Numerical simulations on a nanosecond-pulse exciplex pumped cesium vapor laser

Chenyi Su; Chenyi Su; Xingqi Xu; Xingqi Xu; Jinghua Huang; Bailiang Pan

doi:10.1364/OE.398615

1. Introduction

Diode-pumped atomic alkali vapor laser (DPAL) has grown as an indispensable light source in modern scientific research since it has dual advantages of solid-state and gas-phase lasers, which mainly includes high outputting power, efficient energy conversion, good beam quality and excellent thermal management [1]. The concept of alkali vapor laser was proposed by Schawlow and Townes in 1958 [2]. However, due to the limitation of the pump source, it did not attract widespread attention until the realizations of Ti: Sapphire pumped Rb vapor laser in 2003 [3] and diode-pumped Cs vapor laser in 2005 [4], respectively. So far, the outputting power of DPAL has been increased from mW to kW through the applications of efficient pump configurations, high-power pump sources and cooling systems [1,3,5–9], proving its potential in high-power laser region.

However, the mismatching between alkali atomic absorption and laser diodes array (LDA) emission leads not only a low energy conversion efficiency but also a severe thermal accumulation in the vapor cell [10]. In general, the linewidth of LDA is two orders of magnitude greater than that of alkali atomic absorption, which means thousands of torr of noble gas is required as a broadening medium in an efficient high-power DAPL [10]. In 2008, Readle et al. experimentally reported a new class of alkali laser, excimer pumped alkali vapor laser (XPAL), whose laser-active medium is excimer pair of alkali atom and noble gas [11]. Especially, its resonance absorption linewidth is broadened to nanometer scale [12]. Palla et al. presented a theoretical model for XPAL systems and compared it with the experimental results, and further simulated its operating performance [13–15]. The results showed that XPAL has the potential of high-power laser in CW (continuous-wave) operation [15]. However, the severe heat accumulation in the operation process hindered the experimental research of XPAL [16]. Therefore, the method of introducing a super-fast gas flow into the vapor cell has been proposed, and the results showed that the sonic-level gas flow is required to control the thermal effects in a CW XPAL, which is difficult in experiment t [16–21]. On the other hand, the operation mode of QCW (quasi-continuous-wave) pulse pumped can effectively suppress the thermal effect [22,23]. Thus, the research of nanosecond-pulse pumped XPAL is very significant for the development of high-power XPAL.

Inspired from the methodology of “ab initio” and “ray tracing”, a physical model for the time evolution of kinetic processes and laser beam propagation are established in this paper, in which the non-stationary rate equations is solved by ordinary differential equation (ODE). On basis of the validation of our simulation result by comparing with experimental data, we further depict the microphysical picture in the ignition process of short-pulse pumped XPAL system, providing an insight into the influence of pump energy, cell temperature and buffer gas on its output characteristics.

2. Description of model

2.1 Laser kinetic processes and rate equations in the high-power XPCsL systems

An energy diagram of high-power XPCsL is depicted in Fig. 1, giving a clear physical picture of the kinetic processes of XPCsL: the laser-active state, ${X^2}\Sigma _{1/2}^ + $, is formed due to the gradient force of intermolecular potential between Cs and Ar gases (thermal equilibrium), providing a resonance with blue-shifted D2 line of Cs. Afterwards, the excited excimer trends to dissociate into unbounded Cs and Ar atoms, respectively. Cs atom at ${6^2}P_{3/2}^{}$ state can decay to the ground state directly with emitting a photon at 852.3 nm, or with the help of relaxation medium, Cs can relax to ${6^2}P_{1/2}^{}$ state firstly and return to ground state with releasing a photon at 894.6 nm (Fig. 1, left-bottom). However, as described in the rate equations, excited Cs can be further pumped to higher excited states by secondary photoexcitation, and photoionization occurs. It should be noted that the cross-sections for secondary photoexcitation and photoionization are reduced to constants since their enegry gaps are both off-resonant with incident frequency, and the three higher excited states (${6^2}D_{3/2}^{},\textrm{ }{6^2}D_{5/2}^{},\textrm{ }{8^2}S_{1/2}^{}$) are merged to one whose degeneracy coefficient is also reduced to a constant (the summation of three levels) [12,15]. Based on the atom physics, the simplifications here can be without loss of generality but improve the efficiency of simulation calculations.

