Two-stage excitation model of diode pumped rare gas atoms lasers

Sichen Long; Yingxiong Qin; Hanyuan Chen; Xiangxi Wu; Meigui Li; Xiahui Tang; Tao Wen

doi:10.1364/OE.27.002771

1. Introduction

In 2012, as a new two-stage excitation laser, a diode pumped rare gas atoms laser (DPRGL) [1] was proposed and demonstrated by Han and Heaven et al. The level structure and dynamic process of DPRGLs are analogous to those of diode pumped alkali metal lasers (DPALs). Particles involved in the laser cycle are produced by gas discharge (the first stage excitation). As the lower pump level, metastable 1s₅ state (Paschen notation) is excited to 2p₉ state by the pump light (the second stage excitation). Then, population inversion is produced by collisional relaxation from 2p₉ state to 2p₁₀ state. Finally lasing process occurs from 2p₁₀ state to metastable state.

DPRGLs overcome the shortcomings of DPALs, and combine the advantages of solid lasers and gas lasers [1]: (1) high quantum efficiency of ~90%; (2) chemical inertness of rare gases that could not react with the buffer gas; (3) excellent beam quality; (4) high optical conversion efficiency, which means DPRGLs are potential high-energy laser candidates.

Based on the advantages, DPRGLs have attracted the attention of many researchers around the world [1–6]. In 2015, Rawlins et al. realized the first cw DPRGLs output by using linear micro-discharge array. Stable and cw laser oscillation at 912.3 nm was observed in the experiment with the optical conversion efficiency of 55% [6].

At the same time, researchers have developed the dynamic process of DPRGLs [7–12]. In 2013, Demyanov et al. theoretically put forward the three-level model for cw DPRGLs systems. This model makes known that optical conversion efficiency reaches ∼60% at the optimal temperature of 300K, Ar content of 1%, and the pressure near atmospheric [7]. In 2015, a five-level longitudinal model of DPRGLs was proposed by Yang et al. Results show that the optical conversion efficiency can reach 50-60%, and the relatively low relaxation collision rate between 1s₄ and 1s₅ levels can be compensated by increasing the concentration of 1s₅ state [8]. On the basis of Yang et al.’s model, the transverse model of DPRGLs was developed by Gao et al [10]. In 2018, Eshel et al. developed the discharge model of DPRGLs system and established dependence of the metastable Ar (1s₅) density on gas pressure, gas temperature, electron density and electron temperature. This discharge model characterized the necessary geometric size for a high-energy DPRGL system of 100 kW output [11]. At the same year, the model of optically pumped Xe lasers was put forward, of which the results showed that the ternary mixtures Xe: Ar: He have the potential for high total efficiency [12].

In DPRGLs system, metastable atoms are produced by gas discharge [13–15]. Massines et al. reported that the reduced electric field is ~7.5 Td under the dielectric barrier discharge in He [23]. Mikheyev et al. researched the dielectric barrier discharge in rare gas mixtures and found that the reduced electric field is ~9.4 Td in Xe and He mixture at the pressure of 600 Torr and room temperature [13,14]. In fact, discharge conditions cannot only influence the excitation efficiency of excited states, but also indirectly affect the optical conversion efficiency by change of concentration of metastable atoms. In order to achieve peak total efficiency, it is important to adjust parameters to balance discharge excitation efficiency and optical conversion efficiency.

In this paper, a complete dynamic model of DPRGLs systems is established, including discharge dynamics and stimulated radiation, taking Ar-He gas mixture as an example. The aim of our model is to optimize total efficiency rather than optical conversion efficiency, and to obtain energy loss mechanism. A comparison with Rawlins et al.’s experimental results is analyzed to verify the validity of our model. Parameters of DPRGLs systems are optimized, through the numerical simulation and analysis of total efficiency and output power density. Finally, our model provide the optimal design scheme of the DPRGLs system at pressures of 0.5 atm and above 1 atm.

2. Model description

The energy levels and the kinetic processes of DPRGL systems are illustrated in Fig. 1 [16]. In our two-stage excitation model, a longitudinally pumped rare gas laser is considered, following Yang et al.’s model [8]. The two-stage excitation means that the first stage is discharge excitation with the red dotted line, and the second stage is diode pump with the red solid line. In this paper, the subscripts i, j = 1, 2, 3, 4, 5 represent the 1s₅, 1s₄, 2p₁₀, 2p₉, 2p₈ levels respectively, corresponding to the five levels involved in the particle cycle. The subscript i = 0 denotes the ground state of Ar atom. n_i and g_i are the number density and degeneracy of the i state, respectively. A_ij represents spontaneous emission rate of transitions from the i state to the j state. k_ij and k_ij^d are rate coefficients from the i state to the j state by collisional relaxation and discharge, respectively. S_p denotes pump rate and S_out denotes induced radiation rate.

