## Spatio-temporal theory of lasing action in optically-pumped rotationally excited molecular gases |

Optics Express, Vol. 19, Issue 8, pp. 7513-7529 (2011)

http://dx.doi.org/10.1364/OE.19.007513

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### Abstract

We investigate laser emission from optically-pumped rotationally excited molecular gases confined in a metallic cavity. To this end, we have developed a theoretical framework able to accurately describe, both in the spatial and temporal domains, the molecular collisional and diffusion processes characterizing the operation of this class of lasers. The effect on the main lasing features of the spatial variation of the electric field intensity and the ohmic losses associated to each cavity mode are also included in our analysis. Our simulations show that, for the exemplary case of methyl fluoride gas confined in a cylindrical copper cavity, the region of maximum population inversion is located near the cavity walls. Based on this fact, our calculations show that the lowest lasing threshold intensity corresponds to the cavity mode that, while maximizing the spatial overlap between the corresponding population inversion and electric-field intensity distributions, simultaneously minimizes the absorption losses occurring at the cavity walls. The dependence of the lasing threshold intensity on both the gas pressure and the cavity radius is also analyzed and compared with experiment. We find that as the cavity size is varied, the interplay between the overall gain of the system and the corresponding ohmic losses allows for the existence of an optimal cavity radius which minimizes the intensity threshold for a large range of gas pressures. The theoretical analysis presented in this work expands the current understanding of lasing action in optically-pumped far-infrared lasers and, thus, could contribute to the development of a new class of compact far-infrared and terahertz sources able to operate efficiently at room temperature.

© 2011 OSA

## 1. Introduction

1. T. Y. Chang and T. J. Bridges, “Laser actions at 452, 496, and 541 *μm* in optically pumped CH_{3}F,” Opt. Commun. **1**, 423–426 (1970). [CrossRef]

_{3}F) gas. Since then, a rich spectrum of laser lines through the 50

*μ*m – 2 mm wave-length region have been obtained from a variety of different molecular gases [2–10

10. R. L. Crownover, H. O. Everitt, D. D. Skatrud, and F. C. DeLucia, “Frequency stability and reproductibility of optically pumped far-infrared lasers,” Appl. Phys. Lett. **57**, 2882–2884 (1990). [CrossRef]

6. H. O. Everitt, D. D. Skatrud, and F. C. De Lucia, “Dynamics and tunability of a small optically pumped CW far-infrared laser,” Appl. Phys. Lett. **49**, 995–997 (1986). [CrossRef]

6. H. O. Everitt, D. D. Skatrud, and F. C. De Lucia, “Dynamics and tunability of a small optically pumped CW far-infrared laser,” Appl. Phys. Lett. **49**, 995–997 (1986). [CrossRef]

_{2}pump lasers may combine to produce truly compact submillimeter-wave laser sources in the near future.

6. H. O. Everitt, D. D. Skatrud, and F. C. De Lucia, “Dynamics and tunability of a small optically pumped CW far-infrared laser,” Appl. Phys. Lett. **49**, 995–997 (1986). [CrossRef]

*E*-field) intensity distribution inside the metallic cell and the ohmic losses associated with the penetration of the

*E*-field inside the metallic walls are also included in our theoretical approach. As we show in this work, both effects are particularly important for a realistic description of the operation of compact OPFIR lasers based on a metallic cavity which waveguides the emitted laser frequency.

_{3}F) gas confined in a cylindrical copper cavity, and find that the model compares well against experimental results. Our calculations show that, for this exemplary system, the region of maximum population inversion concentrates near the cavity walls. The outcome of these calculations also implies that, as a consequence of this population inversion distribution, the lowest threshold intensity corresponds to the cavity mode whose

*E*-field intensity profile presents an optimal spatial overlap with the population inversion distribution, while at the same time, the ohmic losses associated to that cavity mode are minimized. In the case of the considered cylindrical cavity both features are satisfied by the TE

_{01}mode. We also analyze how the threshold intensity depends on the gas pressure and the cavity radius. The results from these simulations are explained in terms of the corresponding variation of the overall gain of the system and the ohmic losses associated with the lowest threshold mode. Remarkably, we predict that as the cavity size is varied, the interplay between both factors yields an optimal value of the cavity radius for which the threshold intensity is minimized for a large range of gas pressures.

## 2. Theoretical framework

_{3}F [6

**49**, 995–997 (1986). [CrossRef]

3. W. H. Matteson and F. C. De Lucia, “Millimeter wave spectroscopic studies of collision-induced energy transfer processes in the ^{13}CH_{3}F laser,” IEEE J. Quantum Electron. **19**, 1284–1293 (1983). [CrossRef]

8. H. O. Everitt and F. C. De Lucia, “A time-resolved study of rotational energy transfer into A and E symmetry species of ^{13}CH_{3}F,” J. Chem. Phys. **90**, 3520–3527 (1989). [CrossRef]

*A*– and

*E*– type), but without loss of generality they will be combined here to describe CW OPFIR laser operation at low pressure [8

8. H. O. Everitt and F. C. De Lucia, “A time-resolved study of rotational energy transfer into A and E symmetry species of ^{13}CH_{3}F,” J. Chem. Phys. **90**, 3520–3527 (1989). [CrossRef]

*k*

_{wall}=

*vλ*

_{MFP}/

*R*

^{2}[16].