Fig. 1. Diagram of energy states in high-power XPCsL system

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The rate equations for high-power XPCsL can be written as

(1)$$\begin{array}{c} \frac{{d{n_0}}}{{dt}} ={-} {k_{01}}{n_0}[Ar] + {k_{10}}{n_1} + {R_2} + {k_{pen}}{n_3}{n_5} + {k_{pen}}{n_4}{n_5} + {L_{D2}}\\ \textrm{ } + {\gamma _q}{n_3} + {A_{30}}{n_3} + {k_{35}}n_3^2 + {L_{D1}} + {A_{40}}{n_4} + {k_{45}}n_4^2, \end{array}$$

(2)$$\frac{{d{n_1}}}{{dt}} = {k_{01}}{n_0}[Ar] - {k_{10}}{n_1} - F,$$

(3)$$\frac{{d{n_2}}}{{dt}} = F - {k_{23}}{n_2} + {k_{32}}{n_3}[Ar],$$

(4)$$\begin{array}{c} \frac{{d{n_3}}}{{dt}} = {k_{23}}{n_2} - {k_{32}}{n_3}[Ar] - {R_{34}} - {L_{D2}} - {\gamma _{31}}{n_3} - 2{k_3}_{ai}n_3^2\\ \textrm{ } - 2{k_{35}}n_3^2 - \sum\limits_{i = p,D1,D2} {F_i^{(35)}} - {k_{pen}}{n_3}{n_5} + {A_{53}}{n_5}, \end{array}$$

(5)$$\frac{{d{n_4}}}{{dt}} = {R_{34}} - {L_{D1}} - 2{k_{45}}n_4^2 - \sum\limits_{i = p,D1,D2} {F_i^{(45)}} - {k_{pen}}{n_4}{n_5} + {A_{54}}{n_5} - {\gamma _{41}}n4,$$

(6)$$\begin{array}{c} \frac{{d{n_5}}}{{dt}} = {k_{35}}n_3^2\textrm{ + }{k_{45}}n_4^2 - {k_{pen}}({n_3}{n_5} + {n_4}{n_5}) - {A_{53}}{n_5} - {A_{54}}{n_5}\\ \textrm{ + }{R_2} + {R_{3b}} - {F_{ion}} + \sum\limits_{i = p,D1,D2} {(F_i^{(35)} + F_i^{(45)})} \end{array}$$

(7)$$\frac{{d{n_6}}}{{dt}} = {F_{ion}} + {k_{pen}}({n_3}{n_5} + {n_4}{n_5}) - {R_1} - {R_{3b}},$$

(8)$$\frac{{d{n_7}}}{{dt}} = {k_{3ai}}n_3^2 - {R_2} + {R_1},$$

where ${n_i}(i = 0,1,2,3,4,5,6,7)$ denotes the population densities of energy state, subscripts of 0–7 represent different atomic states including atomic ground state, ${6^2}{S_{1/2}}$, excimer ground state, ${X^2}\Sigma _{1/2}^ + $, excimer excited state, ${B^2}\Sigma _{1/2}^ + $, atomic D2/D1 states, ${6^2}P_{3/2}^{}$, ${6^2}P_{1/2}^{}$, atomic higher excited states, ${6^2}D_{3/2,5/2}^{}({8^2}S_{1/2}^{})$, ionized ground state, $C{s^ + }$ and associative state, $Cs_2^ + $, respectively. The mixing rate between energy levels is due to the thermal collision. In Boltzmann statistics, the relaxation rate, ${R_{34}}$, can be expressed as

(9)$${R_{34}} = {\gamma _{34}}({n_3} - 2{n_4}\exp ( - \frac{{\Delta {E_{34}}}}{{{k_b}T}}))$$

where ${k_b}$ is Boltzmann constant, $\Delta {E_{34}}$ is the energy gap between Cs D2 and D1 states. The collisional coefficient, ${\gamma _{ij}}$ is determined by the buffer gas and can be obtained as follows,

