Modeling of scaling of a diode longitudinally pumped metastable rare gas with a master oscillator power amplifier

Hanyuan Chen; Sichen Long; Xiahui Tang; Xiangxi Wu; Wei Wang; Yingxiong Qin

doi:10.1364/OE.27.012504

1. Introduction

Recently, significant progress has been made in the development of a hybrid gas phase/solid state laser systems: diode pumped alkali vapor lasers (DPALs) [1–7]. However, there exist two main challenges for DPALs: active chemical properties of the gain medium (usually alkali vapor and hydrocarbons) and precise temperature control (the effective gain medium concentration is vaporized by heating). To solve these issues of DPALs, Han and Heaven et al. proposed an optically pumped metastable rare gas (Rg*) laser (OPRGL) in 2012 [8]. The gain medium of OPRGLs is their usage of rare gases (Rg) only, which makes these lasers inherently chemical inert. OPRGL is an alternative to DPAL because of their similar lasing mechanism, which combines the advantages of solid lasers and gas lasers. As a two-step pumping scheme, OPRGL includes Rg electrical discharge and optical pumping processes. When a diode laser is used as pump source, diode pumped rare gas lasers (DPRGLs) can be regarded as beam conversion systems with a wavelength in the near infrared range. In DPRGLs, a high-power laser with excellent beam quality is produced by a diode laser with poor beam quality at room temperature. In addition, DPRGLs have some other characteristics: high quantum efficiency, the narrow line-width (~MHz), medium recyclable flow heat dissipation, light weight and compactness.

Based on the advantages as mentioned above, DPRGLs have attracted increasing interest of researchers around the world. In recent years, a series of experiments and theoretical models have shown that DPRGLs are of high efficiency, high power and excellent beam quality of DPRGLs in different ways [8–16]. In 2015, Rawlins et al. have firstly realized a cw Ar* laser by microwave frequency microplasma discharge [10]. The cw laser power is 22 mW of a single OPRGL oscillator with narrow linewidth and high beam quality (Gaussian beam profile), which meets the requirements of seed laser in a cw MOPA laser system.

At the same time, researchers have done a lot of theoretical work. Demyanov et al. have established cw OPRGLs model with optical conversion efficiency of 60% based on three-level scheme under consideration of the discharge power density to sustain sufficient Ar* density in 2013 [12]. Subsequently, Yang et al. and Gao et al. developed a longitudinally and transversely diode pumped double-pass model by five-level scheme, respectively [13,14]. And we have established a complete dynamic model of DPRGLs systems with high efficiency and high power output [15]. To increase the power of these hybrid laser systems including DPRGLs and DPALs, till now, higher intensity and narrow-banded (~20 GHz) pump sources have been employed, such as multiple diode laser arrays or stacks of diode laser arrays [16,17]. Besides, similar to in traditional solid-state and fiber laser systems, there are two common ways for DPRGL power scaling: one is to extract power from a single oscillator, the other is to use the master oscillator power amplifier (MOPA) configuration.

Until now, diode pumped alkali amplifiers (DPAAs) have been demonstrated in both experimental and theoretical [18–23]. Zhdanov et al. have firstly presented Cs DPAA with 21.6 dB amplification in 2008 [18], and the first Rb DPAA with 7.9 dB amplification has been demonstrated in 2010 [19]. However, to date, all the high power DPRGL systems are performed using single oscillator, the DPRGA systems may be required for power scaling. Compared with a single oscillator, as an alternative power scaling approach, diode pumped rare gas amplifier (DPRGA) or DPRGA chain configuration is beneficial for the power scaling with excellent beam quality and reduces thermal effect of high power lasers.

In this paper, based on a five-level scheme, a kinetic model of a DPRGA system is established by taking Ar-He gas mixture as an example in order to research the characters of DPRGAs. Then, the validity of our model is verified by comparing with Rawlins et al.’s experimental results. Finally, the influence of optimization factors on single-pass amplification factor and efficiencies are analyzed in the simulation, which are helpful for designing an efficient high power DPRGA system.

In Section 2, the physical model and numerical approach of a single-pass DPRGA based on five-level scheme are introduced. In Section 3, validity of the model is verified by a comparison with Rawlins et al. experimental results. In Section 4, some key factors of laser characteristics are calculated and analyzed.

2. Physical model of DPRGA description

For ease of understanding and explanation, the schematic diagram of an Ar* laser MOPA system is shown in Fig. 1. The seed laser, denoted as I_sl, is produced by the Ar* oscillator, and is injected in amplifier with pump light to obtain the output laser I_laser. Both of oscillator and amplifier contain a homogeneous gain medium, namely helium and metastable Ar* atoms, which is produced by discharge near atmosphere. The difference between oscillator and amplifier is that the later one only considers the single-pass transmission without cavity mirror.

