## Abstract

Calculations on the influence of compressible gas effects on the performances of exciplex pumped alkali vapor lasers (XPALs) with supersonic flow system are made. Modified configuration of experimental apparatus for XPALs is designed in this paper as well. On the basis of fluid mechanics and thermodynamics, theoretical derivations for the boundary of temperature gradient which actually is the position of shock wave plane are firstly deduced and presented. And our simulated results show that XPALs have broadband of not only in absorption spectrum but also in pump power, which will be helpful for researchers to find the way to make very high-power gas lasers. In addition, the simulating method to obtain the operating temperature of XPALs is given, and predicted the value, at the conditions of this paper, is 1500K with optimized parameters of *Pp* = $3\times {10}^{6}$ W, ${T}_{stag}=513$ K, initial Mach number = 2.5.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Diode pumped alkali vapor laser (DPAL) is regarded as an alternative solution in significantly high-power regime. Since it was firstly realized by Krupke et al in 2003 [1], DPAL has attracted many researchers’ attentions for its some advantages over laser diodes (LDs) and ordinary gas lasers, which mainly includes high output power with good beam quality. However, the development of DPALs is indeed hampered because of some difficulties. The principle one of them is the mismatch of the absorption of metal atoms and the stimulated emission of commercial laser diode arrays (LDAs) in spectral width [1]. Many fundamental works have shown that the natural alkali atomic absorption can be broadened to less than 1GHz by Doppler broadening but the output performance of commercial LDA in spectrum is about 1THz [2]. To address this spectral difference, researchers presented that helium can be added into the cell to broaden the atomic absorption to 10GHz and volume Bragg granting (VBG) can be used to decrease the output linewidth of LDAs to 100GHz. Bogachev et al have obtained the DPAL with output power of 1000 watts [3]. Nevertheless, the spectral difference is not absolutely eliminated, it will become critical when the output power of DPAL reaches higher watts. From then on, scientists have ever sought for more modifications on theories and experimental designations. In 2008, researchers in university of Illinois Urbana-Champaign presented a new class of photoassociation laser [4–6], excimer pumped alkali vapor laser (XPAL). In their work, stimulated emission output of Cesium atoms and Krypton has been firstly realized, and absorption profiles of 5 nm and quantum efficiencies of 98% have been obtained in oscillator, showing good compatibility of XPAL system with commercial LDAs. Then Palla and Carroll did valuable theoretical studies on the atomic associative and dissociative kinetic processes [7, 8], demonstrating that a BLAZE-V model to describe the math calculations on XPAL systems. Then Carroll and Verdeyen et al put forward a simple equilibrium model to predict the XPAL system with CW operating [9]. The thermal effect of XPAL system has been investigated by Pan’s team, demonstrating that very large heat loading is generated by the multiple transitions and dramatically steep gradient of temperature is built in the cell, which may make the experiment apparatus melt down and stimulated emission quenched. So, they suggested to introduce the gas flow into the system to reduce the temperature gradients. However, they found that good cooling effect would be gathered when the velocity of gas flow reached hundreds of meters per second [10] and the compressibility of gas would take an important role in performance. Pan’ team applied the fluid methodology to the CW XPAL system, and obtained some profound simulated results [11]. In their work, they drawn conclusion that stimulated emission of XPALs with supersonic flow has a potential to be quenched and subsonic flow has a better heat tolerance and presented a “nozzle-diffuser” closed circle flowing experimental setup [12]. Nevertheless, the influence of quench is not analyzed in detail. It is known to all, shock wave is caused by some discontinuity in supersonic flow, where the physical characters at the two sides are different. Therefore, in this paper, we developed previous theoretical work on XPALs with super-fast flow and derive the characteristics of supersonic flow. Particularly, the position of the shock wave in the supersonic flow, which affects the output performance deeply, is going to be figured out.