(10)$${\gamma _{ij}} = {n_{{C_2}{H_6}}}{\sigma _{ij}}{\left[ {3{k_b}T\left( {\frac{1}{{{m_{Ar}}}} + \frac{1}{{{m_{{C_2}{H_6}}}}}} \right)} \right]^{1/2}},$$

where ${\sigma _{ij}}$ is the corresponding collisional cross-section between states $i \to j$. ${n_{{C_2}{H_6}}}$ is the number density of ethane. ${m_{Ar}}$ and ${m_{{C_2}{H_6}}}$ are molar masses of Ar and ${C_2}{H_6}$, respectively. Since the absorption line of Cs-Ar excimer is very broad, the wavelength dependent cross-section for the optcial pump, ${\sigma _p}$, should be taken into account. In integral form, pump rate is

(11)$$F = \frac{{\int {{I_p}(\nu ){\sigma _p}d\nu } }}{{h{\nu _p}}},$$

where ${I_p}$ is the intensity of pulsed pump beam in Gaussian profile,

(12)$$\begin{array}{ll} {I_p}(0,\nu ,t) &= {E_p} \cdot {f_{beam}}(0) \cdot {f_{pulse}}(t) \cdot \\ &\textrm{ }\frac{{\sqrt {4\ln 2/\pi } }}{{\Delta {\nu _p}}}\exp [ - 4\ln 2 \cdot {(\frac{{\nu - {\nu _p}}}{{\Delta {\nu _p}}})^2}], \end{array}$$

where ${E_p}$ denotes the energy of the pump light. ${f_{beam}}$ and ${f_{pulse}}$ indicate the distribution of the transmitted beam in the cavity and the gaussian profile of the pump beam, respectively. ${\nu _p}$ and $\Delta {\nu _p}$ are the central frequency and the FWHM of the pump beam, respectively.

(13)$${f_{beam}}(x,r) = \frac{{\ln 2}}{{\pi {w_p}^2(x)}}\exp [ - \ln 2 \cdot {(\frac{r}{{{w_p}(x)}})^2}].$$

In order to improve the computational efficiency without loss of accuracy, $r = {w_{p0}}$ in ${f_{beam}}(x)$ is taken to describe the light intensity distribution in the x-axis direction. And ${w_p}(x)$ is the radii of the pump beam on the y–z plane:

(14)$${w_p}(x) = {w_{p0}}\sqrt {1 + {{[\frac{{{c_1}(x - L/2)}}{{k({w_{p0}}/2)}}]}^2}} .$$

${f_{pulse}}$ can be expressed as

(15)$${f_{pulse}}(t) = \sqrt {\frac{{4\ln 2}}{{\pi {{({\lambda _{FWHM}}/2)}^2}}}} \exp [ - \ln 2{(\frac{{t - {t_c}}}{{{\lambda _{FWHM}}/2}})^2}],$$

where ${\lambda _{FWHM}}$ is the FWHM of the pump beam (we choose ${\lambda _{FWHM}} = 4.3 \times {10^{ - 9}}s$ as the pulse duration). ${t_c}$ is the time when the pump light reaches the apex. The time variable, t, in Eq. (15) turns the rate equation to non-stationary, indicating that the time derivatives in the left-hand side of Eqs. (1)–(8) are no longer zero. The time-evolution lasing processes in the high-power XPCsL can be re-written as

(16)$$\begin{array}{ll} {L_{D1}} &= L_{D1}^ +{+} L_{D1}^ -{=} \frac{{dn_{D1}^ + }}{{dt}} + \frac{{dn_{D1}^ - }}{{dt}}\\ &= c{\sigma _{D1}}({n_4} - {n_0})(n_{D1}^ +{+} n_{D1}^ - ) + {A_{40}}{n_4} - \frac{{{\alpha _{pass}}c(n_{D1}^ +{+} n_{D1}^ - )}}{{dx}}, \end{array}$$

