the DR-OPO, both the signal and the idler waves resonate and the output THz power scales as cavity finesse squared. The power levels achieved were in the order of >10 μW, tunable over the 1-4.5 THz range.
Regarding the spectral purity of DFG sources, it is quite straightforward that the linewidth of the emitted THz radiation depends on the linewidth and on the phase relation between the pumping sources. For this reason, the advent of optical FCS in the visible/near IR region allowed to transfer the spectral properties of the finest frequency standards directly to the generated DFG radiation.

High-Resolution Spectroscopy
In the THz spectral region, most rotational transitions of light molecules fall, which are of high interest for applicative research (mainly astrophysics and atmospheric physics) as well as for fundamental studies. Moreover, since absorption coefficients normally scale as either the square or the cube of frequency for hν ≪ kT, THz transitions are much more sensitive tools (compared to millimetric transitions) to detect neutral/ionized atoms/molecules. In spite of the very large scientific interest, high-resolution spectroscopic studies in the submillimetric range started much later than in the neighboring microwave and infrared spectral regions, due to the lack of both adequate coherent sources and detectors.
The introduction of optically pumped, fixed frequency Far-InfraRed (FIR) lasers and of liquid-He cooled bolometric detectors (with typical Noise Equivalent Powers-NEP in the range 10 −12 -10 −14 W/Hz 1/2 ) was the first step toward the development of high-resolution spectroscopy in this region. Laser magnetic resonance (LMR) spectrometry achieves a wider tunability; however, such technique only works on paramagnetic species, tuning the transition of interest into resonance with an FIR laser line by a magnetic field. In this range, LMR proved to be one of the most sensitive spectroscopic techniques, because it is intracavity in nature, with a minimum detectable absorption in the order of 10 −10 . Nevertheless, the accuracy of frequency measurements is limited to only several MHz, due to the uncertainty in the extrapolation at zero magnetic field and the lack of reproducibility in the center of the FIR laser line.
The first measurements in a broad range of the THz region were performed using a Tunable Far Infrared (TuFIR) spectrometer. In this setup, both CO 2 lasers are frequency stabilized onto a 4.3-μm wavelength saturated fluorescence signal from low-pressure CO 2 cells [98]. These frequencies have been measured with an accuracy better than 5 kHz [99,100]. A stability of about 25 kHz is obtained for each locked CO 2 laser, and the overall (statistics + systematic) frequency uncertainty of the generated FIR radiation was estimated to be 35 kHz at the best. The spectrometer sensitivity is limited by the FIR power and the sensitivity of the detector. FIR powers up to a few hundreds of nanowatts are generated with 150 mW from each laser and 6-10 dBm of microwave power applied to the MIM diode. For the best contacts, minimum detectable absorptions are around 10 −4 in a 1-s integration time, corresponding to a minimum detection coefficient of 10 −6 for a 1-m long cell. With such a spectrometer, mainly operated at NIST, Boulder, CO, USA, at LENS, Florence, Italy, and later on in Japan [101], a large number of rotational and ro-vibrational transitions of light molecules and free radicals were measured with high accuracy [64,[102][103][104][105][106][107][108][109][110][111][112][113]. In addition, atomic fine structure transitions could be measured [114][115][116], as well as transition dipole moments of molecules of atmospheric interest, with an accuracy up to 10 −4 [117,118].
In spite of the scarce reproducibility of MIM diodes fabrication, pioneering studies on the breakdown of the Born-Oppenheimer approximation could be systematically performed as well as a careful analysis of the TuFIR sensitivity [119]. Sophisticated setups could be used notwithstanding the low emitted powers, such as Faraday spectroscopy, based on the measurement of tiny birefringence induced in a sample gas by a longitudinal magnetic field (from which the name: Faraday effect) [120,121]. Later studies on the limiting bandwidth of the MIM diode proved that frequencies beyond 9 THz could be generated by using laser lines belonging to rare isotopes of CO 2 [122].
