Optically Pumped Terahertz Molecular Laser: Gain Factor and Validation up to 5.5 THz

Quantum cascade laser‐pumped terahertz (THz) gas lasers are at the edge of revolutionizing THz science where powerful yet tunable sources have long been lacking. Maybe one of the last remaining drawbacks to a wider use of these instruments lies in the lack of available databases of potentially lasing transitions for users. A new figure of merit, the molecular gain factor is proposed, that allows to discriminate transitions by their lasing potential. Using this factor, catalogs of THz laser lines of ammonia, both 14NH3 and 15NH3 , up to 10 THz are reported. Demonstration of the use of these two catalogs, and of the pertinence of the molecular gain factor, is made by experimentally observing 32 laser lines of 14NH3 and 5 lines of 15NH3 up to 5.5 THz. Prospects to generalize the use of this molecular gain factor to a wide range of molecules are discussed.

of development, [24][25][26][27] it appears to have the capacity to generate a virtually endless number of THz laser frequencies by exploiting a large range of molecular gases as the gain medium. This property makes this THz molecular laser a promising local oscillator for heterodyne experiments, provided that reliable laser linelists exist. [28] Available catalogs of pump and laser pairs of frequencies have traditionally been built empirically using time-consuming procedures [29] and the number of lines remains far from exhaustive. Nowadays, molecular databases, developed for atmospheric and astrophysical applications, are complete enough to enable a systematic prediction of accurate pump and laser frequencies.
Among the possible gain media, ammonia (NH 3 ), as a high gain molecule with many possible emission frequencies, has focused particular interest. [23,24,27] Only a handful of laser transitions have been generated in NH 3 , mainly in the 1970s and 1980s (see Tables S1 and S2, Supporting Information), [27,[30][31][32][33][34][35] due to a limited number of coincidences between the pump frequencies emitted by the CO 2 and N 2 O lasers and the MIR absorption spectrum of NH 3 . Nevertheless, NH 3 is ideally suited for the generation of optically pumped THz laser lines because of 1) its large permanent dipole moment of about 1.5D; 2) its strong vibrational transition dipole moments (e.g., 0.2D for the sQ (3,3) transition of the fundamental ν 2 band [26] ); and 3) its "umbrella" inversion, allowed by tunnelling effect, resulting in a splitting of rotational levels (a property that has enabled the observation of the first maser effect [36] ), yielding a number of THz transitions between the tunnelling states. Using 14 NH 3 as gain medium, the highest CW conversion efficiency in the THz has recently been obtained. [26] It is also worth mentioning that NH 3 possesses about 9 bars of vapor pressure at room temperature, making it relatively convenient to handle. Finally, NH 3 is a well-known prototype molecule from a molecular physics point of view: its spectroscopic characterisation has been very well documented, another advantage of using this molecule to build the new generation of THz molecular lasers.
As THz molecular lasers pumped by a QCL will potentially generate a huge quantity of new THz frequencies, a method that would allow comparison and classification of laser lines by their lasing potential is more than ever needed. The aims of the present paper are threefold: 1) define a purely molecular-based figure of merit (i.e., excluding experimental and technical parameters proper to each laser design), hereafter called the molecular gain factor G m , that allows identification of the MIR/THz transitions pairs exhibiting the strongest lasing potential; 2) provide a list of possible THz laser lines that can be obtained using a MIR pump laser in the 10 μm range and ammonia as gain medium (both 14 NH 3 and 15 NH 3 ); 3) verify experimentally the validity of both our figure of merit and catalogs by detecting various laser lines up to 5.5 THz. In the following, after defining the molecular gain factor, we report two laser catalogs, for 14 NH 3 and 15 NH 3 , that have been built using available spectroscopic information on the HITRAN database. [37] In the catalogs, each entry contains the MIR pump and THz laser frequencies-and associated quantum numbers (QN) involved in the transitions-and the associated molecular gain factor. Using these catalogs, we were able to measure 32 laser lines of 14 NH 3 and 5 of 15 NH 3 , up to 5.5 THz, using a new prototype of NH 3 laser (described in Section 4.4). This represents a significant extension of the observed laser lines of both molecules in CW regime, previously limited to 5 and 12 lines for 14 NH 3 and 15 NH 3 , respectively (see Tables S1 and  S2, Supporting Information). In the Results section, we provide an extract of the catalogs (the full catalogs from 100 GHz to 10 THz are given in the Supporting Information) together with our measurements. Perspectives of the present work concerning the laser catalogs and the THz molecular laser are discussed.