(17)$$\begin{array}{ll} {L_{D2}} &= L_{D2}^ +{+} L_{D2}^ -{=} \frac{{dn_{D2}^ + }}{{dt}} + \frac{{dn_{D2}^ - }}{{dt}}\\ &= c{\sigma _{D2}}({n_3} - 2{n_0})(n_{D2}^ +{+} n_{D2}^ - ) + {A_{30}}{n_3} - \frac{{{\alpha _{pass}}c(n_{D2}^ +{+} n_{D2}^ - )}}{{dx}}, \end{array}$$

where ${L_{D1}}$ and ${L_{D2}}$ are respectively represented lasing rates of 895 nm and 852 nm, and ${n_{D1}}$, ${n_{D2}}$ represent the number densities of photons of ${D_1}$ and ${D_2}$ transitions. Superscripts “+” and “-” indicate the lasing beams in positive and negative directions (as shown in Fig. 2). Furthermore, number density of photon can be turned into flux (intensity) by

(18)$$I_{D1}^ \pm{=} h \cdot {\nu _{D1}} \cdot c \cdot n_{_{D1}}^ \pm ,$$

(19)$$I_{D2}^ \pm{=} h \cdot {\nu _{D2}} \cdot c \cdot n_{_{D2}}^ \pm ,$$

Fig. 2. Sketch of an optical system of a XPCsL.

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In short pulse-duration, laser oscillation in the cavity is not as stable as in CW operation, so that spontanesous emission has an important influence on the short pulse laser system, the rates, ${A_{30}}$ and ${A_{40}}$ are determined by

(20)$${A_{40}} = \frac{{\phi (\nu )}}{{s(\nu )V{\tau _{40}}}},$$

(21)$${A_{30}} = \frac{{\phi (\nu )}}{{s(\nu )V{\tau _{30}}}},$$

where V is the volume of a segment, $\nu $ and $\tau $ are the frequency and lifetime of corresponding energy levels, respectively. $\phi (\nu )$ is the uniformed Lorentzian profile of the pressure broadening. $s(\nu )$ is a function of state density, indicating the mode number in a longitudinal frequency scale, which can be expressed as

(22)$$s(\nu ) = \frac{{8\pi {\nu ^2}}}{{{c^3}}},$$

where c is the vaccum speed of light.

${\alpha _{pass}}$ denotes the coefficient of single-pass loss in the cavity,

(23)$${\alpha _{pass}} ={-} \frac{1}{2}\ln ({R_{OC}}{R_{HR}}),$$

where ${R_{OC}}$ and ${R_{HR}}$ are the reflectivities of output coupler and high reflector.

Secondary photoexcitation and ionization in high-power XPCsL can severely influence the lasing power and energy conversion efficiency, and promote the formation of low-temperture plasma. In the non-stationary rate equations, Eqs. (1)–(8), the rates of pump-induced excitation, $F_p^{(35,45)}$, laser-induced exciatation, $F_{D1,D2}^{(35,45)}$, and ionization processes, ${F_{ion}}$ can be computed as follow,

(24)$$F_p^{(35,45)} = \sigma _p^{(35,45)}{n_p}c(3{n_{3,4}} - {n_5}),$$

(25)$$F_{D1,D2}^{(35)} = \sigma _{D1,D2}^{(35)}c(n_{D1,D2}^ +{+} n_{D1,D2}^ - )(3{n_3} - {n_5}),$$

(26)$$F_{D1,D2}^{(45)} = \sigma _{D1,D2}^{(45)}c(n_{D1,D2}^ +{+} n_{D1,D2}^ - )(6{n_4} - {n_5}),$$

(27)$$\begin{array}{ll} {F_{ion}} &= c[{\sigma _{p\_ion}}{n_p} + {\sigma _{D1\_ion}}(n_{D1}^ +{+} n_{D1}^ - )\\ &\quad + {\sigma _{D2\_ion}}(n_{D2}^ +{+} n_{D2}^ - )][3({g_5}{n_5} - {g_{ion}}{n_{ion}})], \end{array}$$

where the degeneracy coefficient of Cs ion is zero due to its capacity of receiving electron is infinite. The kinetic processes involved in the simulation can be found in Table 1.

Table 1. Kinetic processes in the XPAL system.

