## Abstract

Optically pumped molecular gas amplifiers having a gain medium contained in a hollow-core optical fiber are investigated with numerical modeling to understand the primary physical processes that affect amplifier output and efficiency. A comparison of computational results with experimental measurements of incident pump, absorbed pump, and emitted mid-IR from a pulsed, acetylene-filled, hollow-core fiber amplifier [ Opt. Exp. **25**, 13351 (2017)] is used to explore the effects of various physical processes on pulsed amplifier operation. Single frequency, one-dimensional, time-dependent models are shown to align with experimentally measured lasing thresholds and ratios of absorbed pump to emitted laser energy but significantly over predict the amount of incident pump energy that is absorbed. A two-dimensional, time-dependent model that assumes Gaussian spectral and radial intensity profiles for the pump is developed and shows an improved ability to capture pump absorption. Results indicate that 1D, time-dependent models have utility in guiding experiments but the significant influence of the pump and laser propagation modes and the pump spectral characteristics on efficiency, threshold, and signal output must be explicitly included in high-fidelity high-power modeling.

## 1. Introduction

High power fiber lasers that operate in the eye-safe regions of the mid-infrared have numerous applications in research, industry, and defense. Thulium-doped, pulsed, master oscillator power fiber amplifiers have achieved peak powers of more than a kilowatt at 2 *μ*m [1], but the potential mid-IR output of solid-core fiber lasers is constrained by absorption in silica. Molecular gas lasers are able to reach further into the mid-infrared, but the long lengths required for interaction between the gain medium and pump light prohibit traditional gas laser configurations. Hollow-core optical fiber gas lasers (HOFGLAS) are optically pumped gas lasers where the gas gain medium is contained in the hollow core of an optical fiber and may provide a hybrid configuration combining advantages of fiber and gas lasers conducive to robust, scalable mid-IR sources.

Initial experimental demonstrations of HOFGLAS showed pulsed lasing initiated by spontaneous emission in an acetylene-filled hollow-core photonic crystal fiber [2, 3]. Hollow-core, photonic crystal fibers were used due to their ability to transmit near- and mid-infrared light with low loss [4–9]. Acetylene has multiple ro-vibrational transitions in the 3.1 – 5.0 *μ*m range that are suitable for lasing and permit line-tunable emission [10,11]. Other suitable gain media include HCN, CO, CO_{2}, and I_{2} [2, 3, 12]; recent experiments have shown that high-repetition rate [13–15], continuous wave (CW) [11,12,16], and higher efficiency [17–19] configurations are possible.

The large number of gain media and fiber configurations can be explored through numerical modeling and theory. Numerical models are commonly used in the fiber and gas laser community to guide experimental work, optimize designs, and expand theoretical knowledge of laser operation. The physical descriptions requires of solution a challenging system of coupled, three-dimensional, time- and frequency-dependent, nonlinear differential equations for the excited state populations, electromagnetic fields, heat distribution, and material response of the gain media. Practical computational approaches make use of simplifying assumptions and operator splitting based on the different timescales for some of the physical effects [20]. However, many frequently-used approaches are applied to either fiber or gas lasers and rely on physical assumptions that are not equally valid in both configurations. High-fidelity modeling of the physical processes in HOFGLAS can assist the maturation of the technology and may benefit other applications of gas-filled hollow-core optical fibers. For example, development of optical frequency references in hollow-core fibers [21,22] may benefit from estimates of broadening and energy transfer during molecule collisions with core wall collisions.

A time-dependent HOFGLAS model for low power nanosecond scale pulsed operation is developed here to investigate the dominate physical effects in HOFGLAS. The coupled system of differential equations for the exited state populations and the square of the electric field are reduced to one dimension through simplifying assumptions and solved. Subsequent solutions relax the simplifying assumptions to approximate the spectral and spatial properties. The state and intensity equations include terms for state-to-state rotational energy transfer (RET), vibrational energy transfer (VET), fiber attenuation, and molecular collisions with the core walls in addition to radiation absorption and emission terms. The model results are compared to the published experimental HOFGLAS data for mid-IR in [19] to assess the impact and utility of the simplifying assumptions and to assess which measurable experimental parameters are needed to enhance agreement between models and experiment.

