Time-dependent simulations of a CW pumped, pulsed DC discharge Ar metastable laser system

Pengfei Sun; Duluo Zuo; Pavel A. Mikheyev; Pavel A. Mikheyev; Jiande Han; Michael C. Heaven; Michael C. Heaven

doi:10.1364/OE.27.022289

1. Introduction

Optically pumped rare gas lasers (OPRGLs) have been the subject of several experimental and theoretical studies since they were first demonstrated in 2012 [1–18]. These lasers are similar in concept to diode pumped alkali lasers (DPALs) [19–21], but have the advantage of utilizing a chemically inert lasing medium. The heated alkali metal vapors used in DPALs are chemically aggressive, attacking the walls and optical windows of the containment system. Unsaturated hydrocarbons (typically methane or ethane) are often used as energy transfer agents in DPAL’s, and there is chemical degradation resulting from photo-induced reactions of the hydrocarbons with the alkali metal vapors. In contrast, OPGRLs use inert rare gas atoms (Rg = Ne, Ar, Kr or Xe) as the lasing species, and collisional energy transfer is accomplished using He as the relaxation agent. Hence, the entire system is chemically inert and well-suited for closed-cycle operation.

OPRGLs utilize transitions between metastable electronically excited states of rare gas atoms. These states are initially populated by means of a low-power electric discharge (this step is the functional equivalent of heating the alkali metal to obtain sufficient vapor pressure in the DPAL systems). OPRGLs are, in essence, three-level systems. The most important processes are shown in Fig. 1, where the levels have been labeled using Paschen notation. Optical pumping is achieved by excitation of the 2p₉-1s₅ transition. A population inversion between 2p₁₀ and the 1s₅ level is generated by rapid collisional relaxation from 2p₉ to 2p₁₀. The energy transfer agent for this step is He. The system then lases on the 2p₁₀-1s₅ transition. Note that the 1s₄ and 2p₈ levels are included in Fig. 1 as they have some peripheral involvement in the kinetics. As 2p₈ is close in energy to 2p₉, there is forward and backward energy transfer between these levels (both are subject to rapid collisional relaxation to 2p₁₀). The 1s₄ level receives some population by means of radiative relaxation from 2p₁₀. The population diverted by this channel is returned to the lasing cycle by Rg(1s₄) + He→Rg(1s₅) + He energy transfer. The present study is focused on the Ar laser, where the pump and lasing wavelengths are 811.5 nm and 912.3 nm, respectively. The scaling potential for this system was indicated by an experimental study that used pulsed laser excitation, producing Ar laser pulses with an intensity of 27 kW cm⁻² [1].

Fig. 1 Three-level diagram of OPRGL.

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Continuous-wave (CW) lasers have been demonstrated using high-frequency micro-discharge [3] and pulsed DC discharge techniques to generate Ar metastables [2]. A laser that employed a micro-discharge array provided an output laser power of 22 mW with an optical-to-optical power conversion efficiency of 55% (ratio of the output laser power and the absorbed pump power) [3]. Pulsed discharges have also proved to be efficient in producing Ar metastables in Ar/He mixtures at total gas pressures near 1 atm. Short duration pulses (<200 ns) applied with a repetition frequency of 100-200 kHz have yielded sustained Ar(1s₅) number densities near 2x10¹³ cm⁻³. A diode pumped device that used this approach gave a CW output power of 4.1W with a power conversion efficiency of 31% [2].

Several computational models for OPRGLs have been developed, with most of the effort focused on the Ar/He system [3–6,11,12,16,22, 23]. Demyanov et al. [4] reported the first computational study of a CW OPRGL. Electron impact excitation and ionization processes were simulated using electron energy distribution functions that were obtained by solving the Boltzmann equation. Stationary-state solutions of three level laser rate equations were then examined. Based on the results, Demyanov et al. [4] concluded that 60% total efficiency, defined as the output power divided by the sum of the discharge and pump powers, could be achieved in near atmospheric pressure gas mixtures at ambient temperature (300 K). In other simulations the discharge was not modeled. The kinetic equations were solved under the assumption of a fixed number density for the metastables. To analyze the performance of their CW Ar laser system, Rawlins et al. [3] applied a steady-state model for a four-level system consisting of the Ar(1s₅), Ar(1s₄), Ar(2p₉) and Ar(2p₁₀) levels. A key observation from the experiment was that the gas temperature within the micro-discharge was approximately 600 K. Modeling the system with energy transfer rate constants that were determined at 300 K yielded poor agreement with the experimental results. Good agreement was obtained with the assumption that the rate constants for 1s₄→1s₅ and 2p₉→2p₁₀ followed Arrhenius temperature scaling over the 300-600 K range.

Yang et al. [6] developed a five-level model based on stationary-state solutions of the rate equations for the Ar(1s₅), Ar(1s₄), Ar(2p₈), Ar(2p₉) and Ar(2p₁₀) levels. Optical excitation was represented by a longitudinally averaged pump laser in a double-pass configuration. For a pump intensity of 2-4 kW/cm², helium pressure of 1-2 atm, and Ar(1s₅) number density of (1-5) × 10¹³ cm⁻³, they concluded that the system could reach 50%-60% optical power conversion efficiency. The concentration of Ar(1s₅) was chosen to be consistent with the micro-discharge laser system of Rawlins et al. [3]. In a subsequent study, to better characterize the conditions of the micro-discharge plasma, Hoskinson et al. [11,12] used diode laser absorption spectroscopy, photography and fluid dynamic modeling to determine the characteristics both perpendicular and parallel to a flat substrate that supported a single split-ring electrode.

Returning to the dynamics of the discharge, Emmons and Weeks [5] established zero- and one-dimensional (transversally resolved) models to analyze kinetics of species in a pulsed (20 μs duration) DC discharge. Key reaction pathways were identified. Both models predicted time-dependent densities, electron temperatures, current densities and reduced electric fields. In the positive column of the discharge the zero- and one-dimensional models were in good agreement.

Models including both discharge dynamics and optical physics were reported recently. Eshel and Perram [22] extended Yang et al.'s [6] five-level laser model to include electron impact excitation of the ground state and the five levels of the laser system. Electron temperature and density were treated as free parameters, determined by comparing the model predictions to experimental results. Long et al. [17] established a two-stage model, with the first stage being discharge excitation and the second stage optical pumping. The ground state and five excited levels were considered. Rate coefficients for electron collision processes were obtained using the BOLSIG + computational code [24]. The optical excitation and lasing kinetics were simulated using the model of Yang et al. [6].

The models presented in references [22] and [17] were developed for steady-state conditions. However, the most successful approach (to date) for the generation of a larger volume plasma relied on a repetitively pulsed discharge. Clearly, time-dependent models are required for analyses of lasers that use this type of discharge. Other limitations of the previous models include longitudinal averaging of the radiation fields and species number densities. In this paper, we present time-dependent models of a longitudinally pumped laser in zero-dimension and one spatial dimension that include pulsed discharge excitation, species densities, optical pumping and laser output. The models are evaluated by comparing to the experimental results of Han et al. [2].

2. Model description

2.1. Zero-dimensional model

The ZDPlasKin (Zero-Dimensional Plasma Kinetics) [25] software package was used to simulate the time evolution of the species densities in the pulsed discharge based on the equations:

a A+ b B \to a'A+ c C [+ δε]

\frac{d n_{i}}{d t} = \sum_{j = 1}^{j \max} Q_{i j} (t)

R_{j} = k_{j} {[n_{A}]}^{a} {[n_{B}]}^{b}

Q_{A_{j}} = (a' - a) R_{j}, Q_{B_{j}} = - b R_{j}, Q_{C_{j}} = c R_{j}

where (1) represents a reaction indexed by j that involves species A, B and C. δε is the energy gained or lost in the reaction. n_i is density of species i, with i = 1-18 corresponding to Ar(1s₅), Ar(2p₁₀), Ar(2p₉), Ar(1s₄), Ar(2p₈), e^-, Ar, Ar⁺, Ar₂⁺, Ar₂*, Ar(hl), He, He⁺, He*, He₂⁺, He₂*, HeAr⁺ and e^-(W), in which the last entry represents electrons removed through ambipolar diffusion to the walls. Ar(hl) represents all energy levels of Ar above 2p₉. It is assumed that the cross-section for excitation of Ar(3d₆) can be applied for all excitations to the ensemble of all higher energy levels. k_j is the rate coefficient, R_j is the rate, and Q_ij is the source term for species i corresponding to the contribution from reaction j. The gas temperature is considered to be constant in our model (δε terms were not included). The DVODE solver within ZDPlasKin carries out the time integration of rate equations. The Boltzmann equation solver (BOLSIG + ) [24] generates the electron temperature and reaction rate coefficients using the updated reduced electric field E/N (where E is the electric field and N is the total particle density) for each time step. The electron impact reaction cross sections used for these calculations were downloaded from LXCat (https://fr.lxcat.net/home/). The ZDPlasKin and BOLSIG + programs are widely used for plasma simulations and considered to be reliable by the plasma physics community at large. The simulations carried out in the present study yielded physically reasonable results, indicating that the communication between the ZDPlasKin and BOLSIG + programs was working correctly. We did not attempt to make an independent assessment of the validity of these codes.

The discharge system was represented by a simple circuit with a voltage supply, a ballast resistor and a pair of parallel plate electrodes, as shown in Fig. 2. Electrons were considered as the only current carriers. The voltage applied across the positive column V_pc was approximated by the expression

V_{p c} = V_{0} - e \cdot R \cdot A_{g} \cdot n_{e} \cdot v_{d r} - V_{c}

where V₀ is the voltage provided by the generator, e is the elementary charge, R is the resistance of the ballast resistor, A_g is the electrode area, n_e is the electron density, v_dr is electron drift velocity and V_c is the cathode fall voltage. The reduced electric field is updated using the relationship

\frac{E}{N} = \frac{V_{p c}}{N \cdot d_{g}}

, where E is electric field in the transverse direction, N is total density of neutral species, and d_g is the electrode gap.

Fig. 2 Equivalent electrical circuit for the pulsed discharge.

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The set of discharge reactions and their rate coefficients were mainly derived from Emmons and Weeks [5]. This set includes electron impact, recombination, two-heavy-body, three-heavy-body and radiative reactions. A few changes were made to adapt the model to the specific laser system under consideration. Optical pumping and stimulated emission processes were included. We used Arrhenius temperature scaling of the rate constants for simulations at different gas temperatures for the Ar(1s₄) + M → Ar(1s₅) + M and Ar(2p₉) + M → Ar(2p₁₀) + M, M = Ar or He transfer processes. The scaling relationships used are given in Table 1.

Table 1. Arrhenius equations for temperature dependent energy transfer rate constants

View Table

The rate of ambipolar diffusion loss R_a [21] was calculated using the expression

R_{a} \approx \frac{μ_{+} T_{e}}{Λ^{2}}

where μ₊ is the ion mobility, T_e is the electron temperature, and Λ is the characteristic length defined by d_g/π. The mobility of Ar⁺ was calculated by Blanc's law [26], and it was assumed that the mobilities of

{Ar}_{2}^{+}

and HeAr⁺ could be approximated by the same value. The mobilities of He⁺ and

{He}_{2}^{+}

were set to 10.7 and 16.2 cm²/V∙s, respectively [27]. The rate coefficient for the decay of radiatively trapped Ar(1s₄) → Ar(¹S₀) was set to 5.6 × 10⁵ s⁻¹ [7].

Standard methods for calculating beam intensities, and rates of pump and laser transitions in a longitudinally pumped double-pass configuration were applied [21]. Non-stationary solutions were sought as the densities of Ar(1s₅), Ar(2p₉) and Ar(2p₁₀) were changing with time due to discharge excitation, optical pumping and laser transitions. The average intracavity laser intensity (I_ave) and rates for the pump transition (W_pump) and laser transition (W_laser) were also time-dependent. These variables were determined by

\begin{matrix} W_{p u m p} = η \int d ν \cdot \frac{I_{p i n} (ν)}{l_{g} \cdot h ν_{p}} {1 - \exp [σ_{p} (ν) \cdot (n_{3} - \frac{g_{3}}{g_{1}} n_{1}) \cdot l_{g}]} \\ \cdot {t_{p} + t_{p}^{3} r_{p} \cdot \exp [σ_{p} (ν) \cdot (n_{3} - \frac{g_{3}}{g_{1}} n_{1}) \cdot l_{g}]} \end{matrix}

W_{l a s e r} = σ_{l} (ν_{l}) \cdot \frac{(n_{2} - \frac{g_{2}}{g_{1}} n_{1})}{h ν_{l}} \cdot I_{a v e}

\frac{d I_{a v e}}{d t} = {t_{s} t_{l}^{4} r_{l} \cdot \exp [2 l_{g} \cdot σ_{l} (ν_{l}) \cdot (n_{2} - \frac{g_{2}}{g_{1}} n_{1})] - 1} \cdot I_{a v e} \cdot \frac{c}{2 l_{c}} + \frac{n_{2} \cdot c^{2} \cdot σ_{l} (ν_{l}) \cdot h ν_{l}}{S \cdot l_{g}}

Where η represents the spatial overlap efficiency of the pump and laser beams, I_pin(ν) is the pump intensity that enters the laser cavity, l_g is the length of gain medium, and hν_p and hν_l are photon energies of the pump and laser radiation. The output beam was assumed to be single mode. Ϭ_p(ν_p) and Ϭ_l(ν_l) are cross sections of the pump and laser transitions. Collisional broadening coefficients (He at 300 K) for the pump and laser transitions were 3.2 × 10⁻¹⁰ cm³s⁻¹ [18] and 7.5 × 10⁻¹⁰ cm³s⁻¹ [3], respectively. n_1, n₂, n₃ are the densities of Ar(1s₅), Ar(2p₁₀) and Ar(2p₉), and g₁, g₂, g₃ are degeneracies of these levels. r_p is the output coupler (OC) reflectivity at the pump wavelength, r_l is OC reflectivity at the lasing wavelength, and l_c is the resonator cavity length. t_p and t_l are single-pass discharge cell window transmissions for the pump and laser wavelengths, respectively. Other intra-cavity single-pass losses, including the scattering loss, were assumed to be located at the high reflector end with transmittance t_s. Due to the strong divergence of diode pump beam in the experiment [2], r_p was assumed to be zero. The absorbed pump intensity I_abs and output laser intensity I_out were calculated by

I_{a b s} = η \int d ν \cdot I_{p i n} (ν) \cdot {1 - t_{p}^{2} \cdot \exp [σ_{p} (ν) \cdot (n_{3} - \frac{g_{3}}{g_{1}} n_{1}) \cdot l_{g}]}

I_{o u t} = \frac{I_{a v e} \cdot t_{l} \cdot σ_{l} (ν_{l}) \cdot (n_{2} - \frac{g_{2}}{g_{1}} n_{1}) \cdot l_{g} \cdot (1 - r_{l}) \cdot \exp [σ_{l} (ν_{l}) \cdot (n_{2} - \frac{g_{2}}{g_{1}} n_{1}) \cdot l_{g}]}{{\exp [σ_{l} (ν_{l}) \cdot (n_{2} - \frac{g_{2}}{g_{1}} n_{1}) \cdot l_{g}] - 1} \cdot {1 + t_{l}^{2} r_{l} \cdot \exp [σ_{l} (ν_{l}) \cdot (n_{2} - \frac{g_{2}}{g_{1}} n_{1}) \cdot l_{g}]}}

2.2. One-dimensional model

In the one-dimensional model, y mesh elements of equal length were introduced longitudinally, labeled as m = 1 to m = y. Calculations with differing values for y indicated that adequately converged results could be obtained for y>40. A value of y = 50 was adopted as a viable compromise between accuracy and computational run-time. The zero-dimensional ZDPlasKin model [25] was applied within each mesh element. Species densities and intensities of pump and laser radiation within each segment were calculated for each time step:

dt = \frac{l_{g}}{y \cdot c}

In the zero-dimensional model, electron density, drift velocity and electron temperature exhibited no significant differences before and after optical pumping, so in each mesh element they were assumed equal to those in the mid-point element, y/2.

Figure 3 shows the propagation direction and mesh elements of the one-dimensional model. t_os, equal to $\frac{l_{c} - l_{g}}{c} + dt$ , is the time it takes the rightward laser light to propagate from the right boundary of mesh = y-1, reflect (turns to leftward propagation), and then to the right boundary of mesh = y. The time interval is the same for the equivalent process at the left side for light propagating from the left boundary of mesh m = 2 and returning to the left boundary of m = 1. In mesh element m, I^R(t,m) represents rightward laser intensity at the left boundary, I^L(t,m) represents leftward laser intensity at the right boundary, and n(t,m) represents species density for the whole element.

Fig. 3 Schematic diagram showing propagation directions and mesh elements.

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Based on light amplification principles [28], the pump and laser transition rates in mesh element m can be derived as follows:

\begin{matrix} W_{p u m p} (t, m) = \int \frac{σ_{p} (ν)}{h ν_{p}} [n_{3} (t - dt, m) - \frac{g_{3}}{g_{1}} \cdot n_{1} (t - dt, m)] \cdot \\ {\frac{[I_{P}^{R} (t - dt, ν, m) + I_{P}^{R} (t - dt, ν, m + 1)]}{2} + \frac{[I_{P}^{L} (t - dt, ν, m) + I_{P}^{L} (t - dt, ν, m - 1)]}{2}} \cdot d ν \end{matrix}

\begin{matrix} W_{l a s e r} (t, m) = \frac{σ_{l} (ν_{l})}{h ν_{l}} [n_{2} (t - dt, m) - \frac{g_{2}}{g_{1}} \cdot n_{1} (t - dt, m)] \cdot \\ {\frac{[I_{L}^{R} (t - dt, ν, m) + I_{L}^{R} (t - dt, ν, m + 1)]}{2} + \frac{[I_{L}^{L} (t - dt, ν, m) + I_{L}^{L} (t - dt, ν, m - 1)]}{2}} \end{matrix}

where

I_{P}^{R}, I_{L}^{R}, I_{L}^{L}

indicate local intensities of rightward pump, rightward laser and leftward laser beams. They are determined by:

\begin{matrix} I_{P}^{R} (t, ν, m) = {σ_{P} (ν) [n_{3} (t - dt, m - 1) - \frac{g_{3}}{g_{1}} \cdot n_{1} (t - dt, m - 1)] \\ \cdot c \cdot dt + 1} \cdot I_{P}^{R} (t - dt, ν, m - 1) \end{matrix}

\begin{matrix} I_{L}^{R} (t, m) = {σ_{l} (ν_{l}) [n_{2} (t - dt, m - 1) - \frac{g_{2}}{g_{1}} \cdot n_{1} (t - dt, m - 1)] \cdot c \cdot dt + 1} \cdot \\ I_{L}^{R} (t - dt, m - 1) + \frac{n_{2} (t - dt, m - 1) \cdot c^{2} \cdot σ_{l} (ν_{l}) \cdot h ν_{l}}{S \cdot l_{g}} \cdot dt \end{matrix}

\begin{matrix} I_{L}^{L} (t, m) = {σ_{l} (ν_{l}) [n_{2} (t - dt, m + 1) - \frac{g_{2}}{g_{1}} \cdot n_{1} (t - dt, m + 1)] \cdot c \cdot dt + 1} \cdot \\ I_{L}^{L} (t - dt, m + 1) + \frac{n_{2} (t - dt, m + 1) \cdot c^{2} \cdot σ_{l} (ν_{l}) \cdot h ν_{l}}{S \cdot l_{g}} \cdot dt \end{matrix}

Where S is the cross-sectional area of the laser beam. Note that the equations for the power densities at m = 1 and m = 50 are different from the above expressions due to transit of the region outside of the gain medium:

I_{P}^{R} (t, ν, 1) = t_{p} \cdot I_{p i n} (t - \frac{l_{c} - l_{g}}{2 c}, ν)

\begin{matrix} I_{L}^{R} (t, 1) = t_{s} t_{l}^{2} ({σ_{l} (ν_{l}) [n_{2} (t - t_{os}, 1) - \frac{g_{2}}{g_{1}} \cdot n_{1} (t - t_{os}, 1)] \cdot c \cdot dt + 1} \cdot \\ I_{L}^{L} (t - t_{os}, 1) + \frac{n_{2} (t - t_{os}, 1) \cdot c^{2} \cdot σ_{l} (ν_{l}) \cdot h ν_{l}}{S \cdot l_{g}} \cdot dt) \end{matrix}

\begin{matrix} I_{L}^{L} (t, y) = r_{l} t_{l}^{2} ({σ_{l} (ν_{l}) [n_{2} (t - t_{os}, y) - \frac{g_{2}}{g_{1}} \cdot n_{1} (t - t_{os}, y)] \cdot c \cdot dt + 1} \cdot \\ I_{L}^{R} (t - t_{os}, y) + \frac{n_{2} (t - t_{os}, y) \cdot c^{2} \cdot σ_{l} (ν_{l}) \cdot h ν_{l}}{S \cdot l_{g}} \cdot dt) \end{matrix}

where I_pin is the input pump intensity. Then the absorbed pump intensity I_Pabs(t) and output laser intensity I_Lout(t) are calculated using the equations:

\begin{matrix} I_{P a b s} (t) = \int (I_{p i n} (t - \frac{l_{c}}{c}, ν) - \\ {σ_{P} (ν) [n_{3} (t - \frac{(l_{c} - l_{g})}{2 c} - dt, y) - \frac{g_{3}}{g_{1}} n_{1} (t - \frac{l_{c} - l_{g}}{2 c} - dt, y)] \cdot \\ c \cdot dt+1} \cdot t_{p} \cdot I_{P}^{R} (t - \frac{(l_{c} - l_{g})}{2 c} - dt, ν, y)) \cdot d ν \end{matrix}

\begin{matrix} I_{L o u t} (t) = (1 - r_{l}) \cdot t_{l} \cdot ({σ_{l} (ν_{l}) [n_{2} (t - \frac{(l_{c} - l_{g})}{2 c} - dt, y) - \\ \frac{g_{2}}{g_{1}} n_{1} (t - \frac{(l_{c} - l_{g})}{2 c} - dt, y)] \cdot c \cdot dt+1} \cdot I_{L}^{R} (t - \frac{(l_{c} - l_{g})}{2 c} - dt, y) \\ + \frac{n_{2} (t - \frac{(l_{c} - l_{g})}{2 c} - dt, y) \cdot c^{2} \cdot σ_{l} (ν_{l}) \cdot h ν_{l}}{S \cdot l_{g}} \cdot dt) \end{matrix}

Equations (13) and (14) define the rates corresponding to the transitions between lower and upper energy levels through absorption and stimulated emission. The rates are proportional to the number densities in related levels and incident radiation coming from both rightward and leftward directions. Equations (15)–(17) represent the energy change of radiation passing through a mesh element. This is composed of the energy released through stimulated and spontaneous emission, and the energy absorbed through excitation. The effect of spontaneous emission, presented as the second term in (16) and (17) is key to initiate the build-up of laser emission. It is neglected for the pump transition, since the loss of population of the upper level is dominated by stimulated emission and collisional energy transfer.

3. Comparison with experimental results

In pulsed discharge experiments that used 21 W pump power [2], the maximum laser output power realized was 4.1 W. Using 19.7 W pump, the output power was 3.8 W with an optical-to-optical power conversion efficiency of 31%. The Ar(1s₅) density was estimated to be on the order of 10¹³ cm⁻³ before optical excitation [2]. To test the validity of our models, the parameters were set close to the experiment values. The pump power was set to 21 W with estimated beam areas of 2, 2.5 or 3 mm² (difficult to determine for the experiment due to the irregular beam shape and anisotropic divergence, combined with unsophisticated focusing optics). For the lineshape of the pump laser, the full width at half maximum for the Gaussian profile was set to 30 GHz. The optical cavity for the Ar* laser was 30 cm long, with a 40% reflective OC. The discharge voltage was 1500 V amplitude, 80 ns duration, and 200 kHz repetition frequency. This pulse train was applied to electrodes of 3 cm length, 0.63 cm wide, with a 0.26 cm gap. We used a triangle voltage profile in simulations, with 40 ns rise from 0 V to 1500 V and 40 ns decay from 1500 V to 0 V, approximating the rising and trailing edges of the voltage pulse. 80% of the supply voltage was used to take into account discharge power loss that resulted in heat dissipation. The cathode fall voltage was calculated to be approximately 160 V. The gas medium was 720 Torr of a 5.3% Ar/He mixture. A 75 Ω ballast resistor was used (Eq. (5)). An optical mode matching efficiency of η = 0.95 was assumed. Single-pass cell window transmissions for pump and laser wavelengths were assumed to be 99%, and other single-pass losses were set at 5%. A temperature shift of roughly 300K was theoretically possible [3], and also consistent with the experimental observations. The microdischarge laser reported in [3] was estimated to be operating with a gas temperature near 600K. The temperature range for the pulsed discharge set-up used for the 4.1 W laser [2] was investigated by means of emission spectroscopy. A trace of N₂ was added to the gas flow and the emission bands of the N₂ C-B transition were recorded at the level of rotational resolution. The local gas temperature was determined from the rotational line intensity distributions. Values of 600 ± 100 K were obtained for the typical lasing conditions. In addition, we have used the ZDPlasKin model to predict the gas temperature in the discharge. In these calculations it was assumed that the dominant heating process was inelastic collisions of the electrons with He. Hence, the rate of thermal energy deposition was calculated using the BOLSIG + package. Heat loss was assigned to the gas flow through the system (at a measured flow rate of 87 cm³ s⁻¹). The balance of these terms, calculated for the 1500 V discharge at 720 Torr, predicted a local gas temperature of 540 K.

Simulations of the discharge and laser performance at gas temperatures of 300, 400, 500 and 600 K were carried out. Results from the zero-dimensional model (average values of periodic properties) are presented in Fig. 4. These simulations, with gas temperatures of 300-600 K and beam areas of 2.0-3.0 mm², were in satisfactory agreement with the experimental results. At 500K, the discharge system can produce an Ar(1s₅) density of 1.78 × 10¹³ cm⁻³ before optical pumping. With optical pumping the output power ranges from 3.58 to 5.04 W, with optical-to-optical power conversion efficiencies of 29.8% to 43.4%. We also used the zero-dimensional model to examine the lower limit for the discharge pulse frequency that could maintain continuous lasing. This was predicted to be 90 kHz, determined by the overall decay of the 1s₅ population.

Fig. 4 (a) Averaged Ar(1s₅) density before optical excitation, (b) averaged output laser power and (c) conversion efficiency. Temperature: 300K-600K. Beam area: 2-3 mm² in (b) and (c). (zero-dimensional model)

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As noted above, some experimental parameters for the CW Ar laser have considerable uncertainties (e.g., pump beam focus diameter and the local gas temperature) and the temperature dependencies of the energy transfer rate constants are not well characterized. As a consequence, the present model is constrained by optimizing the beam focus and gas temperature, with specific assumptions concerning rate constant temperature dependencies (c.f. Table 1), against the available experimental data. Future experimental studies will be conducted to reduce the parameter uncertainties and measure rate constants at elevated temperatures. Modeling can provide guidance as to which parameters are most important. The present intention is to explore the question of whether the proposed models have included the most significant physical processes, as indicated by the ability to replicate the available lasing data using physically reasonable parameter values.

Using the one-dimensional model, 4.7 W laser output was predicted with 40.3% efficiency at 500K and 2.0 mm² beam area, and 3.1 W with 25.4% efficiency at 500K and 3.0 mm² beam area. The one-dimensional model was also tested by comparison with the time-resolved data for the laser output (Fig. 3. of [2]). In the experiment, changing the voltage amplitude to 1300 V, pressure to 710 Torr, OC to 60%, and length of electrodes to 2.5 cm, resulted in an averaged output power of 3.4 W. We achieved a simulated result of 3.6 W in good agreement with the experiment for the stated parameters used in our one-dimensional program. It was assumed that the temperature was approximately 500 K and the beam area was 3.0 mm². Time-resolved absorbed pump power and laser power curves are shown in Fig. 5, along with the temporal behavior of the Ar(1s₅) density and intracavity laser power for the mesh elements m = 1, y/2 and y followed over two discharge periods. Figure 5(a) shows that pump intensity has not reached saturation. Here it can be seen that the laser intensity does not follow the temporal profile of the absorbed pump power in the one-dimensional model. Apart from the fast transients in the laser output due to the discharge pulse, evident just above 0 and 5000 ns, the laser output reaches a maximum at about 2200 ns after each pulse. As the Ar 1s₅ number density reaches a maximum much earlier in the discharge cycle (around 100 ns) the fact that this does not correspond to the lasing maximum shows that the pump does not have enough intensity to invert the full length of the gain medium when the metastable concentration is at its highest values. In Fig. 5(b) the Ar (1s₅) density is lower in the first mesh element, as compared to the densities in elements y/2 and y, due to the more intense optical pumping.

Fig. 5 Temporal behavior of (a) absorbed pump power and output laser power (zero and one-dimensional models), (b) Ar(1s₅) density in mesh elements m = 1, y/2 and y (one-dimensional model), and (c) intracavity laser power in mesh elements m = 1, y/2 and y (one-dimensional model). The time between successive 80 ns discharge pulses is 5000 ns.

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Another consequence of the low pump intensity is the prediction that the rightward going laser intensities at mesh points y/2 and y are quite similar. This occurs because the pump beam arriving in this region has sufficient intensity to induce transparency but not gain. This is also evident in the comparable values for the leftward intensities at mesh points y/2 and y.

Laser power predictions from the one-dimensional model are lower than the zero-dimensional model for certain situations. Non-uniform distribution of species densities and intracavity powers after pumping process indicates the limitations of stationary solutions in zero-dimensional models for simulation of this experiment. However, as the pump power and discharge pulse duration increase, zero- and one-dimensional models tend to give closer temporal calculation results.

4. Conclusions

Zero- and one-dimensional time-dependent models of OPRGLs are established. Simulation results show quantitative agreement with the data from a previous pulsed-discharge CW laser experiment [2]. Inaccuracies still exist, which can be caused by neglect of non-uniform distributions of Ar(1s₅) (in the transverse direction) and gas temperature resulting from the discharge, pump laser power deposition, and poorly defined rate constants for certain reactions. Both of the models presented here can realize temporal simulations of pump and laser powers and species densities, going beyond the limitations of stationary-state solutions. The one-dimensional model removes the approximation of a longitudinally averaged double-pass scheme for dynamic analysis. The level of agreement with the experimental results of [2] indicates that this model can be used to explore the power scaling capabilities of OPRGLs that utilize pulsed discharge Rg metastable production. The present calculations also show that the laser power and efficiency observed in the experiments reported in [2] was limited by the intensity available from the pump laser. Scaling of the model to predict the performance of a kW class laser appears to be feasible and the report of a pulsed Ar laser operating at 27 kW cm⁻² [1] implies that high-power CW OPRGL’s can be constructed.

Funding

Army Research Office (W911NF-17-1-042721); Chinese Scholarship Council (CSC) Scholarship; Ministry of Education and Science of the Russian Federation (3.1715.2017/4.6),

Acknowledgments

Mr. Sun is a PhD candidate at Huazhong University of Science and Technology and a visiting student at Emory University.

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Reaction	Rate Constant/10⁻¹¹ cm³ s⁻¹
	(Temperature in K)

Ar(1s₄) + He → Ar(1s₅) + He	4.3xExp(−1084/T)
Ar(1s₄) + Ar → Ar(1s₅) + Ar	4.3xExp(−2287/T)
Ar(2p₉) + He → Ar(2p₁₀) + He	30xExp(−879/T)
Ar(2p₉) + Ar → Ar(2p₁₀) + Ar	30xExp(−734/T)

Time-dependent simulations of a CW pumped, pulsed DC discharge Ar metastable laser system

Abstract

1. Introduction

2. Model description

2.1. Zero-dimensional model

2.2. One-dimensional model

3. Comparison with experimental results

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (5)

Tables (1)

Equations (22)

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