Down-conversion generation setups have in principle great potential for broadband frequency metrology applications but, to our knowledge, this field has not been fully developed, yet. The groups of Deninger and Sasaki have proposed an InGaAs photomixers and a GaP-based setup, respectively, achieving a resolution of 1 or 15 MHz [123,124]. Only very recently, in Florence, a waveguided lithium niobate-based setup has been mounted to get a broadly tunable THz source, covering the 1 to 8 THz frequency range, which exploits frequency mixing from two tunable CW fiber-amplified telecom lasers in the waveguided crystal. The DFG emission process, confined in a tight 3 × 5 μm section, allows to reach generation efficiencies in the order of 10 −9 W −1 and power levels in the microwatt range. The source, which provides a 25-GHz modehop-free operation along the whole 7 THz generation span, has also been tested for precision spectroscopy of low-pressure methanol vapors, already proving a relative precision in the order of 4 × 10 −9 on the line center determination (De Regis et al., submitted for publication).
THz QCLs are, in principle, suitable candidates for high-precision spectroscopy, and they have already confirmed some expectations. Moreover, setups based on amplitude and frequency modulation have been developed [125]. In particular, in ref. [125], the absolute frequency of the QCL laser was measured with respect to a FIR laser emission, giving an overall accuracy of 8 × 10 −7 in the line center determination.
As mentioned in the previous section, a great step forward has been achieved with the advent of free standing THz FCSs. The THz comb has been successfully used as local oscillator to narrow and measure the frequency of a THz QCL down to the 10 −11 level [33], only requiring a small fraction of the radiation emitted by the laser. With this setup, highresolution spectroscopy has been performed on single methanol transitions, achieving an accuracy of 4 × 10 −9 [126]. Figure 4 shows the spectrum acquired for the considered methanol transition, and the measurements performed at different vapor pressures.
At present, the use of a cryogenic detector, such as the hot-electron-bolometer, is probably the main experimental complication of this approach. The combination of a FCS able to cover a broad spectrum (1-6 THz), with QCL sources that can cover, point-by-point, most of the same range but with unprecedented power-levels, is a very promising perspective for a metrological-grade investigation of the THz region. Among the several possible applications, it is worth citing the development of comb-assisted THz sub-Doppler spectrometers and of absolutely referenced local oscillators for heterodyne THz spectrometers.
Regarding the latter topic, in particular, not only the narrow linewidth of the local oscillator but also, and mostly, the stability over long time periods of its absolute frequency can represent a real breakthrough. A number of demanding new applications will be potentially addressed, Fig. 4 a Acquisition of a methanol transition around 2.55 , Voigt fit, and residuals. b The same acquisition performed at different vapor pressures. The linear fit gives the transition center with an accuracy of 10 kHz. Reprinted with permission from [126], copyright 2014 e.g., spectroscopic interrogation of cold molecules (see, e.g., [127]), as well as precise measurement of the long-term variation of fundamental physical constants from astronomical observation [128,129].
Moreover, the development of new high-bandwidth, low-NEP, terahertz detectors, such as nanowire [130,131] and graphene [132] field-effect transistors, can make possible, in the future, room temperature detection of beat-note signals.
As a matter of fact, the main limitation to the accuracy achieved in [126] is the Dopplerlimited spectroscopy setup used in the experiment. A way to improve the accuracy of the retrieved frequencies has to exploit sub-Doppler resolution, which can be obtained by means of the Lamb-dip technique [133,134] in case of rotational spectroscopy. Despite the fast growth undergone by spectroscopy and non-linear optics in the past decade, the achievement of sub-Doppler resolution is only at its beginning.
This limitation has been recently overcome by multiplied frequency chain setups, which have achieved an unprecedented accuracy of 1 × 10 −9 in the line center determination at around 1 THz [135]. Figure 5 shows the comparison between the Doppler limited and the sub-Doppler acquisition data. A very similar result has been also obtained at much higher frequencies, with an accuracy of 3 kHz on a 2.64-THz line center determination [56]. The bandwidth, agility, and operability of this THz source have enabled wideband, high-resolution spectroscopic measurements of water, methanol, and carbon monoxide with very high signalto-noise ratio.