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2.2 Algorithm methods

The sketch of optical resonator is presented in Fig. 2. The length of cavity L = 1 m and the Cs-Ar cell placed in the center has a length of 6 cm and a diameter of 2.5 cm. The Gaussian pump pulse whose spectral and pulse linewidths are 7 GHz and 4.3 ns respectively are introduced from the output coupler (OC), and then propagate along the x-axis. Unlike continuous-wave (CW) or quasi-CW operation alkali vapor laser, the non-stationary rate equations for XPCsL have to be solved by the methodology of ordinary differential equation (ODE). By using the “Runge-Kutta” method, we are allowed to obtain a series of numerical results. Specifically, in our simulation, $\Delta t\textrm{ = }0.02$ ns is chosen as time accuracy, since it is much longer than the typical time of light–matter interaction(${10^{ - 18}}$s).

The propagation of the laser beam in the cavity (without gain medium) is described as

(28)$$I_{D1,D2}^ + (x + \Delta x,t + \Delta t) = I_{D1,D2}^ + (x,t) \cdot {f_{beam}}(x)/{f_{beam}}(x + \Delta x),$$

(29)$$I_{D1,D2}^ - (x - \Delta x,t + \Delta t) = I_{D1,D2}^ - (x,t) \cdot {f_{beam}}(L - x)/{f_{beam}}(L - x + \Delta x).$$

At the boundary of the cavity and the Cs-Ar cell, we have

(30)$$I_{D1,D2}^ + (0,t + \Delta t) = I_{D1,D2}^ - (0,t) \cdot {R_{oc}} \cdot {T_s},$$

(31)$$I_{D1,D2}^ - (L,t + \Delta t) = I_{D1,D2}^ + (L,t) \cdot {R_p} \cdot {T_s},$$

(32)$$I_{D1,D2}^ - (x - \Delta x,t + \Delta t) = I_{D1,D2}^ - (x,t) \cdot {T_l} \cdot \frac{{{f_{beam}}(L - x)}}{{{f_{beam}}(L - x + \Delta x)}},(x = {x_1},{x_2}),$$

(33)$$I_{D1,D2}^ + (x + \Delta x,t + \Delta t) = I_{D1,D2}^ + (x,t) \cdot {T_l} \cdot \frac{{{f_{beam}}(x)}}{{{f_{beam}}(x + \Delta x)}},(x = {x_1},{x_2}),$$

where ${R_{oc}}$ and ${R_p}$ are the reflectivity of the output coupler and the high reflector, respectively. ${T_l}$ is the single-pass cell window transmission and ${T_s}$ is the intracavity single-pass loss.

The propagation of the pump beam in the cavity is described by

(34)$${I_p}(x + \Delta x,\nu ,t + \Delta t) = {I_p}(x,t) \cdot \frac{{{f_{beam}}(x)}}{{{f_{beam}}(x + \Delta x)}},x < {x_1}\textrm{ }or\textrm{ }x > {x_{2,}}$$

(35)$$ \begin{array}{c} I_p(x + \Delta x,\nu ,t + \Delta t) = \\ I_p(x,t)\exp \{ -\sigma _p[n_1(x,t)-n_2(x,t)]-\sigma _p^{(35)} [3n_3(x,t)-n_5(x,t)] \\ -\sigma _p^{(45)} [6n_4(x,t)-n_5(x,t)]-\sigma _{p\_ion}[3n_5(x,t)]\} ,x_1 < x < x_{2.} \end{array} $$

The iterative calculating ﬂow chart is shown in Fig. 3.

Fig. 3. Flow diagram of iterative operation.

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As shown in the flowchart, the latest coordinate x of the pump and laser beam can be attained by the calculation of time t. If the coordinate x is in the cell, the ode function in the MATLAB can be used to solve the rate equations with the initial values of the number density of energy levels and photons of D1 and D2 transitions at time $t - \Delta t$. After getting the particle number density of the energy levels and photons at time t, the laser intensity at t and the pump intensity at $t + \Delta t$ are obtained; Otherwise, the propagation equation and boundary conditions was used to simulate the propagation of pump and laser beams. In summary, the evolution of the laser intensity over time can be obtained by multiple iterative loops.