*λ*

_{MFP}is the pressure-dependent mean free path and

*v*is the average relative velocity between molecules. At very low pressure (or equivalently, within

*λ*

_{MFP}from the cavity walls), the transport mechanism changes from diffusion to ballistic and the rate becomes pressure-independent so that

*k*

_{direct}=

*v*

_{abs}/

*R*, with

*v*

_{abs}defined to be the average absolute velocity of a molecule. It is clear from these expressions that the wall rates are very different across the cavity, and we explicitly account for this spatial variation in our numerical simulations. On the other hand, the vibrational relaxation between the excited and higher excited pools can be dominated by molecule-molecule collisions, in which case, the transition rates between them are

*k*

_{exc}=

*N*

_{tot}

*vσ*

_{exc}. Here,

*N*

_{tot}is the total density of molecules and

*σ*

_{exc}is the collision cross section measured from experiments.

*J*

_{1},

*J*

_{2}. . . etc. in each vibrational state, only exist in states that are of the same symmetry type and vibrational state as those connected by the pump. The total population of any state

*J*(i.e.

_{i}*N*

_{tot},

*J*) is simply the sum of its nonthermal and thermal parts,

_{i}*N*

_{tot,Ji}=

*N*+

_{Ji}*f*

_{Ji}*N*

_{pool}, where

*f*is the fraction of the total pool population

_{Ji}*N*

_{pool}in state

*J*, and

_{i}*N*is the amount the state population differs from rotational thermal equilibrium because of the pump. Note that

_{Ji}*N*can be positive (excited vibrational state) or negative (ground vibrational state).

_{Ji}*σ*

_{DD}measured from experiments [9

9. H. O. Everitt and F. C. De Lucia, “Rotational energy transfer in CH_{3}F: The Δ*J* = *n*, Δ*K* = 0 processes,” J. Chem. Phys. **92**, 6480–6491 (1990). [CrossRef]

*J*= ±1 at the same

*K*=

*K*as the pumping transition. Lastly, the K-swap process [8

_{i}8. H. O. Everitt and F. C. De Lucia, “A time-resolved study of rotational energy transfer into A and E symmetry species of ^{13}CH_{3}F,” J. Chem. Phys. **90**, 3520–3527 (1989). [CrossRef]

*K*=3n process and the V-swap process [8

^{13}CH_{3}F,” J. Chem. Phys. **90**, 3520–3527 (1989). [CrossRef]

*A*– and

*E*– symmetry thermal pools are combined as they are here) convert the nonthermal molecules to thermal molecules but not vice versa, and thus, allowing the nonthermal states to return to zero when the pump is turned off.

*R*

_{pump}=

*αP*

_{pump}/

*πR*

^{2}

*N*

_{tot}

*f*

_{low}

*hν*

_{IR}, where

*P*

_{pump}is the pump power,

*f*

_{low}=

*f*

_{J1}is the fraction of molecules in the lower ro-vibrational state

*J*

_{1}(as depicted in Fig. 1),

*ν*

_{IR}is the pump infrared frequency, and

*α*, which takes into account both Doppler broadening and mismatch between pump and absorption lines, measures the gas’ ability to absorb the infrared radiation. Hence, pump transition rate may be increased by reducing

*R*, increasing

*P*

_{pump}, or decreasing

*ν*

_{IR}. The last of these, however, is constrained by the need to match the frequency of the laser line with that of a ro-vibrational transition in the molecules, and in general, cannot be varied at will. Using such an implementation, our model accommodates both CW and pulsed pumping with arbitrary spatial profiles.

*J*levels, terms representing the pumping mechanism, K-swapping terms that allow flow of molecules out of the non-thermal levels (with equilibrating effects on non-thermal levels), and the stimulated emission terms at the FIR wavelength. Since we are interested in the steady state distribution of molecules across the cavity, spatial diffusion terms are also added to the rate equations describing each levels. Stimulated and spontaneous transitions at the lasing wavelength can, however, be neglected in the rate equations when operated near the threshold if the spontaneous emission lifetime,

*τ*

_{spont}, is relatively longer than all other transition lifetimes in the considered system. In methyl fluoride, for instance,

*τ*

_{spont}is on the order of 10

*s*or more. In contrast, the next longest lifetime in the model occurs for the vibrational relaxation terms and yet, has lifetime on the order of 1 × 10

^{−3}

*s*or less. The relative magnitude of these radiative terms involving

*τ*

_{spont}in the rate equations are further diminished by the fact that only a fraction of these transitions contributes to the desired laser mode, especially when the cavity is large. With that, the rate equation describing the molecular density of

*J*

_{2}level in the excited vibrational state (i.e.