A rate equation model of gas laser operation in capillaries has been previously used in tandem with HOFGLAS experiments and showed some success in matching the laser threshold, laser output, and temporal pulse shape observed [3,23]. The reported one-dimensional, time-dependent model included RET and VET through a total removal constant and assumed that the pump and laser light were single frequency and uniformly distributed across the gain media. The inclusion of the additional physics and ro-vibrational states here permits investigation of the effect of numerous changes in experimental conditions on HOFGLAS signal output and analysis of the relative effect of various loss processes. A description of the model and relevant equations is given in Sec. 2 and the values used for gas and fiber properties are described in Sec. 3. Model results are compared to experimental results from [19] and used to investigate the variance in laser output under experimental changes in Sec. 4. Broader implications of the simulation results are discussed in Sec. 5 with a brief outline of avenues for future work.

## 2. Model description

A simplified hollow-core optical fiber gas amplifier configuration is sketched in Fig. 1(a). A generic configuration is chosen to resemble pulsed experiments using acetylene reported in [2, 3, 15, 17–19] and the numerical model covers processes that occur in the fiber portion of the experiment. The ends of the fiber are enclosed in containers and fixed adjacent to fully transparent optical windows. The containers and the fiber are filled with acetylene at pressure *P* and temperature *T*. Monoenergetic pump light is incident on one end of an unbent, optical fiber with length *L*, and a circular, hollow core of radius *R*. Lasing in the gas-filled core starts from spontaneous emission or is seeded by an external source.

The model is comprised of differential equations for ro-vibrational population densities when the gas is pumped with light tuned to the P or R branch of |*v*_{1} + *v*_{3}, *j′*〉 ← |0, *j″*〉 at 1.5 *μ*m. The resulting population inversion permits lasing at 3.1 *μ*m along the P and R branches of |*v*_{1} + *v*_{3}, *j′*〉 → |*v*_{1}, *j″*〉. A ro-vibrational energy level diagram depicting these transitions is shown in Fig. 1(b). The partial differential equations for the intensities and population densities of states accessible through radiation and collisions are solved from initial conditions to compute the signal output and the re-distribution of pump energy in the system. The intensities correspond to the square of the electric field and other nonlinear interactions with electric field are neglected for low power operation.

The equations for intensity corresponding to |*v*_{1} + *v*_{3}, *j′*〉 ← |0, *j″*〉, *I*_{p}(**r**, *t*, *ν*), with frequency-dependent cross section *σ*_{p}(*ν*) and Einstein A coefficient *A*_{p} are

*I*

_{l}corresponding to |

*ν*

_{1}+

*ν*

_{3},

*j′*〉 → |

*ν*

_{1},

*j″*〉 with intrinsic cross section

*σ*

_{l}and Einstein A coefficient

*A*

_{l}are

*μ*m and 3.2

*μ*m, state |

*ν*

_{1}+

*ν*

_{3},

*j*〉 has population density

*n*

_{p}, and degeneracy

*g*

_{p}, state |

*v*

_{1},

*j*〉 has population density

*n*

_{l}(

**r**,

*t*) and degeneracy

*g*

_{l}, and the ground ro-vibrational state |0,

*j*〉 has population density

*n*

_{g}(

**r**,

*t*) and degeneracy

*g*

_{g}. Separate equations of form Eqs. (1) and (2) are used for the P and R branches of both transitions. Backwards propagating intensity is estimated by solving equations of form Eqs. (1) and (2) traveling in the opposite direction.

The equations for population density include RET, VET, and collisions with the core walls for all states and radiative transfer for the accessible states. The population densities of the rotational states of the *ν*_{1} + *ν*_{3} vibrational state are represented as

*ν*

_{1}vibrational state as

*j*and

*j′*of a vibrational state,

*k*

^{wc}is the rate constant for VET due to molecules colliding with the core walls,

*k*

^{vet}is the rate constant for VET due molecular self collisions, and

*n*

_{C2H2}is the density of possible collision partners. The total density of acetylene molecules from the ideal gas law is used for

*n*

_{C2H2}. State-to-state RET is excluded for the vibrational ground state which reduces the population available for pumping is limited to the thermal population of |0,

*j″*〉. The population in the |0,

*j″*〉 state is modeled through

The model comprised of Eqs. (1–7) represents a challenging set of coupled, nonlinear differential equations that can be solved using lower-order methods under simplifying assumptions and can potentially be directly solved with finite element, finite-difference, or beam propagation methods. The frequency dependence can be removed by assuming single frequency operation, i.e. that the pump and laser have linewidths much smaller than the absorption linewidth, through *I*(**r**, *t*, *ν*) = *I*^{0}(**r**, *t*)*δ*(*ν* − *ν*_{p}), where *δ* is the Dirac delta function. The spatial dependence can be reduced to the longitudinal dimension, i.e. the pump and signal have a uniform transverse intensity distribution, through *I*(**r**, *t*, *ν*) = *I*^{0}(*z*, *t*, *ν*)Θ(*r* −*r*_{0}), where circular symmetry is assumed and Θ is the Heaviside step function.

In tandem these assumptions reduce the model to one spatial dimension with four single-frequency intensities. Non-uniform transverse intensity profiles are approximated by dividing the cross sectional area into smaller regions with independent population densities and each subjected to locally uniform intensity. The radial profile is mapped onto the smaller regions while conserving the total energy and the population terms in Eqs. (1) and (2) are replaced with sums over the regions. The spectral width of the pump profile can be approximated through a similar process using frequency intervals *dν* but the pump profile is enforced only as an initial condition. The result is a system of loosely coupled equations comprised of a set Eqs. (3–7) for each region and Eqs. (1) and (2) for each frequency interval loosely.

## 3. Calculation of state properties, cross sections, and rates for C_{2}H_{2}

The *v*_{1} and *v*_{3} vibrational modes of acetylene are symmetric and antisymmetric stretching of the C–H bonds. Acetylene energy levels have *ortho* or *para* nuclear spin symmetry for symmetric and antisymmetric states. Conversion between *ortho* and *para* states for rotational transitions is forbidden, thus rotational energy transfer has form |*v*, *j*〉 → |*v*, *j* ± 2*m*〉, where *m* is an integer. The energies of ro-vibrational states are calculated from the model in [24]. Further information on the structure, modes, and ro-vibrational states of acetylene can be found in [24] and the references therein.

The highest rotational state included for each vibrational state is selected such that all higher rotational states contribute less than 1% to RET out of the terminal state for the signal P branch. The initial population density of acetylene molecules is estimated from the ideal gas law, the purity of atomic absorption grade C_{2}H_{2}, and the natural abundance of ^{12}C_{2}H_{2} [25,26]. Initial population densities for each ro-vibrational state are found from the ratio of the state partition function to the total internal partition function. Temperature dependent forms of the total and state partition functions are given in [27,28].

Initial intensities are assumed to be temporal Gaussians with total energy given by the incident pump or seed energy. Radiative cross sections are calculated from the Einstein A coefficients for each ro-vibrational transition through *σ*(*ν*) = [(*c*^{2}/(8*πn*^{2}*ν*^{2})]*ρ*(*ν*)*A*, where *ρ*(*ν*) is assumed to be a Voigt profile and *n* is the index of refraction for acetylene. The index of refraction is near unity for the cases here. The Voigt profile includes pressure broadening using the self-broadening coefficients from HITRAN and Doppler broadening at room temperature. The broadening due to wall collisions can be estimated from gas kinetics as described in [29]. Estimates for the conditions indicate the wall collision broadening is less than 1% of the pressure or thermal broadening for the conditions and is therefore excluded from the absorption profile.

Einstein A coefficients from measured line intensities are available for the |*v*_{1} + *v*_{3}, *j′*〉 → |0, *j″*〉 transitions [25, 26] but not the |*v*_{1} + *v*_{3}, *j′*〉 → |*v*_{1}, *j″*〉 transitions. The A coefficients of the |*v*_{1} + *v*_{3}, *j′*〉 → |*v*_{1}, *j″*〉 transition are estimated from fits of the acetylene line intensities for ro-vibrational transitions in the HITRAN database with energies in the range 50–9900 cm^{−1} [30]. The calculated A coefficients for the transition |*v*_{1} + *v*_{3}, *j′*〉 → |*v*_{1}, *j″*〉 are within 15% of the coefficients for the similar transition |*v*_{3}, *j′*〉 → |0, *j″*〉 for the P and R branches with 6 < *j′* < 14. The difference is smaller than the uncertainty on the fits [30].

Endothermic state-to-state RET constants are estimated from fits of the ro-vibrational state energy *E ^{j}*, through [31]

*k*

_{B}is the Boltzmann constant,

*B*

_{v}is the molecular rotation constant, and

*k*

_{0},

*α*, and

*β*are fit constants. Fit constant values for RET in

*v*

_{1}+

*v*

_{3}are given in [31] but are unavailable for

*v*

_{1}. Investigations of the RET total removal constants for various C–H stretching vibrational states with energies below 7000 cm

^{−1}have reported total removal constants in the range 10

^{−10}− 10

^{−9}cm

^{−3}s

^{−1}which are approximately equal to or greater than the rates of collisions from gas gas kinetics (see, for examples, [32,33]). The

*k*

_{0},

*α*,

*β*for the

*v*

_{1}+

*v*

_{3}state from [31] are used for

*v*

_{1}, in the absence of measured values.

Self-collisions also result in vibrational state changes, though less commonly than RET. VET is computed for each rotational state in the excited states but VET removal constants have been measured for few vibrational states of C_{2}H_{2} and the paths to the ground vibrational state are not known. Estimates of VET based on experimental measurements of ro-vibrational energy transfer through acetylene self-collisions near 6560 cm^{−1} suggest that VET contributes no more than 16% of the total population removal due to self-collisions though smaller contributions are equally possible [31]. In the present work, the VET total removal constant is estimated as one tenth of the sum of the RET constants for a particular ro-vibrational state.

The losses due to collisions with core walls are estimated through *k*^{wc} = *α*^{wc} *β*^{wc} *τ*^{−1}, where *α*^{wc} is the probability that a collision with the wall will induce a transition, and *β*_{wc} is the fraction of the molecules that are able to collide with core walls on the timescale of the pulse. The expectation value for the time between collisions, *τ*, is given by [29] as

*a*is the surface area of the fiber core,

*V*is the volume of the core, and

*m*is the molecule mass. Quenching rates of vibrationally excited acetylene molecules through collisions with silica have not been measured. Measurements of vibrational quenching in acetylene scattered from a LiF surface [34,35] show that almost all scattered acetylene molecules in low vibrational states undergo VET. Much smaller rates are obtained for scattering from conductors. The rates of VET following a collision with silica are assumed to be similar to scattering from LiF leading to the choice of

*α*

_{WC}= 1. The upper limit of the total proportionality constant

*α*

^{wc}

*β*

^{wc}= 1 is used here. This choice is discussed further in Sec. 5.

## 4. Computational results

Model results are compared to experimental data available in the literature for pulsed lasing at 3.2 *μ*m in an acetylene-filled, hollow-core, fiber amplifier pumped at 1.5 *μ*m. The experimental data is displayed in Fig. 3 of [19] and descriptions of the experimental configuration are available in [17–19]. The fiber used in the experiments is described as a 10.9 m length hypocycloidal kagomé fiber with a 60/72 *μ*m diameter core and 0.08 dB/m attenuation at 1.5 *μ*m. It is stated that the fiber modes have not been calculated and that attenuation at 3.2 *μ*m has not been measured. The optical parametric amplifier (OPA) used for pumping is described in [23] though measurements of the spectral profile are not available. Pump absorption results in a population inversion that leads to lasing through amplification of spontaneous emission. The model is used to compute the amount of pump energy absorbed and signal output for various pump energies coupled into the fiber with parameters chosen to approximate the reported properties.

Computational results with all loss rates set to zero and using the narrow-linewidth and uniform radial profile assumptions described in Sec. 2 are compared to experimental results in Fig. 2. Predictably, results from this *lossless, single-frequency* model indicate a slope-efficiency near the maximum gain with all transitions saturated and predict absorbed pump and signal energies at saturation that are larger than reported in the experiment. Model correspondence with the experimentally measured efficiencies is improved by accounting for the loss processes described in Sec. 3. The improvement can bee seen Fig. 3 but the model predicts higher output and absorbed energies at saturation than reported measurements similar to the results that exclude losses.