Fig. 1 The energy levels and the kinetic processes for two-stage excitation systems.

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For the first stage excitation, we assume that the five levels (1s₅, 1s₄, 2p₁₀, 2p₉, 2p₈) can be considered as a whole. Under the steady-state condition, the particles formation rate is equal to their loss rate of the whole. The population of the five-level whole can be generated by discharge excitation. While in the population cycle, there are three main loss processes in the whole: (1) a formation of excimer Ar₂*; (2) the trapped radiation from the 1s₄ state to the ground state; (3) the spontaneous emission processes from 2p₁₀ state to 1s₂, 1s₃ state, and from 2p₈ state to 1s₂ state. For a conservative estimation, the inverse processes of these loss processes and subsequent population transfer are ignored.

In the next few sections, we first analyze the discharge mechanism (Section 2.1) and the optical mechanism (Section 2.2) separately. Then, the rate equations including the two-stage excitation are established (Section 2.3). Finally, the influence of reduced electric field E/N, gas pressure P, Ar fraction y_Ar on total efficiency η_tot is determined (Section 2.4).

2.1 Discharge excitation mechanism

In order to obtain excitation efficiency and rate coefficients of the five levels in continuous discharge, the software BOLSIG + is used [17]. The Boltzmann equation for electron energy distribution function in two-term approximation is solved where elastic scattering, excitation and ionization processes by collisions with electrons are considered. Our model assumes that the electric field and the collision probabilities of reaction processes are spatially uniform. The cross sections for elastic and inelastic collisions for Ar are quoted from [18], and for He are quoted from [19]. After the electron energy distribution functions (EEDF) are calculated, rate coefficients of the related energy levels for electron collision can be obtain by [20]:

k_{0 i}^{d} = \sqrt{\frac{2 e}{m}} \int_{0}^{\infty} ε σ_{i} (ε) f (ε) d ε (i = 1, 2, 3, 4, 5),

where k₀_i^d is the rate coefficient for the electron collision reaction from ground state to the i state and the superscript d represents discharge processes. e/m is the charge to mass ratio of the electron, ε is electron energy in electronic volts, f (ε) is electron energy distribution function, and σ_i (ε) is cross sections of the i state. Major determinants of EEDF are the parameters such as Ar fraction, electron density and reduced electric field E/N, where E is the electric field and N is the neutral gas density. Through the detailed balance, the rate coefficients for their reverse processes are determined.

2.2 Stimulated emission

In this model, the longitudinal pump scheme is used and the pump rate is described as [21]:

\begin{matrix} S_{p} = \frac{η_{d e l} η_{mode}}{l_{g}} \int d λ \frac{1}{h ν_{p}} I_{p} g_{p} (λ) \cdot {1 - T_{p}^{2} \exp [- (n_{1} - \frac{g_{1}}{g_{4}} n_{4}) σ_{14}^{b r o a d e n e d} (λ) l_{g}]} \\ \cdot {1 + R_{p} T_{p}^{2} \exp [- (n_{1} - \frac{g_{1}}{g_{4}} n_{4}) σ_{14}^{b r o a d e n e d} (λ) l_{g}]}, \end{matrix}

where η_del is pump delivering factor, η_mode is mode overlapping factor and these two parameters are set as 1 in our simulation. h is Planck constant, ν_p is pump frequency, I_p is pump intensity, T_p is pump transmission of the cell window, and l_g is the length of the gain medium. R_p is pump reflectivity of the output coupler. If the pump reflectivity R_p is set as 0, our pump scheme is simplified to a single-pass scheme. g_p (λ) is the line shape of pump laser, which is assumed to be Gauss shape in our simulation. σ₁₄^broadened (λ) is the spectrally resolved stimulated absorption cross section, and the expression is [7]:

σ_{14}^{b r o a d e n e d} (λ) = \frac{g_{4}}{g_{1}} \frac{A_{41} λ_{41}^{2}}{8 π} g_{L} (λ) = \frac{g_{4}}{g_{1}} \frac{A_{41} λ_{41}^{2}}{8 π^{2}} \frac{\frac{Δ v_{41}}{2}}{{[(\frac{c}{λ}) - (\frac{c}{λ_{41}})]}^{2} + {(\frac{Δ v_{41}}{2})}^{2}},

where c is the speed of light, λ₄₁ = 811.5 nm and g_L(λ) are central wavelength and line shape for the absorption transition. At the atmospheric pressure, collision broadening is dominant, so the Lorentzian line shape is applied to our calculation. Δν₄₁ is the full-width at half-maximum (FWHM) of the absorption line obtained by Δν₄₁ = 2Nξ₄₁ where ξ₄₁ is pressure broadening coefficient. For the 1s₅ → 2p₉ transition, the pressure broadening coefficients are 2.8 × 10⁻¹⁰ s⁻¹cm³ (300 K) for Ar [22] and 3.2 × 10⁻¹⁰ s⁻¹cm³ (300 K) for He [22].

The expression of the laser rate is give below [21]:

S_{o u t} = \frac{1}{l_{g}} \frac{I_{o u t}}{h ν_{L}} \frac{R_{O C} T_{L}}{1 - R_{O C}} \cdot {\exp [(n_{3} - \frac{g_{3}}{g_{1}} n_{1}) σ_{31} l_{g}] - 1} \cdot {1 + T_{L}^{2} R_{L} \exp [(n_{3} - \frac{g_{3}}{g_{1}} n_{1}) σ_{31} l_{g}]},

where ν_L is laser frequency, I_out is laser intensity, T_L is laser transmission of the cell window and R_L and R_OC are laser reflectivities of back reflector and output coupler, respectively. For the 2p₁₀ → 1s₅ transition, the pressure broadening coefficients are 1.5 × 10⁻¹⁰ s⁻¹cm³ (300 K) for Ar [8] and 6.2 × 10⁻¹⁰ s⁻¹cm³ (300 K) for He [13].

2.3 Rate equations

In this model, the rate equations for two-stage excitation systems of DPRGLs are as follows:

\begin{matrix} \frac{d n_{1}}{d t} = - S_{p} + S_{o u t} + k_{21} N [n_{2} - \frac{g_{2}}{g_{1}} n_{1} \exp (- \frac{Δ E_{21}}{k T})] + n_{3} A_{31} + n_{3} k_{31} N + n_{4} A_{41} \\ + n_{4} k_{41} N + n_{5} A_{51} + n_{5} k_{51} N - n_{1} k_{{Ar}_{2}^{*}} N^{2} y_{Ar} + k_{01}^{d} n N - k_{10}^{d} n n_{1}, \end{matrix}

\frac{d n_{2}}{d t} = n_{3} A_{32} + n_{5} A_{52} - k_{21} N [n_{2} - \frac{g_{2}}{g_{1}} n_{1} \exp (- \frac{Δ E_{21}}{k T})] - n_{2} A_{20} + k_{02}^{d} n N - k_{20}^{d} n n_{2},

\begin{matrix} \frac{d n_{3}}{d t} = - S_{o u t} + k_{43} N [n_{4} - \frac{g_{4}}{g_{3}} n_{3} \exp (- \frac{Δ E_{43}}{k T})] + k_{53} N [n_{5} - \frac{g_{5}}{g_{3}} n_{3} \exp (- \frac{Δ E_{53}}{k T})] \\ - n_{3} A_{31} - n_{3} k_{31} N - n_{3} A_{32} - n_{3} A_{p 10 - s3} - n_{3} A_{p 10 - s 2} + k_{03}^{d} n N - k_{30}^{d} n n_{3}, \end{matrix}

\begin{matrix} \frac{d n_{4}}{d t} = S_{p} - k_{43} N [n_{4} - \frac{g_{4}}{g_{3}} n_{3} \exp (- \frac{Δ E_{43}}{k T})] + k_{54} N [n_{5} - \frac{g_{5}}{g_{4}} n_{4} \exp (- \frac{Δ E_{54}}{k T})] \\ - n_{4} A_{41} - n_{4} k_{41} N + k_{04}^{d} n N - k_{40}^{d} n n_{4}, \end{matrix}

\begin{matrix} \frac{d n_{5}}{d t} = - k_{54} N [n_{5} - \frac{g_{5}}{g_{4}} n_{4} \exp (- \frac{Δ E_{54}}{k T})] - k_{53} N [n_{5} - \frac{g_{5}}{g_{3}} n_{3} \exp (- \frac{Δ E_{53}}{k T})] \\ - n_{5} A_{51} - n_{5} k_{51} N - n_{5} A_{52} - n_{5} A_{p 8 - s 2} + k_{05}^{d} n N - k_{50}^{d} n n_{5}, \end{matrix}

\frac{d I_{o u t}}{d t} = (R_{O C} R_{L} T_{L}^{4} \exp [σ_{31} (n_{3} - \frac{g_{3}}{g_{1}} n_{1}) 2 l_{g}] - 1) \frac{I_{o u t} c}{2 l_{c}} + Δ I_{s p o},

where ΔE_ij is energy difference between the i state and the j state, k is the Boltzmann constant, T is gas temperature, n is electron density and l_c is the length of the resonant cavity. ΔI_spo describing the contribution of spontaneous emission is neglected in our calculation. The reaction processes and their rate constants considered in the rate equations are shown in Table 1. The neutral collision processes and rate coefficients are adopted from the Yang’s model [8]. Two different gas atoms are involved in the collisional relaxation processes, therefore, the Ar fraction can affect the operation characteristics and the total efficiency of the systems. For the rate constants in the rate equations, k_ij = k_ij^Ar × y_Ar + k_ij^He × y_He needs to be an average of k_ij^Ar and k_ij^He weighted by the gas pressure. In order to simulate the DPRLGs systems with different temperatures effectively, the rates are modified by an Arrhenius temperature scaling [5]. The pre-exponential factor is 3 × 10⁻¹⁰ cm³/s and the activation energies are determined by the rates and temperatures in Table 1. The back rate coefficients are determined by detailed balance.

Table 1. Collisional relaxation processes and rate constants involved in DPRGLs model.

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The Einstein spontaneous emission coefficients A_ij are listed in Table 2 and provided by the National Institute of Standards and Technology website (except for A₂₀) [16]. The emission from 1s₄ state to ground state is trapped and it is assumed that radiatively trapped coefficient A₂₀ is equal to 5.7 × 10⁵ s⁻¹ to maintain consistency with previous model [5,8]. For the other emissions from 2p states to 1s states, the radiation trapping is not under consideration because of the low number density of 1s states.

Table 2. Einstein spontaneous emission coefficients A in DPRGLs model^a.

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In the model, it is assumed that the number density of excited states is much less than that of the ground state. The solutions of the rate equations are discussed under the steady-state conditions of d / dt = 0. Combined with Eqs. (2) and (4), the rate equations can be solved to obtain the dependent variables such as number density n_i and output laser intensity I_L, as functions of independent variables such as reduced electric field E/N, gas pressure P and mixture fraction y_Ar.

2.4 Total efficiency

An analytical expression for the discharge excitation rate S_d obtained by rate equations is given as follows:

\begin{matrix} S_{d} = \sum_{i = 1}^{5} S_{d i} = \sum_{i = 1}^{5} k_{0 i}^{e} n N - k_{i 0}^{e} n n_{i} \\ = n_{1} k_{{Ar}_{2}^{*}} N^{2} y_{A r} + n_{2} A_{20} + n_{3} A_{p 10 - s3} + n_{3} A_{p 10 - s 2} + n_{5} A_{p 8 - s 2} = S_{l o s s}, \end{matrix}

where S_di is the discharge excitation rate for the i state and S_loss is the loss rate of the population cycle. Under the steady-state conditions, the particles formation rate of the five levels is equal to their loss rate, based on the conservation of the number density. Given that all the excitation states n_i make a contribution to the laser cycle, the discharge excitation efficiency is equal to the sum of discharge excitation efficiencies: η_d = η_d₁ + η_d₂ + η_d₃ + η_d₄ + η_d₅, where η_di obtained by the BOLSIG + is the discharge excitation efficiency of the i state.

The total efficiency is defined as the ratio of the output power to the two-stage excitation power:

η_{t o t} = \frac{I_{o u t}}{I_{p} + W_{d} \times l_{g}} = \frac{η_{o p t}}{1 + \frac{W_{d} \times l_{g}}{I_{p}}},

where η_opt is optical conversion efficiency. Based on the equation of W_dη_di = S_diΔE_i₀, W_d is discharge power density expressed as below:

W_{d} = \frac{S_{d i} Δ E_{i 0}}{η_{d i}} (i = 1, 2, 3, 4, 5) = \frac{\sum_{i = 1}^{5} S_{d i} Δ E_{i 0}}{\sum_{i = 1}^{5} η_{d i}} = \frac{\sum_{i = 1}^{5} S_{d i} Δ E_{i 0}}{η_{d}} .

3. Validity of the model

Firstly, validity of the model is verified by a comparison between results calculated by our model and obtained from Rawlins et al.’s experiment. The model parameters are set to ensure consistency with the experimental conditions [5]: gain length l_g = 1.9 cm, gas pressure P = 1 atm, mixture fraction y_Ar = 2%, gas temperature T = 600 K, laser reflectivity of back reflector R_L = 99% and output couple R_OC = 85% respectively, microwave discharge power and frequency are 9 W and 900 MHz respectively, pump intensity I_p = 1320 W/cm², pump and laser transmission of the cell window are assumed to be 98%. According to the Hoskinson et al.’s model [24,25], reduced electric field E/N is assumed to be 4 Td which corresponds to mean electron energy of 2.27 eV. For a conservative estimation, pump reflectivity of output couple is assumed to be 0. The line shape of pump laser is assumed to be in Gaussian distribution with pump linewidth Δν₄₁ = 2 GHz.

Based on the literature [5], the absorbed pump power is 40 mW, laser power is 22 mW and effective optical conversion efficiency defined as the ratio of laser intensity to pump absorption intensity is 55%, with electron density in a range from 10¹³ to 10¹⁴ cm⁻³. In our model, the absorbed pump power and laser power are 43.44 mW and 23.06 mW respectively with effective optical conversion efficiency of 53.1%, under the conditions of the electron density of 5 × 10¹³ cm⁻³ and microwave power of 8.4 W; the absorbed pump power and laser power are 54.33 mW and 29.82 mW respectively with effective optical conversion efficiency of 54.9%, under the conditions of electron density of 6 × 10¹³ cm⁻³ and microwave power of 8.5 W. The calculated results are consistent with the experimental results basically.

4. Analysis of the results

In this section, some main characteristics of population density of the gain media are theoretically studied and the influence of several key factors on total efficiency are analyzed, so as to provide theoretical support for the design of high power and high total efficiency laser systems. In the following simulations, if not otherwise specified, the parameters are set as follows: gain length l_g = 1 cm, gas pressure P = 1 atm, mixture fraction y_Ar = 2%, laser reflectivity of back reflector R_L = 99% and output couple R_OC = 70% respectively, pump reflectivity of output couple R_P = 99%, pump and laser transmission of the cell window T_L = T_P = 98%, electron density n = 1 × 10¹³ cm⁻³ and pump linewidth Δν₄₁ = 50 GHz. Given that the lack of collisional rate constants at higher temperature, we provide the calculations at room temperature for a conservative estimation [8].

Firstly, the effect of discharge conditions on the gain medium density is studied. Yang et al. proposed that the bottleneck effect of slow relaxation rate k₂₁ can be compensated by the addition of gain medium density [8]. In their model, the gain medium density is expressed as the initial Ar (1s₅) number density which is a constant. In fact, the gain medium density increases with reduced electric field E/N and Ar fraction, as shown in Fig. 2(a). In addition, the influence of reduced electric field and Ar fraction on output laser intensity is shown in Fig. 2(b). As E/N rises from 8 to 15 Td at mixture fraction of 0.2%, the saturated laser intensity increases from 1.4 to 5.6 kW/cm² and linear performance of the laser is improved. Further calculation is made to obtain the higher output laser intensity for higher pump intensity and higher mixture fraction. At mixture fraction of 2%, pump intensity of 50 kW∕cm² and E/N of 15 Td, the output laser intensity increases to 29.5 kW∕cm². Therefore, the high-energy potential of DPRGLs systems can be promoted by the rise of reduced electric field and mixture fraction.

Fig. 2 Influence of reduced electric field and the mixture fraction on (a) gain medium density and (b) output laser intensity.

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Then, we research the impact of some key parameters on the total efficiency of the DPRGLs systems. The first parameter is the reduced electric field and the results are shown in Fig. 3. There is an optimal reduced electric field to obtain the highest total efficiency for a given pump intensity, which is the result of the competition between the two-stage excitation and non-lasing losses. The similar competition mechanism is analyzed in detail later. Optimal reduced electric field increases with the addition of the pump intensity from 10 to 50 kW∕cm², as higher gain media density is pumped to achieve population cycle at higher pump intensity. The optimal reduced electric field ranges from 10 to 15 Td at pump intensity from 20 to 50 kW∕cm².