*total*molecular density of the level

*J*

_{1}in the ground vibrational state after accounting for the nonthermal density and the contribution from the corresponding fraction of the thermal pool, and

*J*level. Similar definitions hold for

^{th}_{J2, i}two adjacent rotational levels and also the thermal for

*i*= 1,2,3 is the decay lifetime out of

_{j,J2}for

*j*= 1,2 represents the decay into

*D*is the diffusion constant given by the product of the molecule velocity and mean free path. In the same way, the thermal pool equation has terms from the K-swap and spatial diffusion processes, as well as vibrational relaxations due to molecule-molecule collisions and diffusion to wall. The equation describing the excited vibrational pool in Fig. 1 is then given by

*τ*

_{exc,}

*(*

_{k}**r**) for

*k*= 1,2 is the decay lifetime from the excited thermal pool to the ground and higher excited pools, while

*τ*

_{n}_{,exc}(

**r**) is the reverse. As noted earlier, the values of these diffusive lifetimes depend on their distances from the cell wall.

*m*= 1,2,...,

*m*

_{0}in the third term is the lifetime of the transition from each of the

*m*

_{0}rotational levels to the excited pool due to the K-swap molecule-molecule collision process. As in Eq. (1), the last term represents spatial diffusion. Similar equations to Eq. (1) and Eq. (2) are used to represent every nonthermal level or thermal pool that contributes to the lasing action, which in turn depends on their degree of non-equilibrium caused by the pump (see Appendix). This way, the complete set of rate equations describing our laser system obeys the diffusion equation

*∂N*

_{tot}(

**r**,

*t*)/

*∂t*=

*D*∇

^{2}

*N*

_{tot}(

**r**,

*t*), and so, conserves the total molecular density.

*J*levels are set equal to zero before any pumping occurs. The set of equations is then evolved in time until steady state is reached, allowing the temporal development of the lasing action to be tracked.

_{i}_{2}laser) which excites molecules from a specific rotational level in the ground vibrational state into a specific rotational level in an excited vibrational state. Because the excited vibrational level is comparatively empty, these photo-excited molecules create a population inversion between the pumped rotational state and the one immediately below it. This leads directly to stimulated emission between rotational states in the excited vibrational state. These photo-excited non-equilibrium molecules are subsequently rotationally and vibrationally relaxed to the ground state through collisions with other molecules and with the chamber walls, respectively.

*N*–

_{Ji}*N*

_{Ji}_{–1}is greater than the difference in the corresponding pool-contributed thermal populations (

*f*–

_{Ji}*f*

_{Ji}_{–1})

*N*

_{pool}for those states. As the pressure grows, the pool molecules have an increasingly difficult time reaching the walls and de-exciting; thus, the thermal contribution from the pool increasingly quenches the non-thermal population inversion. Pool quenching creates the vibrational bottleneck that was once believed to limit OPFIR laser operation to low pressures regardless of pump intensity. However, it is now understood that as the excited vibrational state thermal pool fills, near resonant collisions among its constituents can excite molecules to even higher-lying vibrational levels, thereby providing an alternate relaxation pathway [6

**49**, 995–997 (1986). [CrossRef]

## 3. Results and discussion

*L*

_{cell}and radius

*R*is filled with a suitable gas that lases at the desired THz frequency, while pump power at a much higher IR frequency enters the system from the front window. Depending on the reflectivities of the front and back windows at THz, lasing output power can then escape from the cavity via both channels. In order to compare our numerical calculations with past experimental results, the cavity modeled in the following simulation has length

*L*

_{cell}= 12

*cm*and front window reflectivity Γ

_{front}= 0.96. The back window is opaque (i.e. Γ

_{back}= 1). Furthermore,

*R*is varied across a range of commercially available values from the smallest possible before modal cutoff at 0.08

*cm*to 1

*cm*. The cavity is then filled with an isotopic isomer of methyl fluoride gas,

^{13}CH

_{3}F, where a total of six rotational levels,

*J*

_{ν}_{0}= 3,4,5 in the ground vibrational state and

*J*

_{ν}_{3}= 4,5,6 in the excited state, are included (see detailed rate equations in the Appendix). Here, the K-swap processes occur at roughly 1–2 times the rates of the rotational transitions, and through this process, the lower and higher adjacent J levels can be modeled as part of their respective pool. For low pressure operation, we include the doubly degenerate

*ν*

_{6}state as our only higher excited vibrational state. As we shall see next, these relatively small number of rotational-vibrational levels are already sufficient to attain reasonable match between numerical predictions and experimental data. The system is uniformly pumped at room temperature (