Future Perspectives
Frequency metrology measurements performed on atoms in the visible region achieve a stability of few parts in 10 −18 [136], while measurements performed in the mid-IR on cold molecules are still below 10 −15 [137]. An even wider gap has to be filled for the far IR spectral region. This gap is essentially technological in nature, and continuous efforts are being made by the scientific community to lower the present limit of 10 −9 [56,126,135]. Moreover, most of the molecular models, trying to predict molecular transition energies in the THz range, are in the range of 10 −9 relative uncertainty. These models can take into account effects such as Reprinted with permission from [135], copyright 2013, American Chemical Society Zeeman shift, AC-Stark shift or blackbody shift, whose contributions are expected to be in the 10 −10 level, and have never been measured before. Systematic observations on a set of relevant molecular transitions would definitely improve the actual knowledge on molecular models and databases. For this reason, new strategies and tools are being developed, some of which will be discussed hereby.

Resonant Cavities
Optical resonators are well-established tools commonly used in spectroscopy [138,139]. They have widespread applications over the whole electromagnetic spectrum, from microwaves [140] to UV [141], while record-level optical finesse were achieved in the visible/near-IR [142], especially thanks to the advent of whispering gallery mode resonators [143]. In this context, the THz portion of the electromagnetic spectrum is still lacking such tools. Design and fabrication of cavities efficiently resonating at THz frequencies is challenging, due to the technological gap of THz materials and optical components, compared to other spectral regions. As concerns cavity mirrors, highly reflective dielectric coatings are scarcely developed at THz frequencies, and at present only, metallic coatings can be used, with a maximum reflectivity limited to 99.6% for gold. In addition, metallic mirrors are not suitable as input/ output couplers, because of unavoidable substrate absorption losses, and also the solutions commonly adopted at microwave frequencies (hole output coupling) have proven to be quite unsuitable for THz [144].
An alternative approach consists in using a wire-grid polarizer (WGP) as input/output coupler: if the electric field component is parallel to the metal wires, electrons are free to flow along the wires, and the incoming field experiences an almost complete reflection. However, a small amount of radiation leaks through the grid, allowing for the coupling of light in and out of the cavity. This approach has been adopted in a very few early experiments in far-IR spectroscopy [145][146][147], where finesses of about 14 and 3 were achieved at 690 GHz and 1.5 THz frequencies, respectively (corresponding to Q-factors of about 7000 and 1500, respectively). Braakman et al. demonstrated that, in the sub-THz range (around 300 GHz), resonant cavities based on WGPs can achieve Q-factors as large as 10 5 [148], sufficient for a relevant enhancement. More recently, in the group led by Paolo De Natale, resonant cavities with different setups, specifically a V-shaped and a ring-shaped cavity, have been developed, and have been coupled to a THz QCL.Q-factors of about 2.5 × 10 5 have been reached, see Fig. 6 and [149]. At the same time, these cavities have been for the first time used to influence the QCL emission, paving the way to cavity linewidth narrowing. Finally, another cavity setup has been developed by the same group, which has reached for the first time an unprecedented Q-factor of 2.5 × 10 6 , based on a bow tie design (Campa et al., submitted for publication).

Cold Molecules
Another intriguing playground, which could revolutionize THz frequency metrology, is the application of the above mentioned coherent sources to cold molecules, in analogy to what is currently happening in the mid-IR regime.
The shot-noise limit in a frequency measurement is given by δν = 1/τ N, where τ is the coherence time for the measurement and N is the number of detected molecules. The advantage offered by using a cold sample of molecule is twofold. First, by cooling the internal degrees of freedom of a molecular sample, the distribution of populated quantum states is narrowed according to the Boltzmann statistics. Thus, N is greatly enhanced if the state under investigation is the ground state, or a state that is energetically close to the ground state, compared with k B T. It is worth noticing that a present limit in molecular spectroscopy is posed by the lack of generally applicable methods to prepare a large population of molecules in highly excited states. Second, cooling the external degrees of freedom yields slower molecules that allow for measurements with longer coherence time τ. Moreover, lower velocities yield reduced Doppler broadenings [150].
The coldest molecular species attainable to date are the ultracold alkali dimers created by associating ultracold atoms [151]. They have been employed, for instance, in the studies of ultracold bimolecular reactions [152,153] and have been trapped in optical lattices to analyze their quantum dynamics, which represents the first step toward using these systems to explore many-body dynamics in regimes that are inaccessible to current theoretical techniques [154]. However, as interesting as these species are in the quest for new physical phenomena, they have not been at the focus of the spectroscopists' attention. Instead, their attention has been directed at less exotic molecular species, cooled by direct methods.
