Optimization of physical conditions for a diode-pumped cesium vapor laser

Guofei An; You Wang; Juhong Han; He Cai; Shunyan Wang; Hang Yu; Kepeng Rong; Wei Zhang; Liangping Xue; Hongyuan Wang; Jie Zhou

doi:10.1364/OE.25.004335

1. Introduction

Diode-pumped alkali lasers (DPALs) have extensively been studied due to the potential to generate high power with good beam quality in the recent years [1–6]. Generally, a DPAL is a three-level laser using alkali vapor as the gain medium (usually being cesium and rubidium). However, the electrons in the n²P_3/2 and n²P_1/2 levels will be further excited to the even higher levels such as n²D₅_/2,3/2 and (n + 2)²S_1/2 by the energy pooling collisions under high power pumping. The electrons in these higher levels can then be ionized through the processes of photo-ionization and Penning ionization causing the diminution of neutral alkali atoms in the vapor cell [7,8].

Although considerable researches on DPALs have been undertaken until now, rare literatures can be found in optimization of configurations of a laser system to the best of our knowledge. Furthermore, the published experimental results on laser output are obviously less than the theoretical ones especially when the pump power is larger than 80 W [9]. It means that some important physical phenomena should not be involved in the early theoretical model. There have been few research groups referring the physical characteristics of the deleterious processes including energy pooling and ionization in a DPAL system [10,11]. Benjamin Q. Oliker et al. reported their simulation results to analyze the effect of deleterious processes on both a pulsed and a CW DPAL with static gain medium [12]. However, the optical-to-optical efficiency of a DPAL is somewhat low at the pump level as high as 100 W. In the recent years, Boris D. Barmashenko’s team reported a series of studies on both flowing-gas and static-gas DPALs including the report of the dependence of the power of a Cs DPAL on the cell length [13–17]. In our previous study, a theoretical model was built to explore the influences of energy pooling and ionization on the output features of a DPAL with a static cesium vapor cell [18]. However, the optical-to-optical efficiency only reaches less than 10% with the relatively large pump power.

In this report, we construct an improved theoretical model to investigate how the physical conditions such as the cell wall temperature, cell length, and cell radius affect the output features of a static-gas DPAL with the purpose of increasing the optical-to-optical efficiency and reducing the negative effects of the deleterious processes. We reveal that the output of a DPAL can be dramatically improved by optimization of the physical conditions, and the influence of deleterious processes can also be decreased at the same time.

2. Theoretical analyses

In this report, we consider the configuration of a typical end-pumped continuous-wave DPAL with static media as schematically shown in Fig. 1. After passing a polarized beam splitter (PBS), a pump beam with power of P_p enters into a cylindrical vapor cell with the length of L and the radius of R in which metallic cesium and buffer gases of ethane and helium have been beforehand sealed. The cavity consists of a high reflector with the reflectivity of 100% and an output coupler with the transmittance of 70%. In this model, a L-type oscillator is selected to separate the pump beam and the laser beam by means of an orthogonal polarization approach.

Fig. 1 Schematic diagram of a static DPAL.

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During the simulation, we made the following assumptions with purpose of the algorithm simplification:

(1) The transverse pump distribution holds out a Gaussian intensity profile and keeps unchanged along the optical axis;
(2) The temperature of every cylindrical annulus is a constant along the optical axis;
(3) The temperature distribution at the transverse section of a vapor cell is symmetrical.

2.1 Analyses of heat transfer

Generally, there is temperature gradient inside an alkali cell due to the heat generated during the process of lasing of a DPAL. The temperature gradient cannot be ignored for that the gain medium (alkali vapor) inside the cell is extremely sensitive to the ambient temperature.

In this model, 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 Fig. 2(a). Every cylindrical annulus is thought as both a heat source and a lasing source in the meantime. An arbitrarily cylindrical annulus was selected to be the jth one, the outer radius r_j and the inner radius r_{j + 1} of this annulus can be expressed by

r_{j} = R - (j - 1) \cdot R / N,

r_{j + 1} = R - j \cdot R / N,

where R is the radius of a vapor cell, N is the total number of segmented cylindrical annuli, respectively.

Fig. 2 (a) Segmented configuration of a vapor cell and (b) schematic illustration of the generated and transferred heat at the transverse section of a vapor cell.

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With respect to the heat transfer between the coaxial cylindrical annuli, we use the same theory of our previous paper [19,20]. The temperature in the jth cylindrical annulus can be expressed by

T (r) = C_{j}^{1} \ln r - \frac{Ω_{j} \cdot r^{2}}{4 K (T_{j})} + C_{j}^{0},

C_{j}^{1} = \frac{Ω_{j} \cdot r_{j}^{2}}{2 K (T_{j})} - \frac{Φ_{j} \cdot r_{j}}{K (T_{j}) \cdot A_{j}},

C_{j}^{0} = T_{j} - C_{j}^{1} \ln r_{j} + \frac{Ω_{j} \cdot r_{j}^{2}}{4 K (T_{j})},

A_{j} = 2 π r_{j} L,

where

K (T_{j})

is the coefficient of thermal conductivity [14],

Ω_{j}

is the volume density of generated heat of the jth cylindrical annulus,

A_{j}

stands for the lateral area of the jth cylindrical annulus,

Φ_{j}

is the heat transferred from the jth cylinder to the (j-1)th one is given by [21]

Φ_{j} = {[- K (T) \cdot A \frac{d T}{d r}] |}_{T = T_{j}} .

We can obtain the temperature in the transverse section of the jth cylindrical annulus by substituting

C_{j}^{1}

and

C_{j}^{0}

into Eq. (3). 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.

Figure 2(b) shows the relationship of the generated and transferred heat at the transverse section of a vapor cell, and the following relationships are tenable:

Φ_{1} = P_{t h e r m a l},

Φ_{2} = Φ_{1} - Q_{1} = P_{t h e r m a l} - Q_{1} .

where

P_{t h e r m a l}

is the total heat transferred out from a vapor cell. 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:

Φ_{j} = P_{t h e r m a l} - \sum_{i = 1}^{j - 1} Q_{i},

where j = 1, 2, …, N.

In order to obtain the exact temperature of each cylindrical annulus, we adopt a circulatory calculation method. Before the calculation, we first set an initial value to $P_{t h e r m a l}$ and $T_{w}$ (temperature at the outside of the first cylindrical annulus), respectively. The volume heat density $Ω_{1}$ and the generated heat $Q_{1}$ can be then calculated. By using $Φ_{1}$ and $Ω_{1}$ , we can deduce the temperature distribution inside the first cylindrical annulus with Eq. (3). Therefore, the temperature of the inner side of each cylindrical annulus can be evaluated. By using Eq. (9), $Q_{j}$ and $Φ_{j}$ can be therefore obtained.

Next, we judge whether $\sum_{i = 1}^{j} Q_{i}$ is equal to the given value of $P_{t h e r m a l}$ or not. If $\sum_{i = 1}^{j} Q_{i}$ is not equal to $P_{t h e r m a l}$ , the evaluation will be repeated by using the next value of $P_{t h e r m a l}$ . Such a calculation process will reach the end until the following equation is satisfied:

P_{t h e r m a l} = \sum_{i = 1}^{j} Q_{i} .

By using the final value of $P_{t h e r m a l}$ , we can obtain the exact temperature value of each cylindrical annulus. The total alkali number density of the jth cylindrical annulus can thus be calculated by [15]

n_{0}^{j} (T_{j}) = {\begin{cases} n_{0}^{1} (T_{w}), j = 1 \\ n_{0}^{1} (T_{w}) (\frac{T_{w}}{T_{j}}), j > 1 \end{cases},

where

T_{w}

is the cell wall temperature,

T_{j}

is the temperature of jth cylindrical annulus,

n_{0}^{1} (T_{w})

is the saturated alkali number density inside the first cylindrical annulus which is adjacent to the inner surface cell wall as expressed by [22]

n_{0}^{1} (T_{w}) = \frac{133.322 N_{A}}{R T_{w}} (10^{8.22127 - \frac{4006.048}{T_{w}} - 0.00060194 T_{w} - 0.19623 \log_{10} T_{w}}),

where R is a constant of proportionality with the value of 8.3143 j/(mol·K) and N_A is Avogadro number, respectively.

2.2 Analyses of laser kinetics

We have investigated the influence of energy pooling and ionization on the physical features of a DPAL and extended the traditional three-level rate equations to a quasi-five-level ones [18]. The referred energy levels of atomic cesium include the conventional three levels (6²P_{3/2, 1/2} and 6²S_1/2), the higher excited levels (6²D_{5/2, 3/2} and 8²S_1/2), and the ionized level (Cs⁺) as shown in Fig. 3. For the process of recombination, only the third-body recombination with participation of electron was studied in our model. The reason is well explained in [18].

Fig. 3 Energy level diagram of atomic cesium.

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In the model, the number densities of the 6²S_1/2, 6²P_1/2, 6²P_3/2, 6²D_5/2,3/2 and 8²S_1/2, as well as ions levels are designated to be n₁, n₂, n₃, n₄, and n₅, respectively. The appropriate laser rate equations of the jth cylindrical annulus (see Fig. 2(a)) in a DPAL are

\begin{array}{l} \frac{d n_{1}^{j}}{d t} = - Γ_{P}^{j} + Γ_{L}^{j} + \frac{n_{2}^{j}}{τ_{D_{1}}} + \frac{n_{3}^{j}}{τ_{D_{2}}} + \frac{n_{4}^{j}}{τ_{4}} + k_{E P 2} {(n_{2}^{j})}^{2} + k_{E P 3} {(n_{3}^{j})}^{2} + k_{P I} n_{4}^{j} (n_{2}^{j} + n_{3}^{j}) \\ \frac{d n_{2}^{j}}{d t} = - Γ_{L}^{j} + γ_{32} (T_{j}) {[n_{3}^{j} - n_{2}^{j}] - [2 \exp (- \frac{Δ E}{k_{B} T_{j}}) - 1] n_{2}^{j}} - \frac{n_{2}^{j}}{τ_{D_{1}}} - 2 k_{E P 2} {(n_{2}^{j})}^{2} - k_{P I} n_{2}^{j} n_{4}^{j} \\ \frac{d n_{3}^{j}}{d t} = Γ_{P}^{j} - γ_{32} (T_{j}) {[n_{3}^{j} - n_{2}^{j}] - [2 \exp (- \frac{Δ E}{k_{B} T_{j}}) - 1] n_{2}^{j}} - \frac{n_{3}^{j}}{τ_{D_{2}}} - 2 k_{E P 3} {(n_{3}^{j})}^{2} - k_{P I} n_{3}^{j} n_{4}^{j}, \\ \frac{d n_{4}^{j}}{d t} = k_{E P 2} \cdot {(n_{2}^{j})}^{2} + k_{E P 3} \cdot {(n_{3}^{j})}^{2} - \frac{n_{4}^{j}}{τ_{4}} - k_{P I} n_{4}^{j} (n_{2}^{j} + n_{3}^{j}) - Γ_{p h o t o i o n i z a t i o n}^{j} + k_{r e c o m b i n a t i o n} {(n_{5}^{j})}^{3} \\ \frac{d n_{5}^{j}}{d t} = k_{P I} n_{4}^{j} (n_{2}^{j} + n_{3}^{j}) + Γ_{p h o t o i o n i z a t i o n}^{j} - k_{r e c o m b i n a t i o n} {(n_{5}^{j})}^{3} \\ Γ_{p h o t o i o n i z a t i o n}^{j} = n_{4}^{j} σ_{p h o t o i o n i z a t i o n} (\frac{I_{l}^{j}}{h v_{l}} + \frac{I_{p}^{j}}{h v_{p}}) \end{array}

where

Γ_{P}^{j}

is the stimulated absorption transition rate caused by pump photons,

Γ_{L}^{j}

is the transition rate of laser emission,

Γ_{p h o t o i o n i z a t i o n}^{j}

is the transition rate of photo-ionization,

γ_{32} (T_{j})

is the fine-structure relaxation rate,

Δ E

is the energy gap between the ²P_3/2 and ²P_1/2 levels with the value of 554 cm⁻¹,

k_{B}

is the Boltzmann constant,

τ_{D 1}

is the D₁ (6²P_1/2→6²S_1/2) radiative lifetime,

τ_{D 2}

is the D₂ (6²S_1/2→6²P_3/2) radiative lifetime,

τ_{4}

is the lifetime of 6²D_5/2,3/2 and 8²S_1/2 levels [7],

σ_{p h o t o i o n i z a t i o n}

is the photo-ionization transverse section,

k_{E P 2} {(n_{2}^{j})}^{2}

denotes the transition rate of electrons from the 6²P_1/2 level to the 6²D_5/2,3/2 and 8²S_1/2 levels,

k_{E P 3} {(n_{3}^{j})}^{2}

denotes the transition rate from the 6²P_3/2 level to the 6²D_5/2,3/2 and 8²S_1/2 levels,

k_{P I}

is the Penning ionization rate coefficient,

k_{r e c o m b i n a t i o n}

is the recombination rate constant, and

I_{p}^{j}

and

I_{l}^{j}

are the pump and laser intensities [10], respectively. A summary of critical parameters used in the calculation are listed in Table 1.

Table 1. Summary of critical parameters in our model

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3. Results and discussions

Table 2 lists five cases in the calculation. Case 1 represents a simple three-level system in absence of excitation to the upper states 6²D_5/2,3/2 and 8²S_1/2 as well as 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 the three processes.

Table 2. Different cases in the model

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We first analyze the influence of the cell radius on the output power in Case 1. Figure. 4(a) shows the output power versus the cell length, while the pump power and cell wall temperature are set to be 100 W and 383 K (110 °C), respectively. Fig. 4(b), Fig. 5, through Fig. 6(d) show the output power versus the cell wall temperature with the cell length of 25, 35, and 60 mm, respectively. The different curves denote different cell radii of 2.5, 5.0, 7.5, and 10.0 mm, respectively. It can be seen from Fig. 4(a) that the maximum shifts to the longer cell length when the cell radius becomes larger. In Figs. 4(b)–6(d), the maximum of each curve shifts to the higher cell wall temperature with the cell radius increasing. It is worth noting that the curves with the cell radii of 2.5 and 10 mm correspond to the maximal and minimal peak output powers, respectively, indicating that the peak value of the output power increases as the cell radius tends to be smaller. Considering that it is difficult to fabricate the vapor cell with small radius (<2 mm) in practice, we mainly discuss the cases with the cell radii of 2.5 and 5.0 mm in this study.

Fig. 4 (a) Output power versus cell length at 110 °C and the influence of cell wall temperature on the output power with the cell length of (b) 25 mm, (c) 35 mm, and (d) 60 mm under the pump power of 100 W, the four curves in each figure denote cell radius of 2.5, 5.0, 7.5, and 10 mm, respectively.

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Fig. 5 Influence of cell length on the output power with cell radius of (a) 2.5 mm and (b) 5.0 mm, and the corresponding cell wall temperatures of the five curves in the figures are set to be 90, 100, 110, 120, and 130 °C, respectively.

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Fig. 6 Influcene of cell wall temperature on the output power with cell radius of (a) 2.5 mm and (b) 5.0 mm under 100 W-pumping.

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Figure 5 shows the output power as function of the cell length at different cell wall temperatures with the cell radii of 2.5 mm and 5.0 mm, respectively. The pump power is set to be 100 W. It can be observed that there is an optimal cell length corresponds to the maximal value of the output power for each curve, which means that there is a one-to-one correlation between the optimal cell length and the optimal cell wall temperature when the cell radius and pump power are assumed to be unchanged. According to Eq. (13), the number density of cesium atoms increases with the cell wall temperature. Since the cesium number density is constant at a certain cell wall temperature, the total number of cesium atoms is determined by the cell length. From Fig. 5, when the cell length is shorter than the optimal value for every curve, the output power cannot achieve the peak because of the shortage of the total number of cesium atoms. However, when the cell length becomes longer than the optimal value, the excessive cesium vapor will lead to the reabsorption of stimulated radiation in the cell and will definitely diminish the output power of a DPAL. In addition, it can be seen that the optimal cell length decreases with the cell wall temperature increasing. We can also see that the optimal values of the output power of all curves in Fig. 5(a) are higher than those in Fig. 5(b), indicating that the Φ2.5 mm-cell would bring about a higher output power compared to the Φ5.0 mm-cell. It is noted that the maximum output powers for the curves increase slightly with the cell length.

Next, we analyze the influence of the cell wall temperature on the output power in Case 1, while the cell radii are assumed to be 2.5 and 5.0 mm, respectively. Figure 6 shows the output power as function of the cell wall temperature. The different linetypes in Fig. 6 denote different cell lengths. Being similar to the above results, the total number of cesium atoms in the vapor cell might be the key factor that influences the final output power. Given a certain cell length, the total number of the cesium atom will change with the cell wall temperature. When the cell wall temperature is lower than the optimal value, the output power cannot achieve the peak value because of the shortage of the total number of cesium atoms. On the other hand, when the cell wall temperature becomes higher than the optimal value, the excessive cesium vapor will also lead to the reabsorption of the stimulated radiation in the cell.

Besides the cell wall temperature, we then analyze the influence of the temperature distribution inside a vapor cell on the output power under the condition of Case 1. Figure 7(a) shows the temperature distribution at the transverse section of a vapor cell while the cell length and pump power are set to be 35 mm and 100 W, respectively. When the cell wall temperature is assumed to be 383 K, the calculated results reveal that the temperature inside the cell rises from 383 K at the edge to 568 K at the center. This is because that heat generated around the cell center is greater than that around the cell wall.

Fig. 7 (a) 2D-temperature distribution at the transverse section of a vapor cell with the cell length of 35 mm in Case 1 under 100 W-pumping. (b) Temperature distribution at the transverse section of a vapor cell with the cell length of 35, 50, 60, and 90 mm in Case 1 under 100 W-pumping.

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Figure 7(b) shows the temperature distributions with the cell length of 35, 50, 60, and 90 mm, respectively. The cell radius, cell wall temperature, and pump power are set to be 5 mm, 383 K, and 100 W, respectively. The vertical and horizontal axes of the graph denote the temperature values and the distance to the center of a vapor cell, respectively. As shown in the figure, the temperature at the transverse section exhibits distinct gradient and achieves the maximum values at the center for each curve, and the shorter cell length leads to a more obvious rise of the temperature. We deduce that the reason why the shorter cell length leads to the higher temperature might result from the higher absorbed pump power density when the temperature distribution is assumed to be uniform along the longitudinal direction. According to Eqs. (12) and (13), the alkali number density of the jth cylindrical annulus is inversely proportional to the temperature of jth cylindrical annulus, which means the higher temperature leads to the lower number density inside a vapor cell [19, 20]. That is to say the alkali number density in the cell with the length of 90 mm is higher than those in the cells with the length of 35, 50, and 60 mm. Therefore, the alkali number density in a vapor cell can be increased by optimizing the temperature, cell radius, and cell length. The optical-to-optical efficiency of the laser system can thus be improved.

To better understand how the cell length and cell wall temperature influence the output power, the 3D and corresponding 2D diagrams where the output power is given as a function of the cell length and cell wall temperature are obtained as shown in Fig. 8. The cell radius and pump power are set to be 5.0 mm and 100 W, respectively. It is evident that there is a ridge on the curved surface corresponding to the peak values of the output power, and every peak value is resulted from a pair of cell length and cell wall temperature. When the pump power is increased to 200 W, the results exhibit the similar rule as that pumped by 100 W as shown in Fig. 9.

Fig. 8 3D (a) and 2D (b) illustrations of the output power of a DPAL changing with the cell wall temperature and the cell length with the cell radius of 5.0 mm under 100 W-pumping.

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Fig. 9 3D (a) and 2D (b) illustrations of the output power of a DPAL changing with the cell wall temperature and the cell length with the cell radius of 5.0 mm under 200 W-pumping.

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Figure 10 shows the 3D diagrams where the optical-to-optical efficiency is calculated as a function of both the cell length and the cell wall temperature. It can be seen that the influences of the cell length and cell wall temperature on the optical-to-optical efficiency have the similar tendency to that on the output power. Two evident ridges on the curved surface in two figures correspond to the peak values of the optical-to-optical efficiency, and such optimum values are related by a correlation of a pair of the cell length and cell wall temperature.

Fig. 10 3D illustrations of the optical-to-optical efficiency of a DPAL changing with the cell wall temperature and the cell length with the cell radius of 5.0 mm under (a) 100 W-pumping and (b) 200W-pumping.

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In Fig. 11, the laser power is given as a function of the pump power in different cases. The cell length, cell radius, and cell wall temperature are set to be 35 mm, 5.0 mm, and 383 K, respectively. It can be observed that all the quantities of these curves monotonically increase with the pump level. It should be noted that the optical-to-optical efficiency with the pump power of 100 W (42.1% in Case 1) is much higher than that with pump power of 200 W (33.3% in Case 1), which is due to that the parameters, such as the cell length, cell radius, and cell wall temperature, merely corresponds to the optimal values with the pump power of 100 W. According to our further calculation, the optimal cell length that with the pump power 200 W should be 70 mm under R = 5.0 mm and T_w = 383 K. By lengthening the cell length from 35 mm to 70 mm, the optical-to-optical efficiency with pump power of 200 W increases obviously from 33.3% to 44.5% (not shown in Fig. 11).

Fig. 11 Output power versus pump power in five cases and the optical-to-optical efficiency versus pump power in Case 1 with the cell length, cell radius, and cell wall temperature of 35 mm, 5.0 mm, and 383 K, respectively.

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As shown in Fig. 11, the process of energy pooling (Case 2) has little influence on the output power compared with Case 1 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 differences can be observed when the pump power exceeds 100 W in Fig. 11, in which the percent drop in output power achieve the value of 0.65% (Case 1→Case 3) and 1.38% (Case 1→Case 4) with the pump power of 200 W, respectively. Moreover, when considering both processes of Penning ionization and photo-ionization in the model, the percent drop in output power is 1.63% (Case 1→Case 5) with the pump power of 200 W.

The corresponding quantified results are listed in Table 3. Compared with Case 1, the most prominent effect on output power is due to Case 5, and the percent drop in output power due to Case 4 is greater than that due to Case 3, while Case 2 has little influence on the output power. It can also be observed that the percent drop decreases with the cell length increasing, especially for that due to ionization, implying that the optimization of a DPAL could effectively reduce the influence of energy pooling and ionization. According to our previous study, the influence of these deleterious processes including energy pooling and ionization would be greater with high pump power [18]. Therefore, optimization should play important role in improving the output features of a high-powered DPAL system.

Table 3. Percent drop in output power due to each case with cell radius of 5.0 mm under pump power of 200 W at 383 K

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Although there are no obvious differences among five cases in the manuscript, the phenomenon should be paid necessary attention because the feasible pump power might reach to several thousand watts or even more for a high-powered DPAL, which will be much higher than the order of pump intensity in Fig. 11 and Table 3. The optimization of physical conditions should be an essential task for design of an even higher-power DPAL.

4. Summary

In this study, we develop a theoretical model to study a static-state DPAL by taking the process of energy pooling and ionization into account. The influences of the cell radius, cell length, and cell wall temperature on output features of a DPAL are systemically investigated. The results show that the optimization of a DPAL can be realized by adjusting the cell wall temperature and geometric dimensions such as the length and radius of a vapor cell when given a certain pump power. By optimizing a DPAL, the optical-to-optical efficiency of the laser system can be dramatically improved and the influences of energy pooling and ionization can be effectively decreased. Considering these results, optimization should be a necessary and effective way to improve the physical features of a high-powered DPAL system.

Actually, the buoyancy effect in a vapor cell results in the asymmetrical temperature distribution [17]. To make our model more perfect, we will create a 3-D theoretical regime by taking the buoyancy effect at the transverse section and heat transfer along the optical axis into account in our future study.

Acknowledgments

We are very grateful to 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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Parameter Description	Value	Reference
Partial pressure of He	478.8 Torr	This work
Partial pressure of C₂H₆	100 Torr	This work
Reflectance of the output coupler	0.3	This work
Pump beam waist ω_p	1.0 mm	This work
Laser beam waist ω_l	0.5 mm	This work
Transverse section of the D2 line σ_D2	1.53 × 10⁻⁹ cm²	[22]
Transverse section of the D1 line σ_D1	2.31 × 10⁻⁹ cm²	[22]
Photo-ionization cross section σ_{photoionization}	2.00 × 10⁻¹⁷ cm²	[7]
D1 radiative lifetime τ_D1	34.9 ns	[22]
D2 radiative lifetime τ_D2	30.5 ns	[22]
Lifetime of 6²D_5/2,3/2 and 8²S_1/2 levels τ₄	60 ns	[7]
Pump FWHM	30 GHz	This work
Pump center wavelength	852.3 nm	This work
D₁ transition wavelength	894.6 nm	[22]
Energy pooling rate constant k_EP2	7.1 × 10⁻¹⁰ cm³/s	[23]
Energy pooling rate constant k_EP3	2.8 × 10⁻¹⁰ cm³/s	[23]
Penning ionization rate constant k_PI	1.0 × 10⁻⁸ cm³/s	[24]
Recombination rate constant k_{recombination}	7.124 × 10⁻²¹ cm⁶/s	[25]

No.	energy pooling	Penning ionization	photo-ionization
Case 1	×	×	×
Case 2	√	×	×
Case 3	√	√	×
Case 4	√	×	√
Case 5	√	√	√

No.	L = 35 mm	L = 50 mm	L = 60 mm	L = 70 mm
Case 2	0.01%	0.01%	0.01%	0.01%
Case 3	0.65%	0.41%	0.24%	0.03%
Case 4	1.38%	0.73%	0.51%	0.17%
Case 5	1.63%	1.24%	0.86%	0.37%

Parameter Description	Value	Reference
Partial pressure of He	478.8 Torr	This work
Partial pressure of C₂H₆	100 Torr	This work
Reflectance of the output coupler	0.3	This work
Pump beam waist ω_p	1.0 mm	This work
Laser beam waist ω_l	0.5 mm	This work
Transverse section of the D2 line σ_D2	1.53 × 10⁻⁹ cm²	[22]
Transverse section of the D1 line σ_D1	2.31 × 10⁻⁹ cm²	[22]
Photo-ionization cross section σ_{photoionization}	2.00 × 10⁻¹⁷ cm²	[7]
D1 radiative lifetime τ_D1	34.9 ns	[22]
D2 radiative lifetime τ_D2	30.5 ns	[22]
Lifetime of 6²D_5/2,3/2 and 8²S_1/2 levels τ₄	60 ns	[7]
Pump FWHM	30 GHz	This work
Pump center wavelength	852.3 nm	This work
D₁ transition wavelength	894.6 nm	[22]
Energy pooling rate constant k_EP2	7.1 × 10⁻¹⁰ cm³/s	[23]
Energy pooling rate constant k_EP3	2.8 × 10⁻¹⁰ cm³/s	[23]
Penning ionization rate constant k_PI	1.0 × 10⁻⁸ cm³/s	[24]
Recombination rate constant k_{recombination}	7.124 × 10⁻²¹ cm⁶/s	[25]

No.	energy pooling	Penning ionization	photo-ionization
Case 1	×	×	×
Case 2	√	×	×
Case 3	√	√	×
Case 4	√	×	√
Case 5	√	√	√

Optimization of physical conditions for a diode-pumped cesium vapor laser

Abstract

1. Introduction

2. Theoretical analyses

2.1 Analyses of heat transfer

2.2 Analyses of laser kinetics

3. Results and discussions

4. Summary

Acknowledgments

References and links

Cited By

Figures (11)

Tables (3)

Equations (14)

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