*T*= 300

*K*) with CW CO

_{2}laser at 31 THz in which molecules are excited between

*J*

_{ν}_{0}= 4 and

*J*

_{ν}_{3}= 5 to produce primary lasing transition at 0.245 THz in

*J*

_{ν}_{3}= 4 – 5 of the

*ν*

_{3}vibrational state, with

*τ*

_{spont}roughly chosen to be 15

*s*[19

19. I. Shamah and G. Flynn, “Vibrational relaxation induced population inversions in laser pumped polyatomic molecules,” Chem. Phys. **55**, 103–115 (1981). [CrossRef]

*ν*

_{3}is comparatively low (

*f*=

*e*

^{−Eν/kT}≈ 1/150), making the condition favorable for the creation of population inversion between

*J*

_{ν}_{3}= 5 and the level immediately below. Lastly,

*K*in Fig. 1 equals 3 for 13CH

_{i}_{3}F and we have approximated the K-swap process as the faster Δ

*K*=3n process where the V-swap rate is slow and primarily equilibrates

*A*and

*E*states in the pool. Other molecules with a slower or absent Δ

*K*=3n process will need to approximate the K-swap process by the V-swap process. The rate constants and other salient molecular parameters used are provided in the Appendix.

*R*= 0.26

*cm*copper waveguide cavity as a function of frequency. This loss prediction is made assuming

*L*

_{cell}≫

*R*so that only the circular plane of the cavity matters and that the copper conductivity is high enough that the wall currents flow uniformly within a skin depth of the surface [18

18. R. Bansal (ed.), *Handbook of Engineering Electromagnetics* (Marcel Dekker, Inc., 2004). [CrossRef]

_{01}, will be the first to lase upon threshold and remains the only mode present for pump power near the threshold, despite the cavity’s ability to support multiple modes at THz. Alternatively, one can ensure single-mode operation even for large pump power by selecting the smallest possible cavity size with radius ∼

*λ*

_{THz}/2, where

*λ*

_{THz}is the wavelength of the THz lasing output. For our purpose, it suffices to simply operate near the threshold for single-mode operation since this also facilitates independent design optimizations of our system based on molecular gas physics and photonics considerations. Figure 2(c) depicts the intensity profiles of the three lowest loss modes of the cavity. It may be noted that the magnitude of ohmic loss scales roughly with the fraction of modal intensity residing near the cell wall, which in turn relates to the amount of penetration, to within a skin depth, into the wall. At low enough pump rate, only losses of TE

_{01}mode (both ohmic losses and leakage through the end mirrors) can be compensated by the gain produced in

^{13}CH

_{3}F while the other modes remain suppressed. Thus, we shall consider TE

_{01}to be the lowest-threshold lasing mode.

*N*(

*λ*

_{THz}) and the

*ν*

_{3}thermal pool to steady state are presented and compared at 250 and 350 mTorr, assuming

*P*

_{pump}to be 10 watts in CW operation. A few initial observations are in order:

*(i)*the radial spatial profile of Δ

*N*(

*r*) is the inverse of that for

*ν*

_{3}pool. This is the pool quenching that leads to the vibrational bottleneck mentioned earlier. Note the rapid removal of molecules from

*ν*

_{3}near the cell wall is critical to maintaining the inversion.

*(ii)*The cell center takes a long time to attain steady state given that the wall rate is slow far away from the wall (i.e. molecules have to travel a long distance to reach the cell boundary).

*(iii)*Similarly, a longer time is required to reach steady state at a higher pressure due to the decreased

*k*

_{wall}near the cell center. In Fig. 3, steady state is attained after 100

*μs*at 250 mTorr while it takes 300

*μs*to do so at 350 mTorr.

*(iv)*The magnitude of the variation across the radial direction of the cavity is greater at higher pressure, as will be explained shortly. While we mostly deal with CW systems in this paper, the ability to track the full temporal development of the laser action allows pulsed systems to be studied as well.

*R*= 0.26

*cm*copper cavity

^{13}CH

_{3}F laser. The experimental configuration was identical to that described in [6

**49**, 995–997 (1986). [CrossRef]

*J*= 4 – 5 lasing transition in

*ν*

_{3}state of 13CH

_{3}F may be written as [20, 21

21. S. L. Chua, Y. D. Chong, A. D. Stone, M. Soljac̆ić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Express **19**, 1539–1562 (2011). [CrossRef] [PubMed]

*N*〉 is the pressure-dependent steady-state effective inversion defined as

**E**

_{0}(

**r**) is the normalized mode profile of the passive cavity (∫

*d*

**r**

*ε*

_{0}

*n*

^{2}(

**r**) |

**E**

_{0}(

**r**)|

^{2}= 1) and

*V*

_{ACT}being the volume of active region of the considered structure. In our case,

*V*

_{ACT}is the entire cavity (note that this implies that the gas’s behavior is less relevant in regions where the modal intensity is small but plays an important role in regions of high field intensity). In the considered case,

*total*molecular density of the upper (

*J*= 5) and lower (

*J*= 4) lasing transition level in

*ν*

_{3}state after accounting for the fractional contribution from the excited thermal pool, and

*J*level.