A conceptually simple method for preparing cold molecules is buffer-gas cooling, pioneered by J. Doyle [155]. Cooling is achieved via collisions with cryogenically cooled helium atoms, and the temperature of the cold molecules is typically around 1 K. A large variety of atoms and molecules has been cooled using this technique, ranging from atoms and dimers to benzonitrile, fluorobenzene, anisole, for instance [156]. Further, buffer-gas cooling Fig. 6 Two different THz resonators setup and acquisitions: V-shaped cavity on the left with 2.4 × 10 5 achieved quality factor and ring-shaped setup on the right, with 2.6 × 10 5 achieved quality factor. Reprinted with permission from [149], copyright 2015 has been used for the production of molecular beams both of gaseous precursors and of laserablated species [157]. If curved guides are coupled to this kind of sources, the subset of molecules that are moving sufficiently slowly are extracted from the output of the source [158]. Spectroscopy on buffer-gas cooled species has been performed both inside the cooling cell [156,159,160] and on the molecular beam [161]. Only very recently, cavity ringdown detection has been successfully used in a beam of buffer-gas cooled of acetylene [162].
Supersonic molecular beams are a classical method to produce cold molecular samples [163]. A molecular beam is generated by letting a gas expand from a high-pressure source into a low-pressure ambient background via a nozzle. Basically, a supersonic molecular beam is a fast-moving beam of internally cold molecules and it facilitates spectroscopic studies by providing molecules in an almost collision-free environment, thus with a strong reduction of inhomogeneous broadening. However, typical molecular beams have speeds of the order of 300-1000 m/s and several techniques have been developed to manipulate and control their motional degrees of freedom.
Stark and Zeeman effects are common methods to manipulate polar and paramagnetic molecules with electric or magnetic fields, respectively [164][165][166]. In Stark decelerators, molecules with a permanent electric dipole moment (EDM) convert part of their kinetic energy into Stark energy upon entering an electric field, if they are in an appropriate quantum state. If the electric field is switched off before the molecule has left the electric field, the lost kinetic energy will not be returned. This process can be repeated over multiple stages until the molecules reach the desired final velocity. Once the average velocity is low enough, molecules can be loaded in a trap [167], for instance. Zeeman deceleration is entirely analogous, except that the force is exerted by a magnetic field on a magnetic dipole moment [168]. As a viable alternative to abruptly switching between different static field configurations, molecules can be captured in traveling potential wells [169,170], directly from the supersonic molecular beam, and then decelerated. These methods are used to prepare a molecular beam in a single quantum state and at a mean speed typically adjustable between 400 and 500 m/s to rest, with a translational temperature tunable from 1 K to 5 mK. High-resolution mid-IR spectroscopy of decelerated species has been performed for NH3 [171] and hydroxyl radicals (OH) [172,173].
In these experiments, an interaction time as long as 1 ms was obtained. In 2008, Stuhl et al. [174] identified a class of diatomic molecules that presented almost-cycling transitions, which can be used for laser cooling. A couple of years later, DeMille group demonstrated the action of a radiative force acting on strontium monofluoride (SrF) [175] and then transverse laser cooling [176] and deceleration [177] of a SrF beam. Finally, in 2014, they were able to trap SrF in a threedimensional magneto-optical trap [178]. In the meanwhile, laser cooling of YO [179] and CaF [180] has also been reported. Typical temperatures reached with all these techniques are in the range of a half to about 100 mK, whereas the densities are in the range of 10 7 molecules per cm 3 . Applying high-precision THz spectroscopy techniques to these cold molecule sources could in principle lead to an actual improvement for accuracy of at least two orders of magnitude.