Molecular Gain Factor
To establish the potential for a pair of pump/laser transitions having sufficient gain to overcome the losses of a resonant cavity, we introduce the molecular gain factor, G m . For one molecule of gain medium and a given pair of pump/laser transitions, the small-signal gain coefficient is expressed as the product of two factors and K a factor depending on the relaxation rate and pump laser performances (expressed in J m À5 ). The three involved energy levels (0, 1, and 2) are ranked by increasing energy thus the pump transition occurs from the level 0 to the level 2 and the THz laser one from 2 to 1. S 0 02 is the line strength of the pump transition (in m 2 s À1 ) and ν 02 its frequency (in Hz), B 21 is the Einstein coefficient of stimulated emission for the THz transition (in m 3 J À1 s À2 ), M is the molar mass of the molecular gas (in kg mol À1 ), and T is the temperature (in K). The constants c, N A , and k B are expressed in SI units. G m is thus expressed in SI units of m 4 J À1 . For the sake of simplicity, the uG m notation will be used in the following when referring to G m units. Details on the derivation of this molecular gain factor and its units can be found in Section 4.2.
Apart from the temperature, G m relies only on molecular parameters and fundamental constants. If the technical parameters (e.g., pump power density, gain medium pressure, and molecular laser design) of the laser are kept constant, the higher the value of G m , the lower the molecular laser threshold and thus the higher the THz output power. Consequently, G m can be used to classify the multitude of possible laser transitions of a given molecule, here two isotopologues of ammonia, against their potential to produce an intense radiation. In addition, this factor can also be used to compare the lasing potential of different molecular species.

Catalogs of MIR-Pumped THz-Laser Transitions of Ammonia
The first step in establishing lists of possible laser lines is to identify all possible pairs of pump/laser transitions, i.e., transitions sharing a common level, as sketched in Figure 1. Two possible laser transitions are distinguishable: those sharing a common level in the excited state and those sharing a common level in the ground state (often called "refilling transition" [13,25] ). The latter type of lasing, which has been found to be significantly less favorable than the former one due to high required pump power, [25] will not be further discussed here.
To establish our catalogs, we have restrained the spectral range of the pump transitions to the rovibrational transitions of the symmetric bending ν 2 of NH 3 , between 560 and 1360 cm À1 for 14 NH 3 and between 533 and 1295 cm À1 for 15 NH 3 , while the frequency range of the paired THz transition has been limited to the 0.1-10 THz (3.3-333 cm À1 ) region. We have further restrained our investigations to the ΔK ¼ 0 transitions, presumably the strongest ones (see Section 4.1). Only the pairs of transitions exhibiting an exploitable molecular gain factor, i.e., within five orders of magnitude of the strongest one (here G m > 1 Â 10 À21 uG m ) are reported. Under these restrictions, a total of 326 potential laser lines accessible via 808 pump/laser combinations for 14 NH 3 (Table S3, Supporting Information) and 285 potential laser lines accessible via 733 pump/laser combinations for 15 NH 3 (Table S4, Supporting Information) are reported in this work. Subsets of these tables, displaying only the THz laser lines with the strongest potential, their paired MIR pump frequency and molecular gain factor, are reported in Table 1.