Fig. 1 Schematic diagram for an Ar* laser MOPA system.

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The energy level scheme for excited rare gases is more complicated than that of the alkalis. The complete energy levels associated with the DPRGA systems contain four 1s-levels (from 1s₂ to 1s₅, Paschen notation) and ten 2p-levels (from 2p₁ to 2p₁₀). In our model, however, the kinetic of the DPRGA model can be described as a five-level structure, including the lower pump level 1s₅, the upper pump level 2p₉, the upper laser level 2p₁₀, and the resonant level 1s₄ and the 2p₈. This simplification was based on the assumptions: one is that the population transfer from 1s₂ and 1s₃ levels is ignored because the intermultiplet transfer data of manifold 1s is incomplete, and the other is that we only consider the 2p₈ level and ignore the 2p₁~2p₇ levels because of the small energy gap between the levels of 2p₈ and 2p₉ (about 0.01 eV). A five-level model based on more complete energy levels and population transfer channels is shown in Fig. 2.

Fig. 2 Illustration of Ar energy levels and transfer channels in the mode.

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The detailed lasing mechanism of DPRGA is as follows: the metastable 1s₅ state of Rg (excepting He), as the common laser and pump lower level, is generated by discharge. Optical pumping on the 1s₅→2p₉ transition with subsequent collisional relaxation to the laser upper level (2p₁₀ state) creates inversion between 1s₅ and 2p₁₀ states undergoing absorption and amplification processes of seed laser.

In Fig. 2, the red dotted and black lines represent the spontaneous emission processes and the collisional relaxation processes, respectively. And the orange and yellow thick lines respectively describe the pumping and lasing processes. In our model, we mainly concern the repopulation of the five related energy level by optical pumping and corresponding laser performance, and the repopulation of these levels by discharge is not included. It is assumed that the pump and seed laser intensities are homogenous along the longitudinal direction (the optical axis). Our model of DPRGA based on rate equations is described as follows:

\begin{array}{l} \frac{d n_{1}}{d t} = - Γ_{p} - Γ_{s l} + Γ_{l a s e r} - k_{A r_{2}^{*}} N^{2} y_{A r} n_{1} + k_{21} N [n_{2} - \frac{g_{2}}{g_{1}} n_{1} \exp (- \frac{Δ E_{21}}{k_{B} T})] \\ + (A_{31} + k_{31} N) n_{3} + (A_{41} + k_{41} N) n_{4} + (A_{51} + k_{51} N) n_{5}, \end{array}

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

\begin{array}{l} \frac{d n_{3}}{d t} = - Γ_{l a s e r} + Γ_{s l} + k_{43} N [n_{4} - \frac{g_{4}}{g_{3}} n_{3} \exp (- \frac{Δ E_{43}}{k_{B} T})] + k_{53} N [n_{5} - \frac{g_{5}}{g_{3}} n_{3} \exp (- \frac{Δ E_{53}}{k_{B} T})] \\ - (A_{31} + A_{32} + A_{p 10 \to s 2} + A_{p 10 \to s 3} + k_{31} N) n_{3}, \end{array}

\begin{array}{l} \frac{d n_{4}}{d t} = Γ_{p} + k_{54} N [n_{5} - \frac{g_{5}}{g_{4}} n_{4} \exp (- \frac{Δ E_{54}}{k_{B} T})] - k_{43} N [n_{4} - \frac{g_{4}}{g_{3}} n_{3} \exp (- \frac{Δ E_{43}}{k_{B} T})] \\ - (A_{41} + k_{41} N) n_{4}, \end{array}

\begin{array}{l} \frac{d n_{5}}{d t} = - k_{54} N [n_{5} - \frac{g_{5}}{g_{4}} n_{4} \exp (- \frac{Δ E_{54}}{k_{B} T})] - k_{53} N [n_{5} - \frac{g_{5}}{g_{3}} n_{3} \exp (- \frac{Δ E_{53}}{k_{B} T})] \\ - (A_{51} + A_{52} + A_{p 8 \to s 2} + k_{51} N) n_{5}, \end{array}