^{th}**E**

_{0}(

**r**) is the normalized field of the TE

_{01}mode and

*n*≈ 1 for the gas. The width of the transition line, Δ

*ν*, is broadened due to a combination of Doppler effects and molecular collisions and has a strong dependence on both pressure and transition frequency. The former is associated with the Gaussian lineshape while the latter, which leads to pressure broadening, is Lorentzian. Hence,

*ν*

_{THz}± Δ

*ν*/2 by assuming a large optical resonator whose dimension is several wavelengths so that its exact shape becomes less significant. We find that this applies in our case since the cylindrical copper waveguide is almost 100

*λ*

_{THz}long and a few

*λ*

_{THz}s in diameter. Moreover, the inversion density, and hence gain, is not saturated by stimulated emission for the entire range of input intensity plotted. As we are only interested in the threshold value in this part of the work, such approach remains valid.

*E*-field penetration into the copper wall or to energy leakage from the front mirror. For

*L*

_{cell}≫

*R*, we can treat the system as one-dimensional along the cylinder axis for loss analysis and quantify its coefficient as 2

*α*

_{ohmic}– ln(Γ

_{front}Γ

_{back})/

*L*

_{cell}, where

*α*

_{ohmic}is the amplitude loss coefficient shown in Fig. 2(b). The magnitudes of these loss coefficients are indicated as horizontal red lines in Figs. 4(a) and 4(b). As such, the input threshold intensity predicted from our model is determined at the intersection between the red and blue lines, which indicates the point at which losses are compensated by the gain at

*λ*

_{THz}so that any further increase in pump intensity saturates the gain and is thereby channeled into coherent output power that escapes from the front window. From Figs. 4(a) and 4(b), the numerical thresholds attained are 0.78 and 2.42 W/cm

^{2}for the laser operated at 100 mTorr and 300 mTorr, respectively.

*α*′, for

^{13}CH

_{3}F is 71 m

^{−1}Torr

^{−1}[14] so that the ability of the pump to excite the gas decreases with increasing pressure. The effect is to reduce the amount of IR power effectively pumping the gas and can be captured by defining an incident pump power

*κ*is a pressure-dependent factor that accounts for the net fraction of incident IR power available for excitation in the

*L*

_{cell}long cavity. This factor is simply given by

*κ*= 0.67 and 0.36 for our 100 mTorr and 300 mTorr setups resulting in actual pump thresholds (

^{2}, respectively. These results compare well with experimentally measured thresholds of 1.1±0.4 and 5.5±1.5 W/cm

^{2}. Possible sources of discrepancy in the comparison include instabilities that make measuring the threshold difficult, deviation from spatially uniform pumping in the experiment, the one-dimensional treatment in balancing gain-loss, and the possibility that the emission was from the refilling transition in the

*ν*

_{0}state. Even so, the current treatment provides adequately accurate predictions of the lasing thresholds. More importantly, this comparison acts to verify our claim that the aforementioned model of the OPFIR is physically robust and rigorous, yet numerically tractable.

*P*

_{out}and input power

*P*

_{in}above threshold: where

*f*

_{out}accounts for the fraction of emitted power that escapes via the front window, and

*P*

_{sat}is the saturation power at THz in which the gain in the system is half its unsaturated value. In this particular system,

*P*

_{sat}=

*h̄ω*

_{THz}∫

_{V}_{ACT}

*d*

**r**Δ

*N*

_{th}(

**r**)/

*τ*

_{eff}where Δ

*N*

_{th}is the inversion at threshold and

*τ*

_{eff}is an effective lifetime that accounts for the various rates between the rotational levels in the lasing

*ν*

_{3}state, and is found to be with The lifetimes in Eq. (6) describe the transition rates in the excited

*ν*

_{3}state where lasing occurs and are defined in accordance to their definitions in Fig. 1 and Eq. (1). With that, the slope

*dP*

_{out}/

*dP*

_{in}is

*N*

_{th}and

^{−6}and 39.2×10

^{−6}at 100 mTorr and 300 mTorr, respectively. Note that the best slope efficiency for the considered

^{13}CH

_{3}F laser is 8 × 10

^{−3}, which is the Manley-Rowe limit when the quantum efficiency is 1. These calculated values compare reasonably well with experimental slope values of (44 ± 22) × 10

^{−6}and (48 ± 24) × 10

^{−6}, for which the greatest source of uncertainty is the calibration of the the Golay cell used to detect the THz radiation. Given this, these results further validate our model.