3. Results and discussion

Figure 4 is the comparison between experimental (a) [11] and simulation (b) results, which can verify the accuracy of the model. With the parameters shown in Table 2, a good agreement of simulation results with experiment is shown up with a ∼7.5 ns of time delay between pump and laser pulse. However, there is a ∼0.9 ns of difference in the FWHM of the lasing signal between simulation and experimental result. The lasing condition is ideal in the simulation while inevitable perturbation in the experiment can decay the density inversion rapidly. As a result, the simulated laser pulse appears earlier and its rising curve was smoother.

Fig. 4. Experimental (a) [11] and simulation (b) results of time evolution of pump and laser pulses.

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Table 2. Parameters of experiment and simulation.

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Figure 5 calculates the effect of cell temperature on output laser pulse. It can be seen from Fig. 5(a) that the peak of laser pulse and the delay time between pump and laser peaks increase with rising the cell temperature. Higher temperature means higher particle number density of Cs, which leads to a higher laser pulse power. However, the time to establish the inversion of the number density between energy levels of Cs also increases. Figure 5(b) shows the peak of each laser pulse first rises and then decreases as the temperature increases, which means, there is an optimum temperature of Cs-Ar cell for a specific pump energy.

Fig. 5. (a) The relationship between the intensity of lasing pulses (D1 line) and time at different temperatures; (b) The maximum intensity of each pulse as a function of temperature. The energy of pump pulse is 6 mJ and the pump pulse is Gaussian profile at each case.

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Figures 6(a) and 6(b) presents the effect of pump energy on laser pulses. As shown in Fig. 6(a), the peak of lasing pulse increases with the pump energy while the delay time between pump and laser pulses decreases with it. When the temperature is fixed, the number density of particles of Cs in the cell is constant. The pump light all have a profile of Gaussian. Based on the above reasons, rising the pump energy can increase the intensity of output laser pulse while make the pump pulse reach the pump threshold earlier in time, resulting in the laser output pulse appearing earlier. In Fig. 6(b), for each specific temperature, the peak of laser pulse increases with increasing pump energy, and then gradually approaches saturation.

Fig. 6. (a) Temporal behavior of laser pulses (D1 line) on different pump energy with cell temperature of 423 K; (b) The maximum intensity of each laser pulse as a function of pump energy at different cell temperature.

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Figure 7 shows the outputting laser of the D1(894.6nm) and D2(852.3nm) lines with different concentration of buffer gas (${C_2}{H_6}$). We can see that D1 line is positively proportioned to the pressure of ethane, and further reaches simultaneously with D2 line when 10-15 Torr of ethane is added. Besides, the D2 line always appears earlier than the D1 line, and the time interval between the peak of D1 and D2 pulses decreases with the addition of ethane. It can be seen in Fig. 7(d), the slope of the curve representing the number density of ${6^2}{P_{1/2}}$ grows with the concentration of ethane, suggesting the promoting effect of buffer gas in the relaxation process. It is worth mentioning that the laser pulses shown in Figs. 7(a)–7(c) are measured at $x = 0$ while the number density in Fig. 7(d) is at $x = {x_1}$, where a 1.6 ns of time difference exists between the two coordinates.

Fig. 7. The ratio of D2 line (852 nm) and D1 line (895 nm) at $x = 0$ as the function of the concentration of ${C_2}{H_6}$. (a) 0 Torr of ${C_2}{H_6}$; (b) 10 Torr of ${C_2}{H_6}$; (c) 15 Torr of ${C_2}{H_6}$; (d) The temporal behavior of the number density of ${6^2}{P_{2/3}}$ and ${6^2}{P_{1/2}}$ at $x = {x_1}$ with different concentration of ${C_2}{H_6}$. The pump energy is 18 mJ and the cell temperature is 423 K.