I_{l a s e r} = I_{s l} \cdot (\exp [Δ n_{31} \cdot σ_{31} (λ) \cdot l_{g a i n}] - 1),

where Δn_ij is defined as Δn_ij = n_i ‒ g_i / g_j × n_j, in which n_i and g_i (the subscripts i, j = 1,2,3,4,5) represent the population densities and degenerate degrees of 1s₅, 1s₄, 2p₁₀, 2p₉ and 2p₈ energy levels, respectively. A_ij and ΔE_ij, between i and j energy level, are the spontaneous emission Einstein coefficients and the energy gaps, respectively, which are provided by [24]. k_ij are the collisional relaxation rate constants, which can be obtained by k_ij = k_ij^Ar × y_Ar + k_ij^He × y_He. y_Ar and y_He respectively represent the mixture fraction of argon and helium. k_ij^Ar and k_ij^He are taken from [10,12,25,26]. The reaction processes and the value of k_ij and A_ij considered in the rate equations are shown in Table 1 and Table 2, respectively. In order to simulate the DPRLAs systems with different temperatures effectively, the rates are modified by an Arrhenius temperature scaling [10]. In our model, gain medium density, as a constant, is expressed as the initial Ar (1s₅) number density which is investigated in detail in our previous work [15]. l_gain is length of the gain medium. k_B is Boltzmann constant, and T is the absolute temperature in Kelvin. σ₃₁(λ) is described the excitation cross sections in the laser emission processes for laser amplifier, which is expressed as below:

σ_{31} (λ) = \frac{A_{31} λ_{31}^{2}}{4 π^{2} Δ v_{31}},

where λ₃₁ = 912.3 nm is the wavelength of the seed laser and the output laser of the amplifier. The spectral linewidth of the laser transition, denoted as Δν₃₁, is described as below [27]:

Δ v_{31} = 2 [y_{H e} \cdot k_{31}^{H e} + y_{A r} \cdot k_{31}^{A r}] \times {(T / 300)}^{0.3} \times N,

where k₃₁^He and k₃₁^Ar are equal to 5 × 10⁻¹⁴ cm³/s and 0.6 × 10⁻¹¹ cm³/s, respectively, in Table 1. N represents the density of particles at atmosphere.

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

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

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In addition, the pump rate, the seed laser transition rate and the laser transition rate in the amplifier are represented by Γ_p, Γ_sl and Γ_laser, respectively. And Γ_p is described as below [28]:

Γ_{p} = \frac{η_{d e l} η_{m o d}}{L} \int d λ \frac{1}{h v_{P}} \frac{d I_{p} (λ)}{d λ} \cdot {1 - \exp [- Δ n_{14} σ_{14} (λ) \cdot l_{g a i n}]},

where η_del and η_mod, assumed to be 100% in calculation, represent the delivery efficiency of the pump beam and the mode-matching efficiency, respectively. hv_p is pump photon energy, in which h is the Planck constant and v_p is pump light frequency. dI_p(λ)/dλ is the spectrally resolved pump light intensity, which can be expressed by

d I_{p} (λ) / d λ = I_{P} \cdot g_{P} (λ),

where I_p denotes the pump intensity, and we assume that the lineshape is shown as g_p(λ) in a Gaussian distribution [14]:

g_{P} (λ) = \frac{c}{λ^{2}} \frac{2}{Δ v_{p}} \sqrt{\frac{\ln 2}{π}} \cdot \exp [- \frac{4 \ln 2}{Δ v_{p}^{2}} {(v - v_{p})}^{2}] .

In Eq. (11), c is the speed of light, Δν_p and ν_p are described, respectively, the spectral linewidth in FWHM and the corresponding frequency with the wavelength of the pump source. In order to obtain higher pumping efficiency, Δν_p needs to match the corresponding Ar* absorption linewidth (FWHM) which is related to absorption cross section σ₁₄(λ) in Eq. (9). Since the density of Ar* is produced by discharge under atmosphere, we assume that the homogeneous broadening (Lorentzian profile) dominates in broadening process. Therefore, σ₁₄(λ) can be given as below:

σ_{14} (λ) = \frac{g_{4}}{g_{1}} \cdot \frac{A_{41} λ_{41}^{2}}{16 π^{2}} \cdot \frac{Δ v_{41}}{{(Δ v_{41} / 2)}^{2} + {(v - v_{41})}^{2}},

where λ₄₁ = 811.5 nm is the pump wavelength, ν₄₁ is the corresponding frequency at 811.5 nm. Δν₄₁ is not only related to the total pressure in discharge but also related to the partial pressure ratio of helium and argon, similar to Eq. (8), which is described as below:

Δ v_{41} = 2 [y_{H e} \cdot k_{41}^{H e} + y_{A r} \cdot k_{41}^{A r}] \times {(T / 300)}^{0.3} \times N,

where k₄₁^He and k₄₁^Ar are equal to 2 × 10⁻¹² cm³/s and 2.5 × 10⁻¹¹ cm³/s, respectively, in Table 1.

