Algorithm for evaluation of temperature distribution of a vapor cell in a diode-pumped alkali laser system (part II)

Juhong Han; You Wang; He Cai; Guofei An; Wei Zhang; Liangping Xue; Hongyuan Wang; Jie Zhou; Zhigang Jiang; Ming Gao

doi:10.1364/OE.23.009508

1. Introduction

With high Stokes efficiency, good thermal performance, narrow linewidth, compact size, non-toxic system, etc., a diode-pumped alkali laser (DPAL) becomes one of the most hopeful high-powered laser sources of the next generation and has been rapidly developed in the last ten years [1–5]. However, the thermal effects will bring about some serious problems in nonuniformity of the temperature distribution for a high-powered DPAL system because the thermal conductivity of a gas-state medium is so small that the generated heat cannot be transferred outside efficiently [6–9]. Unlike a conventional electrically-excited gas-state laser, the number densities of alkali vapors and buffer gases (generally being helium and small hydrocarbons such as methane and ethane) in a DPAL often exhibit inhomogeneous distributions resulting from the temperature gradient inside a vapor cell [10]. Actually, the inhomogeneous of the alkali vapor directly influences the output performances of a DPAL. Furthermore, the nonuniformity of buffer gases affects the line-center cross sections of both the D₁ line (n ²P_1/2 → n²S_1/2) and the D₂ line (n²S_1/2 → n²P_3/2) of an alkali atom as well as the rapid rate of fine-structure mixing (n²P_3/2 → n²P_1/2) [11].

Therefore, diminishing the temperature inhomogeneous is one of the key points to realize a high-powered DPAL with good beam quality. Generally, flowing the gaseous medium in an enclosed system is thought as an effective way to reduce the thermally-induced effects of a DPAL configuration [12]. In our previous study, a systematic model is constructed to explore the radial temperature distribution in the transverse section of a cesium vapor cell with static state [11]. In this report, we improve our theoretical model by uniting the laser kinetics, fluid dynamic, and heat transfer procedures together to analyze the features of a flowing-gas Cs-DPAL system. Until now, only two research groups have referred the physical characteristics of a DPAL with a flowing approach. One is the team led by B. D. Barmashenko and S. Rosenwaks who have undertaken a series of fruitful studies on flowing-gas DPALs [9, 12–15]. The other is the team of National University of Defense Technology of China. Although they developed an effective approach to describe the features of alkali vapor lasers in side-pumped configuration with flowing medium, the effects of heat transfer on the laser performance have not been considered in their study [16]. In this report, we reveal that the temperature gradient can be dramatically alleviated by using the flowing procedure. Improvement on both the laser output and the beam quality in a flowing-gas DPAL should be meaningful for realization of a high-powered alkali vapor laser system.

2. Theoretical analyses

As shown in Fig. 1, a vapor cell which connects to an enclosed circulatory system is divided into many coaxial cylindrical annuli. For the jth cylindrical annulus (an arbitrarily one) among the segments, the outside radius r_j and inner radius r_{j + 1} can be simply expressed by

\begin{array}{l} r_{j} = R - (j - 1) \cdot R / N, \\ r_{j + 1} = R - j \cdot R / N, \end{array}

where R is the radius of the vapor cell, N is the total number of segmented cylindrical annuli, respectively. For an end-pumped structure, the optical axis of these coaxial cylindrical annuli coincides with that of the pump laser beam. Actually, every cylindrical annulus is thought as a heat source and a laser emitting source at the same time in the calculation. In this study, the effective length L and radius R of the vapor cell in a DPAL system are 25 mm and 7.5 mm, respectively. The partial pressure is 500 Torr for helium and 100 Torr for ethane, respectively. The buffer gas and the alkali vapor are forced to flow inside the cell with the velocity of U. Convection and heat radiation are so small that they have been ignored in the system. In addition, the temperature distribution along the axial direction and the effects of both end-windows are also neglected.

Fig. 1 Schematic illustration for segmentation procedure of a flowing-gas cell.

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2.1. Analyses of laser kinetics

The population densities of 6²S_1/2, 6²P_1/2, and 6²P_3/2levels in a three energy-level cesium system of the jth cylindrical annulus can be deduced by solving the following rate equations [17]:

\begin{array}{l} \frac{d n_{1}^{j}}{d t} = 0 = - Γ_{p}^{j} + Γ_{L}^{j} + \frac{n_{2}^{j}}{τ_{D_{1}}} + \frac{n_{3}^{j}}{τ_{D_{2}}}, \\ \frac{d n_{2}^{j}}{d t} = 0 = - Γ_{L}^{j} + γ_{32} (T_{j}) [n_{3}^{j} - 2 n_{2}^{j} \exp (- \frac{Δ E}{k_{B} T_{j}})] - \frac{n_{2}^{j}}{τ_{D_{1}}}, \\ \frac{d n_{3}^{j}}{d t} = 0 = Γ_{p}^{j} - γ_{32} (T_{j}) [n_{3}^{j} - 2 n_{2}^{j} \exp (- \frac{Δ E}{k_{B} T_{j}})] - \frac{n_{3}^{j}}{τ_{D_{2}}}, \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, T_j is the temperature of the jth cylindrical,

τ_{D_{1}}

is the D₁ radiative lifetime,

τ_{D_{2}}

is the D₂ radiative lifetime,

γ_{32} (T_{j})

is the relaxation rate of the fine-structure mixing (n²P_3/2 → n²P_1/2), ΔE is the energy gap between the n²P_3/2 and the n²P_1/2 levels, k_B is the Boltzmann constant, respectively.