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4. Summary

In this article, we established a dynamical model for a short–pulse XPAL system with a nanosecond pump pulse. By solving the non-stationary differential rate equations including higher excited and ionized energy levels, we simulated the laser kinetics properties of a nanosecond-pulse XPCsL and figured out the temporal behavior of laser pulses and the dependences of outputting laser performances on buffer gas, cell temperature and pump energy. The stimulation results were consistent with the experiment data and showed unique properties of a short-pulse XPAL. Increasing the temperature of cell and pump energy can both increase the energy of output laser pulse, but the former causes the laser pulse relatively delay in time domain, whereas the latter makes it appear earlier. Moreover, there is an optimal temperature for a specific pump energy while the outputting energy has its maximum limit for a specific cell temperature. Besides, the output characteristics of the double-line laser with different concentration of ethane are also shown in detail. This work can help to understand the laser dynamics and characteristics of a nanosecond-pulsed XPAL system and facilitate the design of an efficient short-pulse XPAL in experiment.

Funding

China Aerospace Science and Technology Corporation (KM20170269).

Disclosures

The authors declare no conflicts of interest.

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No.	Process	Cross Section/Rate	References
	Thermal associative/ dissociative process
1	$C s (6^{2} S_{1 / 2}) + A r ⇌ C s (X^{2} Σ_{1 / 2}^{+}) A r$	$k_{01}, k_{10}$	[15]
2	$C s (B^{2} Σ_{1 / 2}^{+}) A r + A r ⇌ C s (6^{2} P_{3 / 2}) + A r$	$k_{23}, k_{32}$	[15]
	Pumping
3	$C s (X^{2} Σ_{1 / 2}^{+}) A r + h ν_{p} \to C s (B^{2} Σ_{1 / 2}^{+}) A r$	F	Eq. (11)
	Lasing
4	$C s (6^{2} P_{3 / 2}, 6^{2} P_{1 / 2}) \to C s (6^{2} S_{1 / 2}) + h ν_{l}$	L	Eqs. (16)–(17)
	Spontaneous emission
5	$C s (6^{2} P_{3 / 2}) \to C s (6^{2} S_{1 / 2}) + h ν_{l}$	$A_{30}$	[4]
6	$C s (6^{2} P_{1 / 2}) \to C s (6^{2} S_{1 / 2}) + h ν_{l}$	$A_{40}$	[4]
7	$C s (6^{2} D_{3 / 2}, 6^{2} D_{5 / 2}) \to C s (6^{2} P_{3 / 2}) + h ν$	$A_{53}$	[17]
8	$C s (6^{2} D_{3 / 2}) \to C s (6^{2} P_{1 / 2}) + h ν$	$A_{54}$	[17]
	Relaxation
9	$C s (6^{2} P_{3 / 2}) + C_{2} H_{6} \to C s (6^{2} P_{1 / 2}) + C_{2} H_{6}$	$R_{34}$	[17]
	Quenching
10	$C s (6^{2} P_{3 / 2}) + C_{2} H_{6} \to C s (6^{2} S_{1 / 2}) + C_{2} H_{6}$	$γ_{31}$ , $γ_{41}$	[17]
	Photoexcitation
11	$C s (6^{2} P_{3 / 2}, 6^{2} P_{1 / 2}) + h ν_{p} (h ν_{l}) \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2})$	$F_{p}, F_{D 1, 2}$	Eqs. (24)–(26)
	Energy pooling
12	$2 C s (6^{2} P_{3 / 2}, 6^{2} P_{1 / 2}) \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + C s (6^{2} S_{1 / 2})$	$k_{35}, k_{45}$	[24]
	Photoionization
13	$C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + h ν_{p, l} \to C s^{+} + e$	$F_{i o n}$	Eq. (27)
	Associative ionization
14	$2 C s (6^{2} P_{3 / 2}) \to C s_{2}^{+} + e$	$k_{3 a i}$	[25]
	Penning ionization
15	$C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + C s (6^{2} P_{3 / 2, 1 / 2}) \to C s^{+} + C s (6^{2} S_{1 / 2}) + e$	$k_{p e n}$	[25]
	Three-body collision
16	$C s^{+} + e^{-} + e^{-} \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + e^{-}$	$R_{3 b}$	[26]
	Dissociative recombination
17	$C s^{+} + C s (6^{2} S_{1 / 2}) + C s \to C s_{2}^{+} + C s$	$R_{1}$	[17]
18	$C s^{+} + C s (6^{2} S_{1 / 2}) + A r \to C s_{2}^{+} + A r$	$R_{1}$
19	$C s_{2}^{+} + e \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + C s (6^{2} S_{1 / 2})$	$R_{2}$