In our model, the seed laser is assumed to be in a single frequency and amplified being absorbed. The seed laser transition rate Γ_sl is given by

Γ_{s l} = \frac{η_{d e l} η_{m o d}}{L} \int d λ \frac{1}{h v_{s l}} \frac{d I_{s l}}{d λ} \cdot {1 - \exp [- Δ n_{13} σ_{13} (λ) \cdot l_{g a i n}]},

where σ₁₃(λ) is described as the absorption cross sections for laser amplifier.

The laser transition rate Γ_laser can be expressed as

Γ_{l a s e r} = \frac{1}{L} \cdot \frac{I_{s l}}{h v_{l}} \cdot {\exp [Δ n_{31} σ_{31} (λ) \cdot l_{g a i n}] - 1},

where hv_l is laser photon energy.

In the case of cw operation, the solutions of the rate equations are discussed under the steady-state conditions of d / dt = 0. Combined with Eqs. (9), (14) and (15), the rate equations can be solved to obtain the dependent variables such as number density n_i and output laser intensity I_laser. Then the energy conversion channels can be expressed as follows:

P_{l a s e r} = I_{l a s e r} \cdot S,

P_{f l u o r e s c e n c e} = V \cdot (n_{3} A_{31} Δ E_{31} + n_{3} A_{32} Δ E_{32} + n_{4} A_{41} Δ E_{41} + n_{5} A_{51} Δ E_{51} + n_{5} A_{52} Δ E_{52}),

P_{h e a t} = V \cdot {\begin{cases} n_{3} A_{31} Δ E_{31} + n_{4} A_{41} Δ E_{41} + n_{5} A_{51} Δ E_{51} \\ + k_{21} Δ E_{21} [n_{2} - \frac{g_{2}}{g_{1}} n_{1} \exp (- \frac{Δ E_{21}}{k_{B} T})] \\ + k_{43} Δ E_{43} [n_{4} - \frac{g_{4}}{g_{3}} n_{3} \exp (- \frac{Δ E_{43}}{k_{B} T})] \\ + k_{53} Δ E_{53} [n_{5} - \frac{g_{5}}{g_{3}} n_{3} \exp (- \frac{Δ E_{53}}{k_{B} T})] \\ + k_{54} Δ E_{54} [n_{5} - \frac{g_{5}}{g_{4}} n_{4} \exp (- \frac{Δ E_{54}}{k_{B} T})] \end{cases}},

where P_laser, P_fluorescence, and P_heat are the output laser, fluorescence power and waste heat in amplifier. S is the cross section area of the laser beam. V is the mode volume, which can be expressed by V = l_gain × S.

3. Comparison with Rawlins et al.’s experimental results

In this section, validity of the model is verified by a comparison between the simulation by our model and the experiment results by Rawlins et al. The model parameters are set to ensure consistency with the experimental conditions [10]: gas pressure P = 1 atm, mixture fraction of Ar y_Ar = 2%, gas temperature T = 600 K, pump linewidth Δν₄₁ = 2 GHz. The dependence of gain length on amplification factor at the pump intensity I_p = 1320 W/cm² (corresponding to 800 mW) and the seed laser intensity I_sl = 20 W/cm² (corresponding to 12 mW) is shown in Fig. 3. In literature [10], the single-pass amplification factor in probe experiment is about 9, and in our calculation is 9.84 (that is 9.93 dB in Fig. 3). The calculated results agree qualitatively with the experimental results. The main reason for the relatively larger calculated results is that we assumed the delivery efficiency of the pump beam η_del and the mode-matching efficiency η_mod to be 100% in simulation.

Fig. 3 The calculation results of the gain length on single-pass amplification factor. The amplification factor is defined as A(dB) = 10 log₁₀ (I_laser / I_sl).

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4. Simulation results and discussions

In this section, the influence of several key factors on amplification factor are simulated and analyzed, and the calculations results could provide guidance for designing an efficient single-pass DPRGA system in future. We first analyze the influence of the initial Ar(1s₅) density (Section 4.1) on single-pass amplification factor and efficiencies of DPRGAs separately. Then, with different pump intensities I_p, seed laser intensity I_sl and linewidth of pump laser Δλ, the dependences of the single-pass amplification factor on the gain length l_gain (Section 4.2) are respectively discussed. Finally, the influence of pump and seed laser intensities on single-pass amplification factor is determined (Section 4.3).