*ν*(

*λ*

_{THz}), broadens with increasing pressure so that 〈

*γ*(

*λ*

_{THz})〉 reduces [see Eq. (3)]. Note that the variation of the linewidth with pressure is captured in the top axis of the figure. Since the total loss of the system is pressure-independent, such gain reduction increases the threshold linearly with pressure for each value of

*R*. Second order effects that further raise the threshold at high pressure originate from the vibrational bottleneck: A higher pressure impedes the molecular diffusion to the cell wall and the subsequent decrease in wall collision rates reduces the flow of molecules out of the the excited thermal pool, leading to enhanced absorption and hence, a weaker inversion 〈Δ

*N*〉 and gain. Equivalently, the same effects on 〈Δ

*N*〉 can be understood by recognizing that

*λ*

_{MFP}progressively makes up a smaller region near the cell wall as the pressure increases such that a greater proportion of the molecules are in the non-ballistic regime, which also leads to slower

*k*

_{wall}. Red dashed line plotted against the right axis depicts the pressure dependence of

*λ*

_{MFP}. Over the range of low pressures considered, the vibrational bottleneck is only observed for large radius cells (i.e.

*R*≥ 0.26

*cm*in the current setup) while for smaller cavities, there is no vibrational bottleneck and the threshold dependence on pressure remains approximately linear.

*R*values larger than 0.26

*cm*but decreases with radius for

*R*values smaller than 0.26

*cm*. To understand why

*R*= 0.26

*cm*is nearly optimal, the reader is referred to the inset that illustrates the reduction in ohmic losses of the TE

_{01}mode at 0.245 THz when

*R*grows from the waveguide cut-off value, with the radii of interest examined in the main figure marked as square markers. For small radius, the ohmic loss is high and hence, possesses great influence on the direction of threshold shift. In this regime, when the radius increases, the corresponding fall in ohmic loss has the tendency to lower the threshold. On the other hand, at larger

*R*values where the ohmic losses are low, the vibrational bottleneck increasingly raises or prevents threshold with increasing

*R*and pressure. However, at pressures less than 100 mTorr in large cells (i.e.

*R*≥ 0.26

*cm*), the pressure-dependent threshold lines in Fig. 4(c) converge to the linear behavior observed for cells with

*R*≤ 0.26

*cm*. As we shall see next, this follows directly from the fact that

*λ*

_{MFP}grows large at low pressures, so for a constant intensity and pump rate

*R*

_{pump}, the inversion 〈Δ

*N*〉 and gain exhibit little radial dependence. As a result, variation of

*R*has a much smaller effect on the wall collision rate, and correspondingly on 〈

*γ*〉, at lower pressures. So, for cell radii where the ohmic losses are also low, the threshold remains almost unchanged as

*R*is varied.

*R*= 0.5

*cm*cavity pumped by CW CO

_{2}laser at

*P*

_{pump}= 100 W are studied in Fig. 5. From the intensity-weighted averaging scheme used in Eq. (4), we have already noted that it is most desirable to match the peak of the cavity’s eigenmode intensity to regions where Δ

*N*(

*r*) or

*γ*(

*r*) is maximum. Such consideration plays an important role in designing the cavity itself. Figure 5(a) illustrates the distribution of Δ

*N*(

*r*) within the cavity for values of pressure ranging from 50 mTorr to 450 mTorr. In general, strong inversions are favored near the cell wall where the wall collision rates are the highest, and so, allow rapid depopulation of the excited and higher excited thermal pools to reduce pool quenching. This directly implies an advantage in exciting a lasing mode whose intensity peaks near the cell wall. However, such approach will also have to be weighted against the higher ohmic losses that arise from the field’s enhanced interaction with the cell, in a similar trade-off effects to what was already discussed in Fig. 4(c). The details and treatment of the cavity design will be left for future work. We further note that the magnitude of radial variation in the cell increases with pressure. For instance, at 50 mTorr, a good proportion of the cavity is ballistic [see

*λ*

_{MFP}variation in Fig. 4(c)] so that Δ

*N*is approximately uniform, i.e. Δ

*N*(

*r*

_{cen}) = 0.96Δ

*N*(

*r*

_{wall}) where

*r*

_{cen}and

*r*

_{wall}indicate the position at the cell center and wall, respectively. On the other hand, at 450 mTorr, most of the molecules are in the non-ballistic regime where

*k*

_{wall}∝1/

*r*

^{2}as opposed to

*k*

_{direct}∝1/

*r*. This leads to Δ

*N*(

*r*

_{cen}) to be only 14% of Δ

*N*(

*r*

_{wall}). Indeed, it was found that the maximum inversion for the set of parameters used in this example increases with pressure because of the corresponding increase in

*N*

_{tot}. Figure 5(b) depicts a similar plot for the gain. The same characteristics are observed among the two except that

*γ*(

*r*) is also influenced by the laser transition linewidth which broadens as pressure increases. This decrease of gain with pressure is again consistent with the discussion presented for Fig. 4(c), explaining why most OPFIR lasers use large diameter cavities operating at low pressure.