Fig. 3 Influence of reduced electric field on total efficiency for different pump intensity. The red line corresponds to discharge excitation efficiency, and the others correspond to total efficiency.

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Figure 4 shows the relationship between the total efficiency and the gain length for six different two-stage excitation conditions. The optimal gain length is observed to decrease with the increase of the reduced electric field over the range of 8–15 Td. At pump intensity of 30 kW∕cm², the optimal gain lengths are 3.2 cm, 1.4 cm and 0.6 cm under the first-stage excitation conditions of the reduced electric fields of 8 Td, 10 Td and 15 Td respectively. Under the same discharge conditions, as the pump intensity is increased to 50 kW∕cm², the optimal gain lengths are increased to 5.3 cm, 2.4 cm and 1.0 cm respectively. The longer gain length is required to absorb the higher intensity of pump light. Therefore, when a DPRGLs laser system is designed, the gain length can be optimized to achieve the optimal total efficiency according to the two-stage excitation conditions.

Fig. 4 Influence of gain length on total efficiency for different reduced electric field at pump intensity of (a) 30 kW/cm² and (b) 50 kW/cm². Discharge power densities are 4.9, 7.5, 16.5 kW/cm³ at E/N of 8, 10, 15 Td respectively.

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Another important parameter is the mixture fraction of which the influence on the total efficiency is shown in Fig. 5. The reduced electric field is set as 10 Td and the other parameters are the same as above. The results reveal that there is an optimal mixture fraction for different pump intensity. This phenomenon is caused by the interaction of several processes as shown in Fig. 6 where the pump intensity is 30 kW/cm². The discharge excitation efficiency increases with the Ar fraction. A higher argon fraction could provide sufficient Ar(1s₅) number density to absorb the pump light. In this way, pump absorption efficiency is improved by the increase of the Ar fraction, too. However, a relatively higher argon fraction also increases the non-lasing loss, such as the excimers’ production, which leads to the decrease of effective optical conversion efficiency. The peak total efficiency is the result of a balance between the two-stage excitation efficiency and the effective optical conversion efficiency. At pump intensity of 30 kW/cm², the peak total efficiency reaches 44.7% and the optical conversion efficiency is 55.5% with the Ar fraction of 5.8%. Therefore, the proportion of discharge power to total input power is only 19.5%, which means that the discharge power used to maintain steady state is low. There is a significant deviation of the proportion between our model and Demyanov et al.’s model [7], which is primarily caused by different energy loss mechanisms and different definition of discharge excitation efficiency.

Fig. 5 Influence of Ar fraction on total efficiency for different pump intensity at E/N = 10 Td. Discharge power density is 7.2 kW/cm³ at I_p = 30 kW/cm² and y_Ar = 5.8%.

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Fig. 6 Influence of Ar fraction on total efficiency (η_tot), discharge excitation efficiency (η_d), optical conversion efficiency (η_opt), effective optical conversion efficiency (η_eff) and pump absorption efficiency (η_p) at I_p = 30 kW∕cm² and E/N = 10 Td.

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In Fig. 7, the total efficiency is represented as functions of gas pressure for different two-stage excitation conditions. At the reduced electric field of 15 Td and the pump intensity of 30 kW∕cm², the total efficiency approaches 39.2% with the pressure of 0.83 atm. If the reduced electric field is decreased to 8 Td, the total efficiency and the optimal pressure is raised to 47.7% and 1.75 atm respectively. In addition, higher pump rate increases the maximum rate of the population cycle. Therefore, As the pump intensity is risen to 50 kW∕cm², the peak total efficiency is further increased to 51.5% and the optical conversion efficiency is 62.7% with the corresponding pressure of 2.17 atm. Under the discharge condition of E/N = 8 Td, the apertures of gain medium need to be 5.4 cm² and 3.2 cm² to realize output intensity of 100 kW for the pump intensity of 30 kW∕cm² and 50 kW∕cm², respectively. The results indicate that DPRGLs systems have the advantages of small size, high power output and high total efficiency.