4.1 Influence of the initial Ar(1s₅) density

The gain medium density is a very important factor in designing a DPRGA system. In our model, the gain medium density is expressed as the initial Ar(1s₅) number density which is obtained by discharge. Yang et al. proposed that the bottleneck effect of slow relaxation rate k₂₁ can be compensated by increasing the initial Ar(1s₅) number density (3 × 10¹³ cm⁻³) to realize a cw linear lasing [13]. And the number density of metastable Ar* atoms under conditions of a dc microdischarge was measured in [29] to be 10¹⁴ cm⁻³. As a result, the initial Ar(1s₅) number density is set as the order of 10¹⁴ cm⁻³ in our calculations.

Then we study the influence of the initial Ar(1s₅) density on single-pass amplification factor. In the calculation, the parameters are set as follows: total gas pressure is 1 atm (room temperature) mixture with 2% Ar and 98% He (i.e. y_Ar = 2% and y_He = 98%), the pump intensity is I_p = 50 kW/cm² with linewidth of 50 GHz (FWHM), and the seed laser intensity is I_sl = 100 W/cm². The selection of the main parameters is analyzed in detail. According to the literature [10], properly elevated temperature may be beneficial to laser performance because of faster energy transfer kinetics. Given that the lack of collisional rate constants at higher temperature (also the reliable temperature scaling laws), we provide the calculations at room temperature for a conservative estimation [13]. The simulation results are shown in Fig. 4.

Fig. 4 The initial Ar (1s₅) number density influence on single-pass amplification factor of a DPRGA at different (a) gain lengths, (b) pump intensities, (c) seed laser intensities and (d) pump linewidths.

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In DPRGA system, uniform propagation of pump light in gain medium is beneficial to obtain higher energy laser. The uniform propagation characteristic of pump light in gain medium can be guaranteed by choosing suitable Rayleigh length, that is, by selecting the appropriate gain length. With different gain lengths of 1 cm, 5 cm, 10 cm, 15 cm, 30 cm, the dependences of the single-pass amplification factor on the initial Ar(1s₅) number density are shown in Fig. 4(a). It can be seen from Fig. 4(a) that the equivalence between gain length and initial Ar(1s₅) density. For a certain column Ar(1s₅) density with different gain length, the maximum single-pass amplification factor keeps the same (~22.18 dB) theoretically. But at the same amplification factor, the gain length is inversely proportional to the initial Ar(1s₅) density. At the same time, the Rayleigh length can be expressed as L = πω₀² / λ (ω₀ represents the beam waist of the pump light), which is proportional to the cross section area of pump light at a definite wavelength. However, according to the relationship between power and intensity (P = I × S), the pump power is proportional to the cross section area of the pump light at a certain pump intensity. To summarize, for identical intensities, the gain length is inversely proportional to the initial Ar(1s₅) density and is proportional to pump power. However, the higher initial Ar(1s₅) density poses a challenge to discharge conditions. So, it is necessary to keep a balance between the gain length and the initial Ar(1s₅) density in designing a DPRGA system. In our simulation model, gain length is set as a moderate value of l_gain = 15 cm when we study the influence of other factors on amplification factor.

Then the influence of initial Ar(1s₅) density on laser performance at different pump intensities is shown in Fig. 4(b). For a given pump intensity, we require an optimal initial Ar(1s₅) density to obtain the highest amplification factor, which is the result of a balance between pump absorption and non-lasing losses. To get higher amplification factor, we need a higher pump intensity and higher initial Ar(1s₅) density. With different pump intensities of 5 kW/cm², 10 kW/cm², 30 kW/cm², 50 kW/cm², 80 kW/cm², 100 kW/cm², the amplification factor could reach 11.79 dB, 14.95 dB, 19.90 dB, 22.18 dB, 24.25 dB, 25.24 dB at the optimal initial Ar(1s₅) density of 3 × 10¹³ cm⁻³, 5 × 10¹³ cm⁻³, 1 × 10¹⁴ cm⁻³, 1.6 × 10¹⁴ cm⁻³, 2.4 × 10¹⁴ cm⁻³, 2.9 × 10¹⁴ cm⁻³, respectively.

Figure 4(c) shows the relationship between the amplification factor and the initial Ar(1s₅) density for six different seed laser intensities under the same pump intensity. For the same column Ar(1s₅) density, the amplification factor is decreased with the increase of seed laser intensity. At initial Ar(1s₅) density of 1.6 × 10¹⁴ cm⁻³, the optimal amplification factors are 32.14 dB, 25.16 dB, 22.18 dB, 15.28 dB, 12.42 dB and 4.23 dB under the seed laser intensities of 10 W/cm², 50 W/cm², 100 W/cm², 500 W/cm², 1 kW/cm² and 10 kW/cm² respectively.