Model results from Monte Carlo sampling are presented with experimental results Figs. 3, 4, and 5. Unknown experimental quantities are sampled uniformly from an assigned possible range and experimentally measured quantities are sampled from their associated uncertainty range. The properties assigned a possible range are fiber attenuation at 3.2 *μ*m with range 0.08 − 0.36 dB/m, the mode field diameter of the fundamental mode at 3.2 and 1.5 *μ*m, and the difference between the central frequencies of the Gaussian pump spectral profile and the absorption profile with range −50 to 50 MHz. The ranges selected indicate the effect of the unknowns on the model but are not intended to represent confidence margins on the results. The absorbed pump energy and output at 3.2 *μ*m are computed for incrementally increased values of pump energy coupled into the system. Computed values of signal output from each randomly selected parameter set are fit with a polynomial. The fit function is used to estimate the signal output for absorbed energies that correspond to the experimental data. The mean of the Monte Carlo results are marked with a ‘*’ and error bars indicate the standard deviation.

A comparison of the modeled and experimentally measured [19] relation between absorbed pump energy to the pump energy coupled into the fiber is shown in Fig. 4. Computations with the 1D, single-frequency model including losses show nearly full absorption of the pump energy at all C_{2}H_{2} pressures tested. Experimental measurements show much lower absorption that depends on pressure [19]. The unphysical absorption characteristics may be due to the simplifying assumptions that reduce the spectral profiles of the pump and laser to single frequencies and eliminate the system’s transverse spatial dependence. Results from computations where the pump spectral profile and the radial intensity profiles are approximated with Gaussian functions are also shown in Fig. 4 and correspond more closely with experimental measurements.

The radial intensity profiles at the signal and pump frequencies are representative of the fundamental mode and approximate the impact of the pump spectral shape. However, measurements of these properties are not available and model results with assumed profiles indicate a strong sensitivity of the model, and possibly experiments, to these properties. The relative impact of the simplifying assumptions made in the models is clear in Fig. 6, where all model inputs are the same except those impacted by simplifying assumptions. The lasing threshold and efficiency are markedly different when the pump has a broad spectral width (1 GHz) and the radial intensity profile of the pump and signal are Gaussian with a mode field diameter of 0.7*R _{in}* where

*R*is the radius from the center of the fiber to the inside of the hypocylcloidal kagomé cups.

_{in}## 5. Discussion

Model results shown in Fig. 3 indicate that the efficiency and lasing threshold may be predicted with acceptable uncertainties by a model that uses simplifying assumptions to create a one-dimensional, time-dependent model with narrow spectral linewidths and includes losses. This indicates it may be possible to use a single-frequency, 1D model to appraise prospective gain media and optimize experiment configurations with respect to the lasing threshold and slope efficiency without complete knowledge of the fiber, gas, and pump properties. Computed results with various pump spectral characteristics and radial intensity distributions indicate that the signal output at saturation and the amount of pump energy that is absorbed strongly depend on the pump central frequency, pump spectral width, and the radial intensity profile. Model results for total losses to RET and VET are weakly impacted by the pump spectral and radial intensity profile effects since these losses are more closely proportional to the spatially integrated excited state population densities, though this does not extend to the spatial and temporal distribution of heat from RET and VET.

The Gaussian profiles for the spectral and radial intensity profiles are assumed here to permit exploration of their effects on the output. The discrepancy between the model and experimental results in Fig. 4 may be primarily attributable to differences in the assumed and unmeasured [19] pump spectral profiles. Broader application of model results using the methods described in 2 may be possible. Direct assessment of the agreement in saturation characteristics using measurements of the pump spectral and radial intensity profiles will be needed to allow accurate modeling of other measurable attributes, such as beam quality, backwards propagating signal from spontaneous emission, and the temporal profile of the signal.

Collision timescales for acetylene molecules from kinetic theory are longer than the pulse duration for the 3.6, 7.5, and 9.8 torr cases but statistically permit significant energy transfer. For example, the collision frequency from kinetic theory for C_{2}H_{2} at 300 K and 5 torr is 70 MHz. RET and VET are accounted for in the vibrationally excited states but are not included in calculations of ground state populations. Omitting state-to-state transfer in the ground state limits the population available for pumping to the thermal population of particular ro-vibrational states, which is probable when pumping with nanoscale pulses.