## 2. Description of the model

#### 2.1 Experimental apparatus

The experimental setup for supersonic CW XPALs can be designed as Fig. 1. In fact, this type of experimental device, nozzle, is very ordinary in fluid mechanics, where the supersonic flow would be generated when the flow passes the throat, the narrowest part in the nozzle structure [12].

Figure 1 is a nozzle schematic experimental apparatus for XPALs with supersonic flow, which is oriented from the concept of the Laval nozzle in fluid mechanics. In this setup, the stagnation pressure and temperature of gas mixture are settled as 6.15 atm and 513K, respectively. The gas mixture would flow in the direction that the arrow marks when gas pump is turned on and then reach the local speed of sound at the throat of nozzle. In fluid mechanics, Mach number is used to describe the velocity of sonic-level flow, and it equals to 1, because of the definition, when the gas velocity equals to the speed of sound. It is determined by the geometric parameters of nozzle,

#### 2.2 Kinetic processes for stimulated emission

The kinetic processes of XPALs have been studied in [10]. Four or five energy levels are involved in classical and simplified theory, which are ${6}^{2}{S}_{1/2}$, ${X}^{2}{\text{\Sigma}}_{1/2}^{+}$,$\text{}{B}^{2}{\text{\Sigma}}_{1/2}^{+}$, ${6}^{2}{P}_{3/2}$ and ${6}^{2}{P}_{1/2}$, respectively. Generally speaking, this simplified model can be used to explain the stimulated emission process very well, and it also has good agreement to experiments. However, if CW and higher output power laser is being sought, we need to consider the threshold of pump intensity of XPALs is so significantly high that higher excited energy levels would play roles in the kinetic process. Figure 2 shows the energy levels of a Cs-Ar XPAL involved in our calculations [14].

In our model, higher excited energy levels, ${6}^{2}{D}_{3/2,\text{}5/2}\text{}$ and ${8}^{2}{S}_{1/2}$, can be excited by energy pooling and photoexcitation where both of pump and laser lights take parts. In addition, associative ion energy level, $C{s}_{2}^{+}$, can be formed by associative [15] and Hornbeck-Molnar ionization [16] from lasing upper level and higher excited level, respectively, and it can also be obtained by recombination from ion energy level, $C{s}^{+}$, which is mainly accumulated by photoionization from higher excited energy levels. It is worth to note that three- body recombination from $C{s}^{+}$ is taken into account [17], but simulated results indicate that the proportion in whole recombination is too small to make difference.

The rate equations for the completely kinetic processes are listed as follows,

^{+}, and Cs

_{2}

^{+}, respectively.

*k*and

_{01}, k_{10}, k_{23}*k*are the thermal equilibrium constants between

_{32}*n*,

_{0}*n*and

_{1}, n_{2}*n*.

_{3}*k*and

_{34}*k*denote rates of the energy pooling and penning ionization processes, respectively. And [

_{pen}*Ar*] presents the concentration of Argon in stagnation. Other related parameters and the equations solving method have been studied deeply and demonstrated [11].

#### 2.3 Fluid thermodynamics and energy conversion

Previous work shows that the pump threshold of XPAL systems is dramatically high [10, 18], indicating that even small energy defect is able to generate considerable heat loading which is sufficient to melt the apparatus down and abort stimulated emission. So, Pan’s team has presented a scheme that supersonic flowing gas is applied to XPALs, and simulated results indicate that the heat loading can be effectively removed so that the temperature in the radiation zone keeps relatively low. However, it may not be stable for supersonic flow with heat conversion, especially the shock wave can always be observed. Now, we are going to analyze this situation specifically.

In the XPAL system, the heat loading is raised by multiple transitions. It can be seen from Fig. 2 and deduced by rate equations Eqs. (4)-(10), which is written as

Meanwhile, considering energy balance in the radiative zone, the removed heat of flow is

*R*is the perfect gas constant in mass.