*R*

_{pump}increases when the same pump power is more tightly confined, the gain tends to grow as cavity radii shrink until it eventually saturates (because

*N*

_{tot}is a conserved quantity). Figures 5(c) and 5(d) track the optimum radius size for lasing based on the numerical model developed. An optimum operating pressure exists for each radius where the gain is maximized. Beyond that, vibrational bottleneck begins to play a dominant role, explaining the fall of the gain as pressure rises. At high enough pressure, the lasing action will be quenched. For example, in Fig. 5(c) the cut-off pressure occurs at 380 mTorr for

*R*= 1

*cm*. Because of the role of higher lying vibrational levels, the cutoff pressure is pump intensity dependent such that the greater the pumping intensity, the higher the cut-off pressure. Thus, the optimum pressure and gain results summarized in Fig. 5(d) based on the molecular gas physics greatly favor small-sized cavities, especially if the effects of waveguide ohmic losses can be overcome.

## 4. Conclusions

## Appendix: Rate equations and rate constants of ^{13}CH_{3}F gas lasers

^{13}CH

_{3}F gas lasers. Figure 6 and Eq. (7) to Eq. (15) refer to a specific example of a methyl fluoride gas discussed in Sec. 3, adopted from the general model presented in Fig. 1, Eq. (1) and Eq. (2) of Sec. 2. We have labeled the rotational levels from

*N*

_{1}to

*N*

_{6}, and the thermal pools as

*N*,

_{A}*N*and

_{B}*N*. Hence,

_{C}*τ*

_{12},

*τ*

_{23},

*τ*

_{45},

*τ*

_{56}, and their reverse processes represent rotational transitions due to dipole-dipole collisions,

*τ*

_{1}

*,*

_{A}*τ*

_{4}

*, and similar processes represent K-swap transition lifetimes from the rotational levels (i.e.*

_{B}*N*

_{1}and

*N*

_{4}) into their respective thermal pools (i.e.

*N*and

_{A}*N*), and

_{B}*τ*,

_{AB}*τ*, and their reverse processes are vibrational transitions which arise due to collisions with the cell wall. Readers should note that all notations in the main body of the paper are kept consistent with those used in Fig. 1, Eq. (1) and Eq. (2).

_{AC}**r**,

*t*) being the

*total*molecular density of

*J*= 4 nonthermal level in

*ν*

_{0}state after accounting for the fractional contribution from thermal pool

*A*, and

*g*

_{2}= 9 is its degeneracy. Similar definitions hold for

*g*

_{5}= 11.

^{13}CH_{3}F,” J. Chem. Phys. **90**, 3520–3527 (1989). [CrossRef]

## Acknowledgments

## References and links

1. | T. Y. Chang and T. J. Bridges, “Laser actions at 452, 496, and 541 |

2. | T. K. Plant, L. A. Newman, E. J. Danielewicz, T. A. DeTemple, and P. D. Coleman, “High power optically pumped far infrared lasers,” IEEE Trans. Microwave Theory Tech. |

3. | W. H. Matteson and F. C. De Lucia, “Millimeter wave spectroscopic studies of collision-induced energy transfer processes in the |

4. | M. S. Tobin, “A review of optically pumped NMMW lasers,” Proc. IEEE |

5. | P. K. Cheo (ed.), |

6. | H. O. Everitt, D. D. Skatrud, and F. C. De Lucia, “Dynamics and tunability of a small optically pumped CW far-infrared laser,” Appl. Phys. Lett. |

7. | R. I. McCormick, H. O. Everitt, F. C. De Lucia, and D. D. Skatrud, “Collisional energy transfer in optically pumped far-infrared lasers,” IEEE J. Quantum Electron. |

8. | H. O. Everitt and F. C. De Lucia, “A time-resolved study of rotational energy transfer into A and E symmetry species of |

9. | H. O. Everitt and F. C. De Lucia, “Rotational energy transfer in CH |

10. | R. L. Crownover, H. O. Everitt, D. D. Skatrud, and F. C. DeLucia, “Frequency stability and reproductibility of optically pumped far-infrared lasers,” Appl. Phys. Lett. |

11. | D. Dangoisse, P. Glorieux, and J. Wascat, “Diffusion and vibrational bottleneck in optically pumped submillimeter laser,” Int. J. Infrared Milimeter Waves |

12. | J. O. Henningsen and H. G. Jensen, “The optically pumped far-infrared laser: Rate equations and diagnostic experiments,” IEEE J. Quantum Electron. |

13. | R. J. Temkins and D. R. Cohn, “Rate equations for an optically pumped, far-infrared laser,” Opt. Commun. |

14. | H. O. Everitt, “Collisional Energy Transfer in Methyl Halides,” PhD Thesis (Department of Physics, Duke University, 1990). |

15. | H. O. Everitt and F. C. De Lucia, “Rotational energy transfer in small polyatomic molecules,” in |

16. | L. E. Reichl, |

17. | A. E. Siegman, |

18. | R. Bansal (ed.), |

19. | I. Shamah and G. Flynn, “Vibrational relaxation induced population inversions in laser pumped polyatomic molecules,” Chem. Phys. |