Frequency Referencing Improvement
As previously discussed, the frequency standard in the THz region relies on Global Positioning System (GPS) dissemination of frequency standards, which is ubiquitous at present, providing the most widespread time and frequency reference for the majority of industrial and research applications worldwide. On the other hand, the ultimate limits of GPS presently hinder further advances in high-precision, scientific, and industrial applications relying on this dissemination scheme. This is not yet the case of THz technologies, as the actual frequency accuracy limit of 10 −9 is not yet comparable with the accuracy given by GPS clock, but nevertheless far more accurate standards will become crucial in the next future. A timely exploitation of the continuous advances in the realization of new frequency standards is inherently tied to the uncertainty with which the standard is delivered from metrological institutes to the end users. Presently, GPS allows for traceability to the second in the International System of Units (SI) with a typical fractional frequency uncertainty of 10 −13 on a day average [181]. This remarkable result can be obtained with a very simple equipment, consisting only of a receiver without any feedback from the user to the reference clock infrastructure. However, GPS dissemination severely limits the accuracy of current frequency standards and, even more, that of new optical atomic clocks [182], which have already shown fractional accuracy capabilities of 10 −18 in a few hours of measurement [183][184][185]. This is the main reason why the development and characterization of long-haul optical fiber links (OFLs), rather than microwave satellite links, recently featured a substantial thrust [186][187][188][189][190]. Indeed, in the last years, OFLs have shown undoubted effectiveness in comparing remote atomic clocks beyond the ultimate limit of GPS-based comparisons, attaining precision levels not limited by the frequency transfer method [191][192][193][194][195][196], and are also envisioned as key elements for the development of a new kind of high-accuracy relativistic geodesy [195,197]. Besides optical clock comparisons, short-range OFLs have recently been exploited in order to increase the spectral performance of a local laser and to allow for precision molecular spectroscopy [198]. It has been recently demonstrated that long-haul fiber-based optical frequency dissemination is a reliable tool for remote end users to perform high-precision procedures well beyond the ultimate capabilities of GPS dissemination [199]. The present dissemination of the fiber link in Italy is shown in Fig. 7.

Novel QCL Sources
Single-mode THz QCLs have already shown their huge potential for metrological grade THz spectroscopy [126] but, at the same time, QCL technology is making continuous leaps forward, toward new generation sources. One of the main objectives is to build THz QCLs able to operate at room temperature. In this regard, a very promising kind of source is based on intracavity difference frequency generation between two mid-IR QCLs [200]. These sources are currently the only semiconductor sources that are able to emit multi-mW power and cover the entire 1-5 THz range at room temperature [201,202]. Wide-range frequency tuning spanning several THzs at room temperature based on composite distributed feedback (DFB) QCL arrays [203,204], or external cavity technique [205], has been demonstrated. Monolithic electrical tuning is more convenient for practical applications. Monolithic tuning range of 3.44 to 4.02 THz was achieved with a dual-section DFG QCL waveguide design [206], and was further expanded to 2.6 to 4.2 THz with a three-section DFG QCL waveguide design [207]. Very recently, by utilizing a low-loss buried-ridge waveguide design and highly dissipative epi-down mounting scheme, room temperature CW operation at 3.6 THz was demonstrated with a continuous power of 3 μW [208]. However, the relatively high threshold current density and low wall-plug efficiency (WPE) of the demonstrated devices prevented the room temperature CW operation of the monolithic tunable devices, which has been only recently demonstrated [209 and therein refs]. A careful characterization of these devices, regarding both their tunability characteristics and their spectral purity has been very recently performed by (Consolino et al., submitted for publication), showing that not only these devices are able to work at room temperature, but that they are also suitable for metrological grade applications.
Another interesting and promising technology for THz high-precision spectroscopy is continuous-wave multimode broadband QCLs, whose operation has only recently been demonstrated [210,211]. A broadband gain medium is desirable for a wide range of applications and, in QCL comb formation, is driven by ultrafast non-linearities (four-wave mixing) in the active region itself [212]. At terahertz frequencies, comb operation is also favored by the longer lifetimes of the optical transition upper state at those frequencies [213][214][215]. The ultrabroad Fig. 7 Present day dissemination of the fiber link in Italy, in yellow the already working link, and in green the upcoming connections gain bandwidth is achieved by fully exploiting the quantum engineering of intersubband transitions, integrating in the same laser ridge (resonator) different designs of a quantum cascade structure [216], tailored for different frequencies. First demonstrated in mid-IR QCLs [217], this heterogeneous cascade concept has also been successfully implemented in terahertz QCLs [218][219][220]. These sources show a comb emission regime with hundreds GHz spectral coverage, but their comb-like features and controllability have not been characterized, yet.
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