Finally, the influence of initial Ar(1s₅) density on DPRGA performance at different pump spectral linewidths of 2 GHz, 10 GHz, 20 GHz, 50 GHz, 100 GHz, is shown in Fig. 4(d). The results show that the effect of pump spectral linewidth on DPRGA performance decreases with the increasing of initial Ar(1s₅) density. The reason is that the probability of collision between pump light and Ar(1s₅) particles decreases at lower initial Ar(1s₅) density. Therefore, DPRGA performance becomes more sensitive to the pump linewidth. In addition, the amplification factor decreases with the increase of pump linewidth at the same column Ar(1s₅) density. According to the results shown in Fig. 4, we also can clearly see that the amplification factor increases dramatically with the initial Ar(1s₅) density (in range of 1 ~10 × 10¹³ cm⁻³) to its maximum value, and then it decreases slowly with the increase of the initial Ar(1s₅) density (greater than 1.5 × 10¹⁴ cm⁻³) at a certain condition. Under the condition of l_gain = 15 cm, the aperture of the gain medium need to be 6 cm² to realize 100 kW output for the pump intensity of 50 kW/cm² and the seed laser intensity of 100 W/cm². The results show that a DPRGA is a compact and high power output laser system.

Figure 5 shows the relationship between the efficiencies of DPRGA and the initial Ar(1s₅) density. Parameters are the same as mentioned in Fig. 4. With the increase of the initial Ar(1s₅) density, we need to increase the pump absorption efficiency η_absorb, which, otherwise, will lead to the decrease of the laser extraction efficiency η_opt-abs. For the absorbed pump power, there exist three main exit channels: laser, fluorescence and heat. From Fig. 5, we can see that both the fluorescence efficiency η_fluo-abs and the heat efficiency η_heat will increase as the initial Ar(1s₅) density increases, which induces the decrease of η_opt-abs. The results show that the maximum laser extraction efficiency is about 33%. The phenomenon is caused by the absorption of seed laser and the single-pass pumping process.

Fig. 5 Influence of the initial Ar(1s₅) density on pump absorption efficiency (η_absorb), the laser extraction efficiency relative to total pump power (η_opt-opt), the laser extraction efficiency relative to absorbed pump power (η_opt-abs), fluorescence efficiency relative to absorbed pump power (η_fluo-abs) and heat efficiency relative to absorbed pump power (η_heat-abs). And fluorescence power and heat power are calculated by Eqs. (17) and (18).

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4.2 Influence of the gain length

The gain length is another important factor in designing a DPRGA system. In Fig. 6, single-pass amplification factor is represented as functions of the gain length for different simulation conditions. The initial Ar(1s₅) density is set as 1.6 × 10¹⁴ cm⁻³ and the other parameters are set as what listed in Section 4.1. Firstly, we study the influence of gain length on DPRGA single-pass amplification factor at different pump intensities [Fig. 6(a)]. The results demonstrate that the amplification factor increases obviously with the gain length to its optimal value, and then it decreases slowly with the increase of gain length at a certain pump intensity.

Fig. 6 The gain length influence on single-pass amplification factor of a DPRGA at different (a) pump intensities, (b) pump linewidths and (c) seed laser intensities.

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In addition, we find that the amplification factor increases with the increase of pump intensity at the same column gain length. And the optimal gain length increases with the increase of the pump intensity. The optimal gain lengths are 9.9 cm, 15 cm and 22.1 cm at the pump intensities of 30 kW/cm², 50 kW/cm² and 80 kW/cm² respectively. Therefore, the gain length should be optimized to achieve the optimal amplification factor in designing a DPRGA. In Fig. 6(b), the amplification factor is represented as functions of gain length at different pump spectral linewidth. When the gain length increases, the probability of collision between pump light and Ar(1s₅) particles dramatically increases, and the amplification factor becomes less sensitive to the change of pump spectral linewidth. According to the result of Fig. 4(a), the gain length is inversely proportional to the initial Ar(1s₅) density. To obtain a DPRGA system with highest energy output, it is necessary to keep a balance between gain length and initial Ar(1s₅) density. From Fig. 6(c), it can be seen that the maximum single-amplification factors are 32.14 dB, 22.18 dB and 12.41 dB at the seed laser intensities of 10 W/cm², 100 W/cm², and 1 kW/cm², respectively, under the optimal gain length l_gain = 15 cm.