Fig. 7 Influence of gas pressure on total efficiency for different reduced electric field at pump intensity of (a) 30 kW/cm² and (b) 50 kW/cm². Under the optimal conditions at E/N = 8 Td, discharge power densities are 8.8 and 10.9 kW/cm³ at I_p of 30 and 50 kW/cm² respectively.

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Given that steady-state discharges are difficult to sustain at pressures near and above 1 atm, it is also important to obtain optimal operating parameters at pressure of 0.5 atm although the total efficiency is relatively low. The optimal total efficiency is 26.58% as reduced electric field is 10 Td, gain length is 7.31 cm, gas pressure is 0.5 atm, pump intensity is 30 kW/cm², and the other parameters are the same as above.

For high-efficiency DPRGL systems, it is important to optimize laser reflectivity of output couple. In our model, the dependence of total efficiency on laser reflectivity of output couple is shown in Fig. 8. With the increase of reduced electric field, the optimal R_OC is observed to decrease. At the higher pump intensity, the lower reduced electric field could not provide sufficient gain medium density. Therefore, the total efficiency at the pump intensity of 50 kW/cm² is lower than that of 30 kW/cm² for the reduced electric field of 6 Td, 9 Td and 11 Td. The total efficiency has little change and the curves become flatter in a wide range of the laser reflectivity of output couple for the higher reduced electric field because of the high gain and high excitation intensity. At the reduced electric field of 11 Td and the pump intensity of 30 kW/cm², the total efficiency could reach 45%, as the reflectivity varies from 0.01 to 0.44.

Fig. 8 Influence of laser reflectivity of output couple on total efficiencies for reduced electric fields at pump intensity of (a) 30 kW/cm² and (b) 50 kW/cm². Discharge power densities are 2.9, 6.1, 9.0 kW/cm³ at E/N of 6, 9, 11 Td respectively.

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5. Conclusions

Based on the five-level rate equations and the conservation of the number density, the two-stage excitation model of the DPRGLs systems is established. In our model, both discharge dynamics and optical dynamics are considered to understand the energy loss mechanism and some main characteristics such as high power output and high total efficiency. The simulation results of this model are consistent with the experimental results of Rawlins et al. In this paper, the concentration characteristics of gain medium in steady-state DPRGLs are studied, and the effects of important parameters on the total efficiency are analyzed, such as reduced electric field, gas pressure, Ar fraction and laser reflectivity of output couple.

According to the results, the relatively low relaxation collision rate from 1s₄ level to 1s₅ level can be compensated by increase of the reduced electric field, which improves the high energy potential of the laser. The conditions of 100 kW output are as follows: the pump intensity is 30 kW/cm², the reduced electric field is 8 Td, the gas pressure is 1.77 atm and the gain medium aperture is 5.4 cm². The total efficiency is 47.75% and the optical conversion efficiency is 61.5%. When the pump intensity is further increased to 50 kW/cm², the pressure rises to 2.17 atm, the gain medium aperture decreases to 3.2 cm², the total efficiency increases to 51.5% and the optical conversion efficiency increases to 62.5%. Optimal operating parameters at pressure of 0.5 atm is also obtained: reduced electric field is 10 Td, gain length is 7.31 cm and pump intensity is 30 kW/cm² with optimal total efficiency of 26.58%.

The discharge power used to maintain steady state is low, based on the high total efficiency and the high optical conversion efficiency. The total efficiency of DPRGLs has a peak for each parameter due to the balance between the two-stage excitation efficiency and the effective optical conversion efficiency. Therefore, when DPRGLs systems are designed, it is necessary to make an overall consideration between the discharge excitation conditions and optical pump conditions to obtain the best efficiency by parameter optimization.

Funding

National Natural Science Foundation of China (NSFC) (61575072, 61775067); National Key Research and Development Project of China (2016YFB1100302).