20. | A. Yariv and P. Yeh, |

21. | S. L. Chua, Y. D. Chong, A. D. Stone, M. Soljac̆ić, and J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Express |

**OCIS Codes**

(140.3070) Lasers and laser optics : Infrared and far-infrared lasers

(140.3460) Lasers and laser optics : Lasers

(140.4130) Lasers and laser optics : Molecular gas lasers

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: February 10, 2011

Manuscript Accepted: March 21, 2011

Published: April 4, 2011

**Citation**

Song-Liang Chua, Christine A. Caccamise, Dane J. Phillips, John D. Joannopoulos, Marin Soljačić, Henry O. Everitt, and Jorge Bravo-Abad, "Spatio-temporal theory of lasing action in optically-pumped rotationally excited molecular gases," Opt. Express **19**, 7513-7529 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-8-7513

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### References

- T. Y. Chang, T. J. Bridges, “Laser actions at 452, 496, and 541 μm in optically pumped CH3F,” Opt. Commun. 1, 423–426 (1970). [CrossRef]
- T. K. Plant, L. A. Newman, E. J. Danielewicz, T. A. DeTemple, P. D. Coleman, “High power optically pumped far infrared lasers,” IEEE Trans. Microwave Theory Tech. MT22, 988–990 (1980).
- W. H. Matteson, F. C. De Lucia, “Millimeter wave spectroscopic studies of collision-induced energy transfer processes in the 13CH3F laser,” IEEE J. Quantum Electron. 19, 1284–1293 (1983). [CrossRef]
- M. S. Tobin, “A review of optically pumped NMMW lasers,” Proc. IEEE 73, 61–85 (1985). [CrossRef]
- P. K. Cheo (ed.), Handbook of Molecular Lasers (Marcel Dekker, Inc., 1987), pp. 495–569.
- H. O. Everitt, D. D. Skatrud, F. C. De Lucia, “Dynamics and tunability of a small optically pumped CW far-infrared laser,” Appl. Phys. Lett. 49, 995–997 (1986). [CrossRef]
- R. I. McCormick, H. O. Everitt, F. C. De Lucia, D. D. Skatrud, “Collisional energy transfer in optically pumped far-infrared lasers,” IEEE J. Quantum Electron. QE-23, 2069–2077 (1989).
- H. O. Everitt, F. C. De Lucia, “A time-resolved study of rotational energy transfer into A and E symmetry species of 13CH3F,” J. Chem. Phys. 90, 3520–3527 (1989). [CrossRef]
- H. O. Everitt, F. C. De Lucia, “Rotational energy transfer in CH3F: The ΔJ = n, ΔK = 0 processes,” J. Chem. Phys. 92, 6480–6491 (1990). [CrossRef]
- R. L. Crownover, H. O. Everitt, D. D. Skatrud, F. C. DeLucia, “Frequency stability and reproductibility of optically pumped far-infrared lasers,” Appl. Phys. Lett. 57, 2882–2884 (1990). [CrossRef]
- D. Dangoisse, P. Glorieux, J. Wascat, “Diffusion and vibrational bottleneck in optically pumped submillimeter laser,” Int. J. Infrared Milimeter Waves 2, 215–229 (1981). [CrossRef]
- J. O. Henningsen, H. G. Jensen, “The optically pumped far-infrared laser: Rate equations and diagnostic experiments,” IEEE J. Quantum Electron. QE-11, 248–252 (1975). [CrossRef]
- R. J. Temkins, D. R. Cohn, “Rate equations for an optically pumped, far-infrared laser,” Opt. Commun. 16, 213–217 (1976). [CrossRef]
- H. O. Everitt, “Collisional Energy Transfer in Methyl Halides,” PhD Thesis (Department of Physics, Duke University, 1990).
- H. O. Everitt, F. C. De Lucia, “Rotational energy transfer in small polyatomic molecules,” in Advances in Atomic and Molecular Physics (Academic Press, 1995), Vol. 35, pp. 331–400.
- L. E. Reichl, A Modern Course in Statistical Physics (John Wiley & Sons Inc., 1998).
- A. E. Siegman, Lasers (Univ. Science Books, 1986).
- R. Bansal (ed.), Handbook of Engineering Electromagnetics (Marcel Dekker, Inc., 2004). [CrossRef]
- I. Shamah, G. Flynn, “Vibrational relaxation induced population inversions in laser pumped polyatomic molecules,” Chem. Phys. 55, 103–115 (1981). [CrossRef]
- A. Yariv, P. Yeh, Photonics: Optical Electronics in Modern Communications (Oxford University Press, 2007).
- S. L. Chua, Y. D. Chong, A. D. Stone, M. Soljac̆ić, J. Bravo-Abad, “Low-threshold lasing action in photonic crystal slabs enabled by Fano resonances,” Opt. Express 19, 1539–1562 (2011). [CrossRef] [PubMed]

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