Computations here and in [3, 23, 36] indicate that the impact of collisional excited state quenching on the computed slope efficiency, signal output, and lasing threshold decrease predictably with decreasing gas pressures and that total removal rates may be sufficient to capture this effect in many cases. Detailed RET and VET calculations are likely needed to assess spatially and temporally dependent heating that may effect numerous measurable quantities during high-power operation [36] and when multiple rotational states are pumped simultaneously such as in some CW demonstrations [14]. The approach to RET in the excited states taken here may be more suitable for such models than an approach based on total removal constants.

A recent application of wall collision theory and experimental measurement [37] demonstrated that the wall collision times from [29] can be applied to optically pumped populations in a fiber. The approach in [29] does not include considerations for many possible effects that may contribute significantly to the wall collision rates in hollow-core fibers, such as collisions between molecules, molecules ‘sticking’ to the wall surface, or non-uniform distributions of excited state molecules. The expression for *k*^{wc} in Sec. 3 includes *α*^{wc} and *β*^{wc} to approximate some of these effects based on the fiber dimensions, physical structure, and propagation modes. Obtaining *β*_{wc} through convolution of the normalized probability distribution of the excited molecules with cutoff criteria based gas kinetics over the fiber cross section can provide some insight into the impact of wall collisions on HOFGLAS output.

The molecule densities in the current study are sufficient that the mean free path of the molecules is on the same scale as the transverse fiber dimensions such that the impact of assumptions about the radial intensity profile and the rates of pump energy absorption have a non-negligible impact on *k*^{wc}, which leads to the choice of the upper limit on wall collisions through *β*^{wc} = 1. Further theory and numerical studies of wall collisions in hollow core fiber may permit the tailoring of fiber properties to increase re-population of the ground state during CW operation and increase laser output.

Equations (1–6) are solved numerically using a first order method with fixed incremental steps in the work performed here. Results show the expected asymptotic convergence with respect to decreasing step size. Step sizes of Δ*z* = 50 *μm* and Δ*t* = 10 fs are sufficiently small that further decreases change the results by less than 0.2% and numerical error is less than uncertainties due to model inputs. This numerical method has well-known limitations and is selected because it provides easily obtained estimates of the relative importance of various physical processes, though the small step sizes required for uniform stepping can be computationally expensive. Adaptive higher order methods can be implemented to reduce the computation expense and increase the accuracy. Theoretical studies would likely benefit from refinement of the computational methods used for the spectral and radial intensity profiles.

## 6. Conclusions

A rate equation based model of an optically-pumped gas laser where the gain medium was contained in the core of a hollow fiber was developed. The equations were solved numerically to investigate the relative impact of various physical processes on signal output and pup energy absorption. The computational results indicate that fiber attenuation at the pump and signal frequencies and energy transfer due to molecule-molecule collisions significantly effect the signal output and lasing threshold. Molecules colliding with the core walls are much less significant than collisions between molecules during pulsed operation with nanosecond timescales but are likely to be non-negligible at longer timescales and in continuous wave operation.

The differential equations used to described the physical system can be simplified by assuming single frequency operation, uniform radial intensity profiles, and/or lossless operation. Simplified models are able to place upper bounds on the optical-to-optical efficiency for a given configuration. Estimates of the lasing threshold show improved agreement with experimental measurements as more energy loss channels are included. The simplified models may be useful to guide experimental studies or prioritize media and transitions for experimental consideration.

Model results are not limited to computations of the lasing threshold and slope-efficiency but can include the absorbed pump energy and saturation characteristics. The radial intensity profile and the spectral width of the pump have a significant impact on the absorption and saturation characteristics such that both characteristics are needed for deterministic computations, though computations with limited information can indicate the relative impact of experimental factors. The model results as a whole suggest a challenging and nontrivial interplay between the gas and light propagation physical processes that will likely increase in complexity at higher powers. Capturing this interplay and reproducing experimental measurements with high-fidelity modeling will require detailed knowledge of the gas, fiber, and pump properties but models with simplifying assumptions can produce experimentally useful information based on incomplete knowledge of those properties.

## Funding

National Research Council Research.

## Acknowledgments

This research was performed while an author held an National Research Council Research Associateship award at the Air Force Research Laboratory. Ball Aeropsace was not affiliated with any part of this work.

The authors gratefully acknowledge the assistance of Tony Hostutler in estimating the Einstein A coefficients and the authors of [3,19] for access to their experimental data and fruitful discussions of their methods.

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