For a steady-state laser emission, the number density of every energy-level must satisfy the following two equations [18]:

exp [2 (n_{2}^{j} (T_{j}) - n_{1}^{j} (T_{j})) σ_{D_{1}}^{He-broadened} (T_{j}) l] \times T T^{2} \cdot R_{o c} = 1,

n_{0}^{j} (T_{j}) = n_{1}^{j} (T_{j}) + n_{2}^{j} (T_{j}) + n_{3}^{j} (T_{j}),

where

n_{0}^{j} (T_{j})

is the cesium number density of the jth cylindrical annulus as expressed by [10]

n_{0}^{j} (T_{j}) = n_{0}^{1} (T_{w}) (\frac{T_{w}}{T_{j}}),

where T_w is the temperature of the cell wall,

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

is the saturated alkali number density inside the first cylindrical annulus which is adjacent to the internal surface of the cell wall. By solving Eqs. (2)-(5),

n_{1}^{j}

,

n_{2}^{j}

,

n_{3}^{j}

, and

Γ_{L}^{j}

can be obtained if

T_{j}

is assumed as a known value.

Thus, the generated heat of the jth cylindrical annulus can be then calculated by the following formula [17]:

Q_{j} = V_{L}^{i} \cdot γ_{32} (T_{j}) [n_{3}^{j} - 2 n_{2}^{j} \exp (- \frac{Δ E}{k_{B} T_{j}})] Δ E,

where

V_{L}^{i}

is the volume of the jth cylindrical annulus. The value of

Q_{j}

will be used in the next calculation process.

2.2. Analyses of fluid dynamic and heat transfer

Next, we investigate how heat transfer works in the flowing gaseous medium. To explore the heat characteristics in a flowing-gas cell, we first determine the flowing state inside the cell. Generally, the Reynolds Number Re, as expressed in the following formula, can be used to determine whether the flow is laminar or turbulent [19]:

R e \equiv \frac{2 ρ U R}{μ},

where ρ, μ, and U are the density, viscosity, and velocity of mixed flowing gases in the vapor cell, respectively. In our model, the maximal value of U is limited to 10 m/s. It is easy to deduce the maximum Re as about 1800. For circular tubes, the transition from laminar to turbulent flow occurs over a range of Reynolds numbers from approximately 2300 to 4000, regardless of the nature of the fluid or the dimensions of the pipe or the average velocity [19]. So we believe that only laminar flow exist in our flowing-gas system. Furthermore, the corresponding laminar entrance length x_D can be calculated by [20]:

x_{D} \approx 0.05 R e \cdot (2 R) = 1350 mm .

Since such a value is much longer than the gain medium length L (25 mm), the velocity distribution at the transverse section of the cell can be ignored in the calculation. Thus, flowed heat F_j in the jth cylindrical annulus can be expressed by [20]

F_{j} = \frac{S_{j} \cdot U \cdot n (T_{j})}{N_{A}} \int_{T_{w}}^{T_{j}} C_{p} (T) d T,

where S_j is the cross-sectional area of the jth cylindrical annulus, n(T_j) denotes the number density of the total mixed gases in the jth cylindrical annulus, N_A presents the Avogadro constant,

C_{p} (T)

is the molar heat capacity of the total buffer gases resulting from the temperature T as given by [21, 22]

C_{P} (T) = \frac{P_{H e}}{P_{H e} + P_{C_{2} H_{6}}} C_{P - H e} (T) + \frac{P_{C_{2} H_{6}}}{P_{H e} + P_{C_{2} H_{6}}} C_{P - C_{2} H_{6}} (T),

where

P_{H e}

and

P_{C_{2} H_{6}}

are the partial pressures of helium and ethane,

C_{P - H e} (T)

and

C_{P - C_{2} H_{6}} (T)

are the molar heat capacity of helium and ethane, respectively. The transferred heat Φ_j in the jth cylindrical annulus can be expressed by [20]

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

When the thickness of the segmented cylindrical annulus is small enough, Φ_j can be approximately expressed as

Φ_{j} = - K (T_{j}) \cdot A_{j} \frac{T_{j + 1} - T_{j}}{r_{j + 1} - r_{j}},

where K(T_j) denotes the thermal conductivity of the total buffