## Abstract

In recent years, a diode-pumped alkali laser (DPAL) has become one of the most hopeful candidates to achieve the high power performance. A series of models have been established to analyze the DPAL’s kinetic process and most of them were based on the algorithms in which only the ideal 3-level system was considered. In this paper, we developed a systematic model by taking into account the influence of excitation of neutral alkali atoms to even-higher levels and their ionization on the physical features of a static DPAL. The procedures of heat transfer and laser kinetics were combined together in our theoretical model. By using such a theme, the continuous temperature and number density distribution have been evaluated in the transverse section of a cesium vapor cell. The calculated results indicate that both energy pooling and ionization play important roles during the lasing process. The conclusions might deepen the understanding of the kinetic mechanism of a DPAL.

© 2015 Optical Society of America

## 1. Introduction

In the recent years, diode-pumped alkali lasers (DPALs) have been paid more and more attention due to the potential for their excellent physical features [1–3]. The gain cell of a DPAL usually contains the neutral alkali vapor and several kinds of buffer gases such as helium and hydrocarbon with small molecular weight [4,5]. As the D_{2} (*n*^{2}*S*_{1/2}→*n*^{2}*P*_{3/2}, where *n* = 4, 5, and 6 for K, Rb, and Cs, respectively) and D_{1} (*n*^{2}*P*_{1/2}→*n*^{2}*S*_{1/2}) transition lines are spectrally close, the quantum defect in a DPAL is extremely small [6]. Additionally, the radiative wavelength of a DPAL is located in the near-infrared wavelength region that fits the response band of a commercial semiconductor detector, and a DPAL system is generally compact [7,8]. Because of these merits, DPALs have been becoming one of the most ideal candidates for realizing a high-powered laser system.

Unlike other types of lasers, the density of a gaseous medium inside a vapor cell is extremely sensitive to the temperature [9,10]. When the pump power for a DPAL is small enough, heat generated from the relaxation process between the fine-structure levels of alkali atoms in a vapor cell has almost no effects on laser properties and is usually neglected. However, when high-powered diodes are used to pump the vapor cell, the laser features will be seriously affected, and some physical characteristics, e.g. the number density distribution and ionization rate should become different from those for an ordinary solid-state laser.

Generally, a DPAL is a three-level laser, in which the electrons in alkali atoms are pumped from the *n*^{2}*S*_{1/2} level to the *n*^{2}*P*_{3/2} level. After rapidly relaxing between the fine-structure, electrons fall back to the ground state with stimulated emission. However, alkalis are the most easily ionized atomic species, especially for Rb and Cs. Figure 1 illustrates the energy level diagram of Cs atoms, under the condition of strong pumping the electrons will be excited to the higher levels, 6^{2}*D*_{5/2,3/2} and 8^{2}*S*_{1/2}, by energy pooling collisions [11,12], which will reduce the number densities of the 6^{2}*P*_{3/2,1/2} levels. Then, further ionization processes including photo-ionization and Penning ionization will occur on the 6^{2}*D*_{5/2,3/2} and 8^{2}*S*_{1/2} levels resulting in decrease of the density of neutral atoms. To study the effect of the deleterious processes (energy pooling and ionization), Benjamin Q. Oliker *et al*. proposed a theoretical model [13] and they demonstrated both energy pooling and ionization play an important role in a DPAL system. However, they did not discuss the situation that pumped with high power and the effect of temperature distribution on the physical features of a DPAL. Recently, Boris D. Barmashenko’s team reported a fluid model [14], in which the deleterious processes were included, and they studied the temperature distribution. Nonetheless, in their model only the flowing-gas DPAL was studied rather than the static device which is the subject of the present paper. Therefore, developing a precise model to investigate the effect of the deleterious processes in a static vapor cell is urgently needed.

The theoretical regime of this paper is based on our previous model for investigation of a diode-pumped cesium vapor laser [15–19]. The procedures of heat transfer and laser kinetics are combined together to analyze the deleterious processes of energy pooling and ionization in our theoretical model. By using such a theme, the radial temperature and number density distributions have been evaluated in the transverse section of a cesium vapor cell. In addition, the influence of energy pooling, Penning ionization, and photo-ionization on the physical features of a DPAL is also systematically studied. Until now, there have not any published literatures in the similar research field to the best of our knowledge.

## 2. Theoretical analyses

Similar to our previous research [15], we divide a cylindrical vapor cell into many coaxial cylindrical annuli, in which the temperature is treated as a constant along the optical axis as shown in Figs. 2(a) and (b). Every cylindrical annulus is thought as both a heat source and a lasing source in the meantime. By using a segmental accumulation algorithm in which laser kinetics and heat transfer are simultaneously considered, the radial temperature distribution inside a cesium vapor cell and the output characteristics of a DPAL can be obtained at the same time.

#### 2.1 Analyses of laser kinetics

In the theoretical model, we choose cesium as the laser gain medium, and helium and ethane as buffer gases. In addition, by taking the processes of energy pooling, photo-ionization, Penning ionization, and recombination into account, the three-level rate equation used in our previous studies is extended to a quasi-five-level one. The referred energy levels include the traditional three levels (6^{2}*P*_{3/2, 1/2} and 6^{2}*S*_{1/2}), the higher excited levels (6^{2}*D*_{5/2, 3/2} and 8^{2}*S*_{1/2}), and the ionized level (Cs^{+}) (See Fig. 1).

There are several recombination mechanisms: i) direct radiative recombination; ii) third-body recombination with participation of electron or buffer gas; iii) fast conversion of atomic to molecular ions followed by dissociative recombination of molecular ions. The third-body recombination with electron as the third body is typically faster than that with buffer gas as the third body as well as the radiative recombination. Hence the third-body recombination with buffer gas as the third body and the radiative recombination were not included in our model. For the process of dissociative recombination, the rate constant is said to be much faster than that of third-body recombination. However, there are conflicting reports about the dissociative recombination, and the rate constant is not well characterized in the literature [13]. To perfect our model, we will take the process of dissociative recombination into account in our future study.

As shown in Fig. 2(b), we select an arbitrarily cylindrical annulus (*jth*) among all the segments. The outer radius *r _{j}* and the inner radius

*r*of this annulus can be expressed by

_{j + 1}*R*is the radius of a vapor cell,

*N*is the total number of segmented cylindrical annuli, respectively.

In the model, we designate the number densities of the 6^{2}*S*_{1/2}, 6^{2}*P*_{1/2}, 6^{2}*P*_{3/2}, 6^{2}*D*_{5/2,3/2} and 8^{2}*S*_{1/2}, and ions levels to be *n _{1}*,

*n*,

_{2}*n*,

_{3}*n*, and

_{4}*n*, respectively. Therefore, the total number density

_{5}*n*can be expressed as

_{0}*n*=

_{0}*n*+

_{1}*n*+

_{2}*n*+

_{3}*n*+

_{4}*n*. Next, the population distribution and output features of the

_{5}*jth*cylindrical annulus (see Fig. 2(a)) in a DPAL system can be obtained by the appropriate rate equations as follows [20]:

*jth*cylindrical, ${\gamma}_{32}\left({T}_{j}\right)$ is the fine-structure relaxation rate, $\Delta E$ is the energy gap between the

^{2}

*P*

_{3/2}and

^{2}

*P*

_{1/2}levels with the value of 554 cm

^{−1}, ${k}_{B}$ is the Boltzmann constant, ${\tau}_{D1}$ is the

*D*(6

_{1}^{2}

*P*

_{1/2}→6

^{2}

*S*

_{1/2}) radiative lifetime, ${\tau}_{D2}$ is the

*D*(6

_{2}^{2}

*S*

_{1/2}→6

^{2}

*P*

_{3/2}) radiative lifetime, ${\tau}_{4}$ is the lifetime of 6

^{2}

*D*

_{5/2,3/2}and 8

^{2}

*S*

_{1/2}levels [21] (we assume that the lifetimes of 6

^{2}

*D*

_{5/2,3/2}and 8

^{2}

*S*

_{1/2}levels are of the same value to simplify calculation), ${\sigma}_{photoionization}$ is photo-ionization transverse section, the ${k}_{EP2}{\left({n}_{2}^{j}\right)}^{2}$ denotes the transition rate of electrons from the 6

^{2}

*P*

_{1/2}to the 6

^{2}

*D*

_{5/2,3/2}and 8

^{2}

*S*

_{1/2}levels, ${k}_{EP3}{\left({n}_{3}^{j}\right)}^{2}$ denotes the transition rate from the 6

^{2}

*P*

_{3/2}level to the 6

^{2}

*D*

_{5/2,3/2}and 8

^{2}

*S*

_{1/2}levels, ${k}_{PI}$ is the Penning ionization rate coefficients, ${k}_{recombination}$ is the recombination rate constant, and ${I}_{p}^{j}$ and ${I}_{l}^{j}$ are the pump and laser intensities given by [22], respectively. A summary of critical parameters of this model are listed in Table 1.

#### 2.2 Theoretical analyses of heat transfer and radial temperature distribution

Generally, the differential equation of thermal conductivity in the cylindrical coordinate system is given by [23]

where $K\left({T}_{j}\right)$ is the coefficient of thermal conductivity [24], and ${Q}_{j}$ is the heat generated from the*jth*cylinder as expressed by

*jth*cylindrical annulus, ${\Omega}_{j}$ is the volume density of generated heat of the

*jth*cylindrical annulus.

Next, the temperature in the *jth* cylindrical annulus can be obtained after undertaking the integral calculation on both sides of Eq. (4) [15]:

By substituting ${C}_{j}^{1}$ and ${C}_{j}^{0}$ into Eq. (6), one can obtain the temperature in the transverse section of the *jth* cylindrical annulus in which the temperature is assumed as a constant along the optical axis. The temperature of the inner side of the *jth* cylindrical annulus, ${T}_{j+1}$, can also be calculated and is then used as the boundary condition in the calculation of the *(j + 1)th* cylindrical annulus. By employing a circulatory calculation, we can therefore deduce${T}_{2}$, ${T}_{3}$, *…, *${T}_{N}$. The radial temperature distribution in the transverse section of a vapor cell is then obtained.

The relationship of the generated and transferred heat at the transverse section of a vapor cell is shown in Fig. 3. In the figure, ${P}_{thermal}$ is the total heat transferred out from a vapor cell, ${\Phi}_{j}$ is the heat transferred from the *jth* cylinder to the (*j-1)th* one, and the following relationships are tenable:

Therefore, through a recursive calculation of ${Q}_{1}$, ${Q}_{2}$, …${Q}_{j-1}$, the heat transferred from the *jth* cylindrical annulus to the *(j-1)th* one can be obtained as follows:

*j = 1, 2, …, N*.

At the beginning of the calculation, we first offer an initial value to${P}_{thermal}$. According to Eq. (9), ${\Phi}_{1}$ is then obtained. The temperature at the outside of the first cylindrical annulus, ${T}_{w}$, is approximately assigned as the temperature of this cylindrical annulus since the thickness of the segmented annulus is small enough. Therefore, the volume heat density ${\Omega}_{1}$ and the generated heat ${Q}_{1}$ can be calculated. By using ${\Phi}_{1}$ and ${\Omega}_{1}$, we can deduce the temperature distribution inside the first cylindrical annulus with Eq. (6). The temperature of the inner side of the first cylindrical annulus, ${T}_{2}$, can be then calculated and is used as the initial conditions in evaluating the temperature distribution of the second cylindrical annulus. By using Eq. (10), ${\Phi}_{2}$ is thus evaluated. Through the circulatory calculation, ${Q}_{j}$ and ${\Phi}_{j}$ can be therefore obtained.

Next, we judge whether $\sum _{i=1}^{j}{Q}_{i}$ is equal to the given value of ${P}_{thermal}$ or not. If $\sum _{i=1}^{j}{Q}_{i}$ is not equal to${P}_{thermal}$, the evaluation will be repeated by using the next value of ${P}_{thermal}$. Such a calculation process will reach the end until the following equation is satisfied:

By using the final value of ${P}_{thermal}$, we can obtain the temperature distribution in the transverse section.

## 3. Results and discussions

In our model, the cell length and cell radius is 25 mm and 7.7 mm, respectively. The transverse pump distribution holds out a Gaussian intensity profile and the waist radius of the pump is set to be 1 mm. The critical parameters of this model are listed in Table 1.

The dominant processes corresponding to even-higher levels in our model are energy pooling, Penning ionization, and photo-ionization. To systematically investigate the influence of these processes on the physical features of a DPAL, we list five cases in Table 2. Case 1 represents a simple three-level system investigated in the absence of excitation to the upper states 62D5/2,3/2 and 82S1/2 and their ionization. Case 2 denotes the state with considering the process of energy pooling. Case 3 illustrates the processes of energy pooling and Penning ionization. Case 4 describes the occurrence when energy pooling and photo-ionization are considered. Case 5 reflects the circumstance including all of these three processes.

We first analyze the temperature distribution at the transverse section of a vapor cell. Figure 4 shows the temperature distribution with different pump power in Case 1 and Case 5, respectively. The vertical axis of the graph denotes the temperature values, and the horizontal axis denotes the distance to the center of a vapor cell. It can be seen that the temperature at the transverse section exhibits the distinct gradient and achieves the maximum values at the centre for each curve, and the higher pump power leads to a more obvious rise of the temperature for both Case 1 and Case 5.

Figure 5 shows the temperature distribution in different cases under 1500 W pump power, and the inset is the enlarged center part (r < 1.25 mm). It can be found that the five curves close to each other on the edge of a vapor cell. When close to the center, there is almost no difference between Case 1 and Case 2, which means that energy pooling has little effect on the radial temperature distribution. For Case 3 and Case 4 at the center of a cell, it can be observed the temperature descend compared to the curves of Case 1 and Case 2. More interesting is that the curves of Case 4 and Case 5 are nearly the same throughout the transverse section, indicating that the influence of Penning ionization on temperature distribution is much smaller than photo-ionization.

To better study the influence of energy pooling, Penning ionization, and photo-ionization on the physical features of a DPAL, we then analyze the population distributions inside a cesium vapor cell. The total alkali number density of the *jth* cylindrical annulus is given by [25]

*R*is a constant of proportionality with the value of 8.3143

*j/*(

*mol·K*) and

*N*is Avogadro number, respectively. From Eqs. (13) and (14), ${n}_{0}^{1}\left({T}_{w}\right)$ is a constant with a certain

_{A}*T*, and the total alkali number density of the

_{w}*jth*cylindrical annulus ${n}_{0}^{j}\left({T}_{w}\right)$ is inversely proportional to the temperature of

*jth*cylindrical annulus ${T}_{j}$.

Figure 6 shows the population distribution at the transverse section of a vapor cell with the pump power of 1500 W in different cases, while the waist radius is assumed to be 1 mm. It can be seen that the total density of the cesium vapor *n _{0}* increases with the radial position

*r*, which is due to the fact that the temperature at the central area is higher than that near the cell wall. Some inflection points, which locate in the range from about 1.5 to 2.0 mm, can be clearly observed in the curves for

*n*and

_{1}*n*in all the figures from (a) to (e). One can find that population inversion can be achieved in the lasing region, where

_{2}*n*is always larger than

_{2}*n*. Due to the excitation of the process of energy pooling, the number density of 6

_{1}^{2}

*D*

_{5/2,3/2}and 8

^{2}

*S*

_{1/2},

*n*, can be examined, as shown in the figures from (b) to (e). When considering the process of ionization, the number density of ion is calculated to be

_{4}*n*, which is much larger than

_{5}*n*(see Figs. 6(c), (d), and (e)). By comparing Figs. 6(c), (d), and (e), it can be found that the influence of Penning ionization is much smaller than photo-ionization on population distribution, which is similar to the result of the temperature distribution.

_{4}Next, we analyze the population distribution inside a cesium vapor cell with different pump power in Case 1 and Case 5. In Fig. 7, the curves of *n _{0}*,

*n*,

_{1}*n*, and

_{2}*n*decrease obviously when the pump power increases from 100 W to 1500 W in both cases. The reason is that the temperature in a vapor cell with the pump power of 100 W is lower than that for 1500 W. However, for Case 5, the number density of alkali ions

_{3}*n*increases when the pump power increases from 100 W to 1500 W since alkali atoms are easier to be ionized for the higher temperature.

_{5}The total number density of 6^{2}*D*_{5/2,3/2} and 8^{2}*S*_{1/2} *n _{4}* is much smaller than

*n*,

_{1}*n*, and

_{2}*n*. Figure 8 shows the population distribution of

_{3}*n*with pump power increase from 100 to 1500 W in different cases. In Fig. 8, all the curves have inflection points in the range from about 1.5 to 2.0 mm, which is around the lasing boundary. It means that only in the lasing region can the atoms in 6

_{4}^{2}

*P*

_{3/2,1/2}levels be excited to the higher levels by the process of energy pooling. In addition,

*n*exhibits the same changing law with

_{4}*n*when the pump power increases from 100 to 1500 W. It can be seen that ionization reduces number density

_{0}*n*in the lasing region to some extent.

_{4}Taking into account the process of ionization, the population distributions of alkali ions have been evaluated as shown in Fig. 9. It is obvious that the number density caused by photo-ionization is much larger than that by Penning ionization, indicating that photo-ionization plays the leading role in the process of ionization, which is consistent with the result of temperature distribution. In addition, it can be found that the number density caused by photo-ionization (dashed line) increases with the pump power. Such a phenomenon is due to that the higher pump power leads to the higher temperature and higher ionization probability. However, the curves caused by Penning ionization (real line) little decrease with the pump power rising. We deduce that there might be competitive relation between Penning ionization and photo-ionization during the process of ionization and photo-ionization should have stronger ability of ionization. This is also the reason why the curves of Case 4 and Case 5 are nearly the same throughout the transverse section in Fig. 5.

In Fig. 10, the laser power is given as a function of the pump power. It can be observed that all the quantities of these curves monotonically increase with the pump level. Similar to the result of temperature distribution (Fig. 4), the process of energy pooling has little influence on the output power compared with the condition without considering the process of excitation and ionization. Comparing the curve of Case 1 with those of Case 3 and Case 4 in the figure, distinct decreases can be observed when the pump power exceeds 200 W. The differences become larger with the pump power rising and achieve the value of 1.08% and 3.11% with the pump power of 1500 W, respectively. In addition, when considering both processes of Penning ionization and photo-ionization in the model (Case 5), the curve decreases about 3.16% with the pump power of 1500 W. This result proves once more that photo-ionization has greater influence than Penning ionization. Since the parameters in our model are not optimized, the influence of these deleterious processes might be greater in practice. Based on the calculation results, the decrease in the output power resulting from the processes of energy pooling, Penning ionization, and photo-ionization becomes much more serious for an even-higher pump that should not be ignored. Since the optical-to-optical efficiency of a DPAL is not the key point of this study, we do not optimize the main parameters such as the temperature, waist radius, and reflectance of the output coupler. This does not affect our investigation about the influence of the processes of energy pooling, Penning ionization, and photo-ionization on the physical features of a DPAL.

## 4. Conclusion

In this study, we develop a model to calculate a static-state DPAL by simultaneously considering laser kinetics and heat transfer. The continuous temperature and population distributions of Cs and Cs^{+} have been evaluated with different pump power in the transverse section of a cesium vapor cell. The deleterious processes of energy pooling, Penning ionization, and photo-ionization, which result in the decrease of number density of neutral alkali atoms and the increase of the alkali ions, have been taken into account in our model. The results shows that the temperature, number density of alkali atoms, and output power will decrease after considering these deleterious processes, and the situation becomes worse with the pump power rising. According to the calculated results, we deduce that there might be competitive relation between Penning ionization and photo-ionization during the ionization process, in which photo-ionization is the dominant one. Considering these results, the deleterious processes play important role during the lasing process and cannot be ignored in designing a high-powered DPAL system.

## Acknowledgments

We are very grateful to Prof. Salman Rosenwaks and Dr. Boris D. Barmashenko at Ben-Gurion University of the Negev of Israel for their valuable helps in calculation of saturated alkali number densities inside a static alkali vapor cell.

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