It is important to note that the accuracy of the frequencies in the proposed lists and of the molecular gain factor values is intrinsically linked to that of the spectroscopic databases used to build the catalog (here HITRAN). Additional information on this particular point can be found in Section 4.3. In the tables reported in this article, frequencies are those from the HITRAN database; [37][38][39][40][41] experimental values can differ and for some transitions, frequencies found in the JPL database [42] can be more accurate.
In Tables S3 and S4, Supporting Information, each THz laser line is reported together with the possible MIR pump frequencies, the corresponding QN, the molecular gain factors, and the polarization of the THz radiation with respect to the pump polarization. Because of spectroscopic considerations (see Section 4.1), several combinations of pump/laser lines are generally possible. Each of the THz laser lines can theoretically be generated by one of three MIR pumping frequencies, corresponding to the possible absorptions in the P, Q, and R branches of the ν 2 band ( Figure 1). Alternatively, it is possible that a pump Figure 1. Schematic energy-level diagram of 14 NH 3 in the ground and v 2 ¼ 1 states for K ≤ 2 and J up ≤ 3. Levels of "s" symmetry are plotted using dark lines, those of "a" symmetry using gray lines. Illustration of possible pump (black arrows)/laser (red and blue arrows) transitions. Rotation-inversion lines (ΔJ ¼ 1) are represented in red, pure inversion ones (ΔJ ¼ 0) in blue. Table 1. List of the ten 14 NH 3 laser lines in the ν 2 band exhibiting the strongest gain factor in the 0.1-10 THz range. The star symbols indicate the experimentally observed lines. 14  These lines are absent from our catalogs because one of the two lines is not reported in the HITRAN database. The reported frequencies are taken from ref. [51]. For these transitions, the G m factor of the corresponding pair of transitions in 14 NH 3 was scaled by the ratio of molar masses between the two species.
transition shares a common upper state with two THz transitions yielding to a competition between the two laser frequencies. For instance, upper states with J > K, K 6 ¼ 0, and "a" inversion symmetry can generate two laser frequencies by either pure inversion or rotation-inversion ( Figure 1, central panel). Laser emission can either oscillate between the two possible frequencies or, because of gain values and/or laser cavity losses, possibly favor one of them. These transitions are labeled with an asterisk in Tables S3 and S4, Supporting Information, and listed in  Tables S5 and S6, Supporting Information. One can also notice in Tables S3 and S4, Supporting Information, that, for several THz transitions, either the P, or Q, or R pumping branch is missing. This can be due to the restrictions applied to the building parameters of our database (G m cut-off, MIR or THz frequency ranges) or to spectroscopic selection rules. As an example, for J ¼ K energy levels in v 2 ¼ 1, pumping from the R branch is not allowed because of the J ⩾ K requirement, and only a pure inversion laser transition is allowed ( Figure 1). Another example is that of the K ¼ 0 levels which can not be reached using Q-branch transitions and for which no laser pure inversion transition is allowed because of the a ↔ s symmetry selection rule ( Figure 1).
A graphical representation of the two catalogs is proposed in Figure 2. Individual plots for 14 NH 3 and 15 NH 3 are displayed in Figure S1, Supporting Information. On these figures, the possible pairs of IR pump frequencies and THz laser frequencies are plotted with their corresponding molecular gain factors. Such graphical representation of laser frequencies highlights the broadband coverage accessible using NH 3 as gain medium over the entire THz region. Every colored circle corresponds to a potential laser line generated by MIR pumping in the ν 2 band which exhibits a molecular gain factor higher than the chosen cut-off (G m > 1 Â 10 À21 uG m ). The figure reveals a structured distribution of THz frequencies relative to the MIR frequencies resulting from the spectroscopic properties of the molecules. Regions of P-, Q-, and R-branches pump transitions on one hand, and of pure inversion and rotation-inversion laser transitions, on the other hand, are clearly distinguishable (see Figure S1 for a graphical representation of the corresponding areas). This figure also reveals a high density group of high G m laser lines (larger than 1 Â 10 À18 uG m ) accessible around 1 THz thanks to the unique properties of the pure inversion transitions within v 2 ¼ 1 of NH 3 . A zoom displayed on the right panels of Figure 2 and S1, Supporting Information, shows the J and K distribution in this region.