*A*and

_{in}*M*are the cross-section square and Mach number at the inlet of nozzle throat, respectively. Now, we can present the expression on particles population densities to enclose rate equations

_{in}The opposite direction heat conduction in the flow is taken into account and calculated by

#### 2.4 Computation on the position of shock wave plane in the supersonic flow

In the subsection 2.3, we have analyzed the thermodynamic situation in the supersonic flow, which includes the generation and removal of heat. And numerical iteration calculation is carried out by taking energy conservation. In this part, we are going to derive more specific calculation on the supersonic flow in XPAL system.

At the beginning, conservation law of mass is always through the whole fluid mechanics, and it can be expressed by

where ${N}_{a}$ is Avogadro number and $\mu $ is the relative molecular mass of gas mixture, where the Argon gas presents the complete flow system. It is because the partial pressure of Argon plays dominant role in gas mixture. So, [*Ar*] will be used to represent the concentration of the complete gas mixture. $v$ denotes the velocity of flow. Therefore, the differential for logarithmic form of Eq. (19) is likeTake the Euler’s movement equation below into Eq. (20),where $p$ indicates movement pressure. Then we can finally obtain the relationship of cross-section square and Mach number in differential form,In our model, we assume that shock wave plane is generated in nozzle and radiative zone. This is because the discontinuity plane is caused somewhere temperature gradient exists and setting flow still keeps. For those lower radiative power, less energy defect takes place, which discontinuity plane stays in the radiative zone that Eq. (22) would be deduced like

*lsr*” and “

*rs*” indicate the corresponding parameters in the radiative zone and at the right side of shock wave plane, respectively. ${T}_{lsr}$ denotes the stagnation temperature at the first segment of the radiative zone, which is assumed aswhere ${l}_{sw}$ indicates the coordinate for the shock wave plane on the longitudinal direction. ${M}_{rs}$ is the Mach number at the right side of shock wave plane,where ${M}_{ls}$, Mach number at the left side of shock wave plane, is originated from${P}_{rs}$ is the movement pressure at the right side of shock wave, and it is expressed aswhere ${P}_{ls}$ is Mach number at the backwards of shock wave plane, which can be written asIn addition, ${P}_{rs,stag}$ indicates the stagnation pressure loss in the shock wave, and it is illustrated by [13]

However, with increase of pump power, shock wave plane tends to propagate to the nozzle and as the pump power reaches a threshold value, shock wave plane would be swallowed into the nozzle. For such a high pump power, calculations on shock waves are very different. Therefore, we present a calculating method for the position of shock wave plane by integrating Eq. (22)

## 3. Result and discussion

In the previous work, exciplex pumped alkali vapor lasers with supersonic flow have been deeply studied, and results show that XPALs may be quenched when it is pumped by very high power. However, as a potential candidate in significantly high-power laser regime, XPALs are not inevitable to be pumped by high power. Therefore, even if supersonic flow system is not as stable as those lasers with low velocity flow, its good performance is still attractive to researchers.

Figure 4 shows that the dependence of stimulated emission output power on pump power in logarithmic coordinate axis. It indicates that XPALs can be pumped in broadband power domain and reach to its maximum output power of $1.37\times {10}^{5}$ W with $3\times {10}^{6}$ W of pump power, then rapidly jump down to quench with power of pump lights ~${10}^{7}$W.

Figure 5 demonstrates the temperature gradients on radial direction at different power of pump lights, where Fig. 5(a) shows the situation that the XPAL system is at fluorescence phase, and with the increase of pump power, temperature curves intend to be split out. Figure 5(b) tells that the temperatures in operating XPALs are relatively high, but gradients are kept smooth which means good thermal distributions are maintained. Therefore, optical-optical energy conversion efficiency can be enhanced to relatively ideal level, ~3%. It is worth to note that optical-optical efficiency of 3% is still much lower than ideal value, 45%, which is simulated by Huang’ team. This has been discussed in literature [10]. The temperature curve shows that the stagnation temperature is ~1500K when peak outputting is achieved with ${P}_{p}=3\times {10}^{6}$ W.