4.3 Influence of pump laser and seed laser intensities

According to the experimental results of Rawlins et al., the output of the DPRGL is approximately dozens of milliwatts (i.e., W/cm²) [10]. As a high power five-level DPRGA system, the required pump intensity is several kW/cm². In our calculation, the dependence of amplification factor on seed laser intensity at different pump intensities is shown in Fig. 7. It is seen that, for a constant seed laser intensity, the amplification factor increases with the increase of the pump intensity. In addition, when the seed laser intensity is weaker, the amplification factor is larger.

Fig. 7 The gain length influence on single-pass amplification factor of a DPRGA at different pump intensities.

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In DPRGAs, when the seed laser is absent, the Rg* works in recycling mode 1s₅→2P₉→2P₁₀→1s₅ to convert pump photons into laser photons, and the cycling step 2P₁₀→1s₅ is dominated by spontaneous emission. And when the seed laser is present, the cycling step 2P₁₀→1s₅ is dominated by stimulated emission, which is much faster than the spontaneously emission process. Thus, the presence of seed laser accelerates the total atomic recycling rate, including the pump absorption step 1s₅→2P₉. And for a constant initial Ar(1s₅) density, the pump absorption fraction is dramatically enhanced.

5. Conclusions

In this paper, we set up a five-level scheme model for a DPRGA by considering the conservation of the number density. The design of DPRGA is very promising in achieving high power output. By analysis of the kinetic of laser amplification processes, we established a physical model to simulate the output characteristics of the DPRGAs systems. The results predicted by our model agree well with the experiment results in Ref [10]. We simulated and analyzed the influences of the initial Ar(1s₅) density, gain length and the intensities of pump and seed laser on amplification factor. From the simulation results, we can reach the following three conclusions. Firstly, the maximum single-pass amplification is existed under an optimal initial Ar(1s₅) density. Secondly, higher initial Ar(1s₅) density can result in the less sensitiveness of pump spectral linewidth and shorter gain length. Thirdly, the total atomic recycling rate can be accelerated by injecting the seed laser into the system compared with a single oscillator, which is beneficial for obtaining higher power output.

According to the simulation results, the DPRGA system is promising in achieving high power output by keeping a balance of some main parameters. The conditions of the maximum amplification factor of 22.18 dB (i.e.100 kW output) are derived as follows: the pump intensity is 50 kW/cm², the seed laser intensity is 100 W/cm², the gas pressure is 1 atm and the gain length is 15 cm. However, considering the only single-pass pumping in DPRGAs, the laser extraction efficiency of DPRGAs (~33%) is lower than that of a single oscillator (~57%) [13]. Thus, we can provide a high efficiency DPRGA system by multi-pass pumping, and a higher power output by a chain of two or more amplifiers structure.

Funding

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

References

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Reactions	Rate constants k_ij	Temperature	Units	References
Ar(1s₄) + Ar → Ar(1s₅) + Ar	1 × 10⁻¹³	300 K	cm³/s	[13]
Ar(1s₄) + He → Ar(1s₅) + He	1.2 × 10⁻¹³	303 K	cm³/s	[13]
Ar(2p₁₀) + Ar → Ar(1s₅) + Ar	0.6 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₁₀) + He → Ar(1s₅) + He	5 × 10⁻¹⁴	300 K	cm³/s	[13]
Ar(2p₉) + Ar → Ar(1s₅) + Ar	2.5 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₉) + He → Ar(1s₅) + He	2 × 10⁻¹²	300 K	cm³/s	[13]
Ar(2p₈) + Ar → Ar(1s₅) + Ar	1.5 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + He → Ar(1s₅) + He	1 × 10⁻¹²	300 K	cm³/s	[13]
Ar(2p₉) + Ar → Ar(2p₁₀) + Ar	2.6 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₉) + He → Ar(2p₁₀) + He	1.6 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + Ar → Ar(2p₁₀) + Ar	1.1 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + He → Ar(2p₁₀) + He	4 × 10⁻¹²	300 K	cm³/s	[13]
Ar(2p₈) + Ar → Ar(2p₉) + Ar	1.1 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + He → Ar(2p₉) + He	4.5 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(1s₅) + Ar + Ar → Ar₂* + Ar	36 × 10⁻³² × T ^-0.6		cm⁶/s	[13]
Ar(1s₅) + Ar + He → Ar₂* + He	18 × 10⁻³² × T ^-0.6		cm⁶/s	[13]