No.	Process	Cross Section/Rate	References
	Thermal associative/ dissociative process
1	$C s (6^{2} S_{1 / 2}) + A r ⇌ C s (X^{2} Σ_{1 / 2}^{+}) A r$	$k_{01}, k_{10}$	[15]
2	$C s (B^{2} Σ_{1 / 2}^{+}) A r + A r ⇌ C s (6^{2} P_{3 / 2}) + A r$	$k_{23}, k_{32}$	[15]
	Pumping
3	$C s (X^{2} Σ_{1 / 2}^{+}) A r + h ν_{p} \to C s (B^{2} Σ_{1 / 2}^{+}) A r$	F	Eq. (11)
	Lasing
4	$C s (6^{2} P_{3 / 2}, 6^{2} P_{1 / 2}) \to C s (6^{2} S_{1 / 2}) + h ν_{l}$	L	Eqs. (16)–(17)
	Spontaneous emission
5	$C s (6^{2} P_{3 / 2}) \to C s (6^{2} S_{1 / 2}) + h ν_{l}$	$A_{30}$	[4]
6	$C s (6^{2} P_{1 / 2}) \to C s (6^{2} S_{1 / 2}) + h ν_{l}$	$A_{40}$	[4]
7	$C s (6^{2} D_{3 / 2}, 6^{2} D_{5 / 2}) \to C s (6^{2} P_{3 / 2}) + h ν$	$A_{53}$	[17]
8	$C s (6^{2} D_{3 / 2}) \to C s (6^{2} P_{1 / 2}) + h ν$	$A_{54}$	[17]
	Relaxation
9	$C s (6^{2} P_{3 / 2}) + C_{2} H_{6} \to C s (6^{2} P_{1 / 2}) + C_{2} H_{6}$	$R_{34}$	[17]
	Quenching
10	$C s (6^{2} P_{3 / 2}) + C_{2} H_{6} \to C s (6^{2} S_{1 / 2}) + C_{2} H_{6}$	$γ_{31}$ , $γ_{41}$	[17]
	Photoexcitation
11	$C s (6^{2} P_{3 / 2}, 6^{2} P_{1 / 2}) + h ν_{p} (h ν_{l}) \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2})$	$F_{p}, F_{D 1, 2}$	Eqs. (24)–(26)
	Energy pooling
12	$2 C s (6^{2} P_{3 / 2}, 6^{2} P_{1 / 2}) \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + C s (6^{2} S_{1 / 2})$	$k_{35}, k_{45}$	[24]
	Photoionization
13	$C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + h ν_{p, l} \to C s^{+} + e$	$F_{i o n}$	Eq. (27)
	Associative ionization
14	$2 C s (6^{2} P_{3 / 2}) \to C s_{2}^{+} + e$	$k_{3 a i}$	[25]
	Penning ionization
15	$C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + C s (6^{2} P_{3 / 2, 1 / 2}) \to C s^{+} + C s (6^{2} S_{1 / 2}) + e$	$k_{p e n}$	[25]
	Three-body collision
16	$C s^{+} + e^{-} + e^{-} \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + e^{-}$	$R_{3 b}$	[26]
	Dissociative recombination
17	$C s^{+} + C s (6^{2} S_{1 / 2}) + C s \to C s_{2}^{+} + C s$	$R_{1}$	[17]
18	$C s^{+} + C s (6^{2} S_{1 / 2}) + A r \to C s_{2}^{+} + A r$	$R_{1}$
19	$C s_{2}^{+} + e \to C s (6^{2} D_{3 / 2, 5 / 2}, 8^{2} S_{1 / 2}) + C s (6^{2} S_{1 / 2})$	$R_{2}$

Numerical simulations on a nanosecond-pulse exciplex pumped cesium vapor laser

Abstract

1. Introduction

2. Description of model

2.1 Laser kinetic processes and rate equations in the high-power XPCsL systems

2.2 Algorithm methods

3. Results and discussion

4. Summary

Funding

Disclosures

References

Cited By

Figures (7)

Tables (2)

Equations (35)

Optics Express