In addition, the changes of Mach number in the radiative zone play important roles in the influences of output performances, Fig. 6(a) indicate that the Mach number at the first segment in the radiative zone (z = 0, in the Fig. 3), where we can see that with the increase of the optical pump power, the subsonic Mach number that is transferred from supersonic flow has a significant reduction. Specifically speaking, when the pump power is lower than $6\times {10}^{5}$ W, the subsonic Mach number is 0.553, which is directly calculated from Eq. (30) by taking the invariance law of stagnation temperature on the shock wave plane. Then, this subsonic Mach number decreases, which expresses that the shock wave plane moves backwards. Lower Mach number at z = 0, steeper temperature gradient is built between shock wave plane and radiative zone. Figure 6(b) shows that the variation of Mach number in the radiative zone (z>0), we can see that Mach number is close to 0.1 when Pp = $7\times {10}^{6}$ W, which can be deduced that the choking effect of flow has been taken place, and lasing process intends to quench, correspondingly.

The results of simulating calculation on the position of shock wave plane is shown in Fig. 7, where we can see that the shock wave is raised at the contact surface of illuminated zone with lower pump power. But it intends to be rapidly swallowed into nozzle when power of pump lights reaches $2.5\times {10}^{6}$ W, indicating that condition for shock wave in the tube, which is only conducted by Eq. (23), cannot be satisfied. In fact, we have done many attempts for finding the pump power range where the shock wave is raised in the tube. However, results show that this sort of situations require ${T}_{lsr}$ of Eq. (23) to reach over 4000K, which is in conflict with other conditional equations. We find that if the ${P}_{rs}$ of Eq. (23) can be increased a little, the requirement for ${T}_{lsr}$ would reduce a lot. So, if the shock wave is in the nozzle, ${P}_{ex2}$ of Eq. (30) takes the place of ${P}_{rs}$. This is why shock wave is always observed in the nozzle rather than in the tube. However, we still assume that if the spatial distribution of pump lights is eliminated, the first segment in the radiative zone causes larger heat loading, it would result in the existence of shock wave in the tube.

## 4. Conclusion

In this article, we develop the calculating theory on XPALs with supersonic flow system, which mainly takes the effects of compressible gas at super-high flowing velocity. Some literatures have pointed out that sonic-level flow may make lasing process quenched since choking effect of setting flow takes place. Therefore, on the basis of fluid mechanics and thermodynamics, we deduce the conditions that may influence laser performance and obtain the position of the discontinuity plane in the tube, which is seen as the boundary of temperature gradient. And simulated results demonstrate our explorations on the conditions and consequences of choking and quenching effects. We obtain that XPAL has a very broad pump power range, which is from $7\times {10}^{5}$ W to ${10}^{7}$ W and when output power reaches to the maximum, its output power curve intends to jump down to zero rapidly, where the lasing process is aborted. At the same time, we notice that the Mach number along flowing direction drops very low (~0.1), which indicates that choking effect turns up in the tube. Hence, choking effect is indeed harmful to the laser performance and hampers the higher power gas lasers. But if this thermo-choking effect can be eliminated, XPALs has a brighter potential in super high-power laser regime. In addition, we gather the changes of stagnation temperature in the tube and radiative zone, showing that distribution of temperature is relatively smooth, which is good for improving the quality of laser beam. Finally, the formula for calculating the position of shock wave is presented and used in this article, corresponding result shows that shock wave plane is always observed at the contact surface of illuminated zone and in the nozzle. As a candidate of high-power laser device, super-fast flowing medium is a practical path to maintain the beam quality and reduce the operating temperature.

## Funding

Zhejiang Provincial Natural Science Foundation (Grant No. LY14A040005); Aerospace Science and Technology Fund (Grant No. KM20170269).

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