Reactions	Einstein spontaneous emission coefficients (s⁻¹)	References
Ar(1s₄) → Ar(1S₀)	A₂₀ = 5.7 × 10⁵	[10]
Ar(2p₁₀) → Ar(1s₅)	A₃₁ = 1.87 × 10⁷	[24]
Ar(2p₁₀) → Ar(1s₄)	A₃₂ = 5.43 × 10⁶	[24]
Ar(2p₁₀) → Ar(1s₃)	A_p10→s3 = 9.8 × 10⁵	[24]
Ar(2p₁₀) → Ar(1s₂)	A_p10→s2 = 1.9 × 10⁵	[24]
Ar(2p₉) → Ar(1s₅)	A₄₁ = 3.31 × 10⁷	[24]
Ar(2p₈) → Ar(1s₅)	A₅₁ = 9.28 × 10⁶	[24]
Ar(2p₈) → Ar(1s₄)	A₅₂ = 2.15 × 10⁷	[24]
Ar(2p₈) → Ar(1s₂)	A_p8→s2 = 1.47 × 10⁶	[24]

Reactions	Rate constants k_ij	Temperature	Units	References
Ar(1s₄) + Ar → Ar(1s₅) + Ar	1 × 10⁻¹³	300 K	cm³/s	[13]
Ar(1s₄) + He → Ar(1s₅) + He	1.2 × 10⁻¹³	303 K	cm³/s	[13]
Ar(2p₁₀) + Ar → Ar(1s₅) + Ar	0.6 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₁₀) + He → Ar(1s₅) + He	5 × 10⁻¹⁴	300 K	cm³/s	[13]
Ar(2p₉) + Ar → Ar(1s₅) + Ar	2.5 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₉) + He → Ar(1s₅) + He	2 × 10⁻¹²	300 K	cm³/s	[13]
Ar(2p₈) + Ar → Ar(1s₅) + Ar	1.5 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + He → Ar(1s₅) + He	1 × 10⁻¹²	300 K	cm³/s	[13]
Ar(2p₉) + Ar → Ar(2p₁₀) + Ar	2.6 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₉) + He → Ar(2p₁₀) + He	1.6 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + Ar → Ar(2p₁₀) + Ar	1.1 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + He → Ar(2p₁₀) + He	4 × 10⁻¹²	300 K	cm³/s	[13]
Ar(2p₈) + Ar → Ar(2p₉) + Ar	1.1 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(2p₈) + He → Ar(2p₉) + He	4.5 × 10⁻¹¹	300 K	cm³/s	[13]
Ar(1s₅) + Ar + Ar → Ar₂* + Ar	36 × 10⁻³² × T ^-0.6		cm⁶/s	[13]
Ar(1s₅) + Ar + He → Ar₂* + He	18 × 10⁻³² × T ^-0.6		cm⁶/s	[13]

Reactions	Einstein spontaneous emission coefficients (s⁻¹)	References
Ar(1s₄) → Ar(1S₀)	A₂₀ = 5.7 × 10⁵	[10]
Ar(2p₁₀) → Ar(1s₅)	A₃₁ = 1.87 × 10⁷	[24]
Ar(2p₁₀) → Ar(1s₄)	A₃₂ = 5.43 × 10⁶	[24]
Ar(2p₁₀) → Ar(1s₃)	A_p10→s3 = 9.8 × 10⁵	[24]
Ar(2p₁₀) → Ar(1s₂)	A_p10→s2 = 1.9 × 10⁵	[24]
Ar(2p₉) → Ar(1s₅)	A₄₁ = 3.31 × 10⁷	[24]
Ar(2p₈) → Ar(1s₅)	A₅₁ = 9.28 × 10⁶	[24]
Ar(2p₈) → Ar(1s₄)	A₅₂ = 2.15 × 10⁷	[24]
Ar(2p₈) → Ar(1s₂)	A_p8→s2 = 1.47 × 10⁶	[24]

Modeling of scaling of a diode longitudinally pumped metastable rare gas with a master oscillator power amplifier

Abstract

1. Introduction

2. Physical model of DPRGA description

3. Comparison with Rawlins et al.’s experimental results

4. Simulation results and discussions

4.1 Influence of the initial Ar(1s₅) density

4.2 Influence of the gain length

4.3 Influence of pump laser and seed laser intensities

5. Conclusions

Funding

References

Cited By

Figures (7)

Tables (2)

Equations (18)

Optics Express

Abstract

1. Introduction

2. Physical model of DPRGA description

3. Comparison with Rawlins et al.’s experimental results

4. Simulation results and discussions

4.1 Influence of the initial Ar(1s5) density

4.2 Influence of the gain length

4.3 Influence of pump laser and seed laser intensities

5. Conclusions

Funding

References

Cited By

Figures (7)

Tables (2)

Equations (18)

Optics Express

4.1 Influence of the initial Ar(1s₅) density