A second graphical representation, where the G m of both 14 NH 3 and 15 NH 3 is plotted as a function of the THz laser frequency, is presented in Figure 3 (corresponding individual plots are reported in Figure S2, Supporting Information). The change between 14 NH 3 and 15 NH 3 isotopologue results in THz frequency shifts of a few tens of GHz (typically 30-40 GHz) with similar G m values. One can notice the absence of some THz lines of 15 NH 3 , particularly in the 1-2 THz range, as a result of missing transitions from the HITRAN database.

Validation of the Figure of Merit and Catalogs
Using the newly established catalogs to guide searches for laser transitions of ammonia using our QCL-pumped laser set-up, 32 pairs of MIR pump/THz laser lines for 14 NH 3 and 5 laser lines for 15 NH 3 were generated in the range 0.74-5.5 THz. Of these, 30 laser frequencies for 14 NH 3 and all frequencies for 15 NH 3 are reported for the first time in a resonant pumping CW regime.   Note that the generation of the remaining lines were not attempted but they are expected to be intense laser lines as well.
These results demonstrate the reliability of the catalog to predict accurate frequencies and molecular gain factors for potential laser lines as well as the strength of the experimental set-up to generate a large number of THz laser lines. One can notice in Table 2 that, among the 32 pairs of transitions, the same THz line at 979.6498 GHz can be obtained from two different MIR pump transitions: either from a Q-branch pumping at 966.3799 cm À1 or from a R-branch one at 1065.5654 cm À1 . The relative THz power measured for these two pumping schemes follows what expected from the calculated G m values. When multiple pump transitions yield the same laser line, the choice of the MIR pump transition is guided by both the frequency range accessible by the pump laser and the G m value of the pump/laser pair. Alternatively, competitions between laser lines involving the same upper level frequently occur. In the case of 14 NH 3 , the pump frequency at 1027.0470 cm À1 yields two possible laser transitions in the THz range accessible using our setup: 1014.0847 GHz or 2822.2564 GHz. The molecular gain factor of the line at 2822.2564 GHz is 4.4 times larger than that of the line at 1014.0847 GHz. Experimentally, we measured a single emission line at 2.8 THz, as expected from the G m values. Nevertheless, the cavity losses can not be neglected for competing transitions. For example, a pump transition located at 966.4736 cm À1 leads to the detection of a laser line at 945.6048 GHz (with G m ¼ 1.3 Â 10 À19 uG m ), even if the molecular gain factor of the competing transition at 3950.7233 GHz is about three times larger, because the latter is less favorable in our laser setup (plausibly due to larger losses for the higher frequency).
Pure inversion transitions of 14 NH 3 and 15 NH 3 are all located around 1 THz and allow for a deep coverage of that spectral region ( Figure 3). Alternatively, rotation-inversion transitions allow the generation of laser lines over the entire THz region. The remaining transitions observed here belong to this group   Table 2) which corresponds to about 300 μW.
The G m figure of merit is further validated by direct measurements of unsaturated gain values γ 0 for six pure inversion transitions of 14 NH 3 ( Figure 4 and Table S7, Supporting Information, see Section 4.5 for details). One can notice the overall agreement between the gain measured and the calculated molecular gain factor G m . By definition and assuming a uniform power density of the pump P p , the gain is the product of G m by a factor K, a parameter which depends on the relaxation rate Γ (see Section 4.2 for a detailed demonstration). In practice, P p can only be approximated as spatially uniform for small line strengths of the pump transition (S 0 02 ); for large S 0 02 values, an exponential decay of P p along the gas cell is expected, although not taken into account in our G m calculation. Consequently, a linear dependence of γ 0 with G m can only be expected experimentally for small S 0 02 values, i.e., small G m values. Indeed, deviation from linearity is observed on Figure 4 beyond the first three points and the experimental points with high G m exhibit γ 0 values underestimated compared with the linear fit. From the linear regression presented on Figure 4 and considering a pressure of 2 Pa and a pump power density of P p ¼ 255 W m À2 that were used for the measurements, a relaxation rate Γ of about 2.5 Â 10 5 s À1 can be estimated, a value in agreement with the 2.6 Â 10 5 s À1 value calculated in Section 4.5.