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Reactions	Einstein spontaneous emission coefficients (s⁻¹)	References
Ar(1s₄) → Ar(1S₀)	A₂₀ = 5.7 × 10⁵	[5]
Ar(2p₁₀) → Ar(1s₅)	A₃₁ = 1.87 × 10⁷	[16]
Ar(2p₁₀) → Ar(1s₄)	A₃₂ = 5.43 × 10⁶	[16]
Ar(2p₁₀) → Ar(1s₃)	A_p10→s3 = 9.8 × 10⁵	[16]
Ar(2p₁₀) → Ar(1s₂)	A_p10→s2 = 1.9 × 10⁵	[16]
Ar(2p₉) → Ar(1s₅)	A₄₁ = 3.31 × 10⁷	[16]
Ar(2p₈) → Ar(1s₅)	A₅₁ = 9.28 × 10⁶	[16]
Ar(2p₈) → Ar(1s₄)	A₅₂ = 2.15 × 10⁷	[16]
Ar(2p₈) → Ar(1s₂)	A_p8→s2 = 1.47 × 10⁶	[16]

Reactions	Einstein spontaneous emission coefficients (s⁻¹)	References
Ar(1s₄) → Ar(1S₀)	A₂₀ = 5.7 × 10⁵	[5]
Ar(2p₁₀) → Ar(1s₅)	A₃₁ = 1.87 × 10⁷	[16]
Ar(2p₁₀) → Ar(1s₄)	A₃₂ = 5.43 × 10⁶	[16]
Ar(2p₁₀) → Ar(1s₃)	A_p10→s3 = 9.8 × 10⁵	[16]
Ar(2p₁₀) → Ar(1s₂)	A_p10→s2 = 1.9 × 10⁵	[16]
Ar(2p₉) → Ar(1s₅)	A₄₁ = 3.31 × 10⁷	[16]
Ar(2p₈) → Ar(1s₅)	A₅₁ = 9.28 × 10⁶	[16]
Ar(2p₈) → Ar(1s₄)	A₅₂ = 2.15 × 10⁷	[16]
Ar(2p₈) → Ar(1s₂)	A_p8→s2 = 1.47 × 10⁶	[16]

Two-stage excitation model of diode pumped rare gas atoms lasers

Abstract

1. Introduction

2. Model description

2.1 Discharge excitation mechanism

2.2 Stimulated emission

2.3 Rate equations

2.4 Total efficiency

3. Validity of the model

4. Analysis of the results

5. Conclusions

Funding

References

Cited By

Figures (8)

Tables (2)

Equations (13)

Optics Express

Reactions	Rate constants k_ij	Temperature	Units	References
Ar(1S₀) + e^- ↔ Ar(1s₅) + e^-	BOLSIG +	300 K	cm³/s	[18,19]
Ar(1S₀) + e^- ↔ Ar(1s₄) + e^-	BOLSIG +	300 K	cm³/s	[18,19]
Ar(1S₀) + e^- ↔ Ar(2p₁₀) + e	BOLSIG +	300 K	cm³/s	[18,19]
Ar(1S₀) + e^- ↔ Ar(2p₉) + e^-	BOLSIG +	300 K	cm³/s	[18,19]
Ar(1S₀) + e^- ↔ Ar(2p₈) + e^-	BOLSIG +	300 K	cm³/s	[18,19]
Ar(1s₄) + Ar → Ar(1s₅) + Ar	1 × 10⁻¹³	300 K	cm³/s	[8]
Ar(1s₄) + He → Ar(1s₅) + He	1.2 × 10⁻¹³	300 K	cm³/s	[8]
Ar(2p₁₀) + Ar → Ar(1s₅) + Ar	0.6 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₁₀) + He → Ar(1s₅) + He	5 × 10⁻¹⁴	300 K	cm³/s	[8]
Ar(2p₉) + Ar → Ar(1s₅) + Ar	2.5 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₉) + He → Ar(1s₅) + He	2 × 10⁻¹²	300 K	cm³/s	[8]
Ar(2p₈) + Ar → Ar(1s₅) + Ar	1.5 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₈) + He → Ar(1s₅) + He	1 × 10⁻¹²	300 K	cm³/s	[8]
Ar(2p₉) + Ar → Ar(2p₁₀) + Ar	2.6 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₉) + He → Ar(2p₁₀) + He	1.6 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₈) + Ar → Ar(2p₁₀) + Ar	1.1 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₈) + He → Ar(2p₁₀) + He	4 × 10⁻¹²	300 K	cm³/s	[8]
Ar(2p₈) + Ar → Ar(2p₉) + Ar	1.1 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(2p₈) + He → Ar(2p₉) + He	4.5 × 10⁻¹¹	300 K	cm³/s	[8]
Ar(1s₅) + Ar + Ar → Ar₂* + Ar	36 × 10⁻³² × T ^-0.6		cm⁶/s	[8]
Ar(1s₅) + Ar + He → Ar₂* + He	18 × 10⁻³² × T ^-0.6		cm⁶/s	[8]