Conclusions
Among the suitable chemical species easily available in all laboratories, ammonia is an excellent candidate for the development of the next generation of optically pumped molecular lasers. It is obvious that such molecular lasers used as local oscillators (LO) can be valuable for a wide range of applications, starting with mixing of these new laser sources with THz radiations of different origins. Our group has already demonstrated the successful heterodyne mixing of a prototype of QCL-pumped molecular laser with synchrotron radiation around 1 THz [28] and the present results allow us to envision new measurements at higher frequencies. Another potential application for radio-astronomy is worth mentioning. For instance, there is a near-coincidence between the aR(4,2) 14 NH 3 laser line located at 2010.371 GHz with the J ¼ 1 À 0 transition of HeH þ cation at 2010.184 GHz, which was recently observed in a planetary nebula. [43] The frequency difference is sufficiently low to allow for heterodyne detection of this line using Schottky diodes or HEB as mixers. Other coincidences of this type exist and will certainly boost the future development of such LO to support new generation of THz observational platforms.
The relatively basic pump/laser process exploited in this paper can easily be generalized to the numerous molecules possessing relatively intense IR absorption bands and allowing pure rotation transitions in the THz range. In this context, we can foresee that the number of generated THz frequencies will increase by orders of magnitude by using various molecules and their isotopic variants. As shown in the present work, the simple extrapolation of 14 NH 3 gain medium to the 15 NH 3 isotopologue directly allows to double the number of laser lines, while using strictly identical experimental conditions. Obviously, with some instrumental effort, pump frequencies can be adjusted to different spectral regions in order to access other fundamental bands of various molecules, as well as overtones and combinations bands providing even more possibilities to generate THz lasers lines. Such opportunities require to sort out and organize molecules and their energy levels allowing the generation of THz laser processes. Our study demonstrates, using the NH 3 molecule, that the generation of such database is of great help to guide the experimental production of new laser lines. To increase the number of entries in our laser database, we set-up a very versatile code interrogating HITRAN molecular spectroscopy database in order to extract all necessary information (energy levels, intensities, and Einstein coefficients of MIR and THz transitions) to generate new THz lines for any molecules already reported in HITRAN. The description of the method used to generate the code is given in Section 4. However, such laser database relies entirely on the accuracy of the spectroscopic molecular characterization. Even though a few lines are currently missing for 15 NH 3 , our work reveals that the HITRAN database, which is constantly improving, is well suited to predict the position and strength of ammonia laser lines.
Finally, deuterated isotopologues of ammonia are other good candidates for generating new THz laser lines. In particular, as for the H isopotologues, the rotational transitions will allow access to numerous frequencies higher than 1 THz. [44,45] . Establishing their THz line catalogs, however, requires an investment from the spectroscopy community to record and analyze their FIR spectra in order to update the spectroscopic databases.

General Considerations
The principle of MIR-pumped THz lasers is based on a population inversion achieved by pumping population, using a ro-vibrational transition, from a state populated at room temperature to one that is not. In the following, we will restrain our study to the  Table S7, Supporting Information). The black dashed line represents the fit of the three first points for which the power density of the pump can be considered uniform (see text).
www.advancedsciencenews.com www.adpr-journal.com ground and v 2 ¼ 1 states of 14 NH 3 and 15 NH 3 , although the general equations we report can be transferred to virtually any molecule and vibrational state if the hypotheses are fulfilled. In our case, the THz laser is pumped by a MIR laser in the 10 μm range, therefore accurate values for the energy levels involving two vibrational states, namely the ground state (GS) and v 2 ¼ 1 (around 950 cm À1 for both 14 NH 3 and 15 NH 3 ), are required. As a symmetric top species, the rotational structure of NH 3 is described using J (the total rotation angular momentum) and K (the projection of J on the C 3 symmetry axis) quantum numbers. In addition to the conventional energy-level diagram for symmetric tops, NH 3 is subject to the well-known tunnelling motion of the N nucleus through the plane formed by the 3 H atoms. This tunnelling effect splits each rotational level with K 6 ¼ 0 into two components, one symmetric (s) and one antisymmetric (a).
Rotational transitions within a nondegenerate vibrational state (the THz emission in our case) are categorized as pure inversion or rotation-inversion following the respective selection rules: By convention, the inversion symmetry located to the left of the arrow corresponds to the higher energy level. For a given J, the "a" level lies at higher energy than the "s" one (e.g., see Figure 1).
Selection rules for the ro-vibration transitions within the v 2 ¼ 1← GS (IR absorption) are: It is worth noting that "forbidden" ΔK ¼ 3 transitions exist and exhibit intensities about two orders of magnitude lower than the corresponding ΔK ¼ 0 transition. [46] These transitions are beyond the scope of this article, in part because in their current state, databases do not allow to reproduce them satisfactorily, [46] and because the expected G m values should be rather small. A schematic representation of the energy-level diagram of the lowest rotational states of NH 3 is displayed in Figure 1 for the ground and v 2 ¼ 1 states, the latter containing less than 1% of the ground state population at room temperature. Examples of rotation-inversion (ΔJ 6 ¼ 0, left panel), pure inversion (ΔJ ¼ 0, right panel), and an example of competition between THz laser lines from a single upper level (middle panel) are given on Figure 1. For simplicity, the quantum numbers associated with each transition are usually given in the absorption convention a=s low P/Q/RðJ low , K low Þ with P, Q, R corresponding to ΔJ ¼ J up À J low ¼ À1, 0, þ 1, respectively, and "up" and "low" to the upper and lower levels involved in the transition.
The large intensities of MIR and THz transitions of NH 3 , owing to the large ν 2 transition moment and permanent dipole moment, respectively, are necessary to obtain high gain values of the laser. In the MIR, intense rotation-vibration transitions ensure an efficient population transfer from the GS to v 2 ¼ 1. This is reinforced by the fact that NH 3 being a light molecule, the moments of inertia are relatively small leading to a relatively limited numbers of rotational levels populated at room temperature in the ground vibrational state, and hence a significant population within each of these levels. In addition, the pure inversion and rotation-inversion transitions occurring within the excited states are intense, allowing laser effects at low threshold.

Derivation of the Molecular Gain Factor of a Pair of Pump/Laser Transitions
To initiate a laser process, the small-signal gain, or unsaturated gain, must exceed the total losses. [47] This parameter is critical to realise a laser regardless of any consideration of the characteristics of the cavity. We derive here the small-signal gain of an optically pumped molecular gas far from saturation, at low pressure in the Doppler regime, and under thermal equilibrium prior to pumping.
We consider a three energy levels scheme with the levels labelled 0, 1, and 2 (E 0 < E 1 < E 2 ). The MIR pumping transition occurs between levels 0 and 2, with E 2 À E 0 ¼ hν 02 , and the THz laser one between levels 2 and 1, at energy E 2 À E 1 ¼ hν 21 , where the energies are expressed in J, h is the Planck constant in J s, c is the speed of light in m s À1 , and ν are the frequencies of the transitions expressed in Hz (or s À1 ). SI units will be used throughout the following demonstration. At room temperature and under thermal equilibrium, the populations in levels 1 and 2 can be considered null (N 1 ¼ N 2 ¼ 0) prior to pumping if the energy of these levels is high compared to k B T, which is the case for the rotational levels that can be accessed using a 10 μm (%5 k B T) QCL pump laser. The optical MIR pumping from level 0 to 2 thus yields an inversion of population between the levels 2 and 1, and the THz photons flux is amplified by stimulated emission.
The laser gain along a distance d of gain medium at a frequency ν is given by: where I 0 and I are the optical intensity, before and after crossing the gain medium (in W m À2 ), respectively, and α 21 ðνÞ is the gain coefficient (in m À1 ). In the case of a small-signal gain or unsaturated gain regime, the gain coefficient α 21 can be approximated as α 21 ðνÞ ¼ γ 0 ðνÞ. The small-signal gain coefficient is expressed as: [47] γ 0 ðνÞ ¼ σ 21 where σ 21 is the stimulated emission cross section of the THz transition (in m 2 ), B 21 is the Einstein coefficient for the stimulated emission of the THz transition (in m 3 J À1 s À2 ), and gðνÞ is the lineshape function (in s). The populations N 2 and N 1 are expressed in m À3 . At the center of the laser transition, under thermal equilibrium assumption (N 1 ¼ 0), and assuming a Doppler linewidth, Equation (7) becomes: where M is the molar mass of the molecular gas (in kg mol À1 ), N A is the Avogadro constant (in mol À1 ), k B is the Boltzmann constant (in kg m 2 s À2 K À1 ), and T is the temperature (in K).
For a small length of a cell containing molecules pumped by a uniform power density P p (in W m À2 ) under steady-state condition: where R p is the pumping rate (in s À1 m À3 ), Γ is the relaxation rate (in s À1 ), and α 02 is the absorption coefficient of the pump transition (in m À1 ). At low pressure, one can consider that relaxation is mainly due to collisions to the walls of the cell and that the effect of molecular collisions is negligible; Γ is then independent of the pair of transitions considered. α 02 can be expressed as [48] α 02 ¼ S 0 02 gðνÞN where S 0 02 is the line strength (in m 2 s À1 ) and N is the total population in m À3 . S 0 02 contains all necessary information on level degeneracies, Einstein coefficient, and population (and is thus a function of the temperature).
Assuming the pump frequency is precisely tuned at the center of the pump transition and a Doppler profile, by combining the previous equations, Equation (8) becomes: The small-signal gain coefficient appears as the product of two factors: with and G m in expressed in m 4 J À1 (uG m ), and K in J m À5 . While the K factor depends on the experimental conditions and pump laser performances, the G m factor depends almost exclusively (at the exception of the temperature) on spectroscopic parameters and on the choice of molecular gas. This molecular gain factor can thus be used as a figure of merit to classify the possible laser transitions of various gain media by their potential to lase.

Establishment of a Catalog of Ammonia THz Laser Transitions
Assuming the population inversion criterion is met for all pumped energy levels (and that the levels in the upper vibrational state are empty prior to pumping), one needs to possess several parameters in order to predict the list of possible THz laser frequencies using various IR pump transitions, and to calculate their molecular gain factors: 1) a list of vibrational transitions of the species (ν 02 and involved quantum numbers); 2) their corresponding spectral line strengths (S 0 02 ); 3) a list of pure rotational transitions within the lasing state (ν 21 and involved quantum numbers); and 4) the Einstein coefficient B 21 of these rotational transitions.
In the case of 14 NH 3 and 15 NH 3 , all these parameters can conveniently be retrieved from the HITRAN online database [37] which was used to build the two catalogs in the present work. Listing all possible pairs of pump/laser lines by identifying all pump and laser transitions sharing the same upper state, is rather trivial; calculating the G m values, however, require some care. Firstly, all entries in the HITRAN database are in CGS units. Secondly, the Einstein absorption coefficient A 21 is reported for each transition and the corresponding B 21 coefficient thus must be calculated. Thirdly, the S ij (S ij ¼ S 0 ij =c Â 10 2 ) values are reported at T ¼ 296 K thus, for a laser operating at a significantly different temperature, the spectral line intensities have to be recalculated (see Ref. [48]). Here, all G m calculations have been performed at T ¼ 296 K. Finally, the S ij values reported in HITRAN are weighted according to the natural terrestrial isotopic abundances I a while in the present work the molecular gain factor is established for a given isotopologue (thus at 100 % abundance).
Expressed as a function of parameters directly extracted from HITRAN, Equation (13) becomes: with G m in uG m , S 02 in cm molecule À1 (HITRAN unit), A 21 in s À1 ,ν in cm À1 , M in g mol À1 , and I a unitless (I a < 1). In our catalogs, we also report the polarization axis of the THz laser beam relative to the IR pump polarization. If jΔJj 02 ¼ jΔJj 21 , the THz polarization axis is parallel to the IR pump one; otherwise it is perpendicular. [49] The frequencies of the THz laser transitions given in our tables correspond to the values given in HITRAN database. The corresponding uncertainty depends on the particular line and isotopologue, and typically ranges from 0.3 to 300 MHz for 14 NH 3 and 15 NH 3 . Specific values can be retrieved from the HITRAN database.

Detection of Ammonia Laser Lines
A schematic representation of the optically pumped molecular laser set-up is given in Figure 5. This laser is an upgraded version of the one presented in Ref. [23]. One major evolution concerns the optical components (waveguide, input and output couplers) which are now placed under vacuum, in an 1.3 m-long cylindrical chamber. The cavity is composed of a 8 mm-internal diameter copper tubing terminated by flat input and output couplers, and the whole cylindrical chamber is filled with 10-30 μbar of a molecular gas acting as gain medium for the laser. This particular design, not as compact as the one presented in Ref. [26], was optimized to enable THz lasing in a wide bandwidth, from 0.7 to about 5 THz. An external cavity QCL (MIRcat, DRS Daylight Solutions) operating between 920 and 1330 cm À1 , with a power of about 100 mW, is used for optical pumping of the molecular gas yielding population inversion. Here, the QCL allows to resonantly excite the rotational levels of the NH 3 ν 2 state. The QCL emitted frequency is measured by a wavemeter (Bristol Instruments 771B-MIR) with an accuracy of 1 ppm (i.e., 0.001 cm À1 at 1000 cm À1 ). The QCL frequency is adjusted by modifying its temperature and/or its current. Some transitions are difficult to access due to mode hops in the external cavity QCL. The optics allowing the injection of the MIR radiation into the THz laser cavity consist of two gold-coated flat mirrors and a single ZnSe lens. The output window is made of TOPAS (cyclic olefin copolymer) and the THz laser radiation is collected and collimated using a Zeonex (cyclic olefin polymer) lens located outside the cylindrical chamber. Acting on the distance between input and output couplers allows us to tune the cavity in resonance with a THz transition, previously identified using our new catalogs. The emitted frequency is verified by scanning the cavity length of the molecular laser in order to measure the distance between successive Fabry-Perot modes, providing a rather good estimate of the laser wavelength.
The resulting THz laser power is measured using a pyroelectric detector (Sensor und Lasertechnik GmbH, THZ 10 HS, calibrated at PTB) by optically chopping the QCL radiation. Obviously the stability of the MIR radiation, both in terms of frequency and intensity, reflects directly on the power oscillations of our THz laser. While rapid oscillations were observed in our previous prototype, the new design presented in this work allows relatively long-term stability of the emitted power (weak fluctuations measured over 45 min). An example of power stability recording at 5.048 THz is shown in Figure S3, Supporting Information. Four typical spectra of THz laser emission for lines around 1 THz are shown in Figure S4, Supporting Information. The typical discrepancy between the measured peak frequency and the value given in the HITRAN database is less than 1 MHz. The typical linewidth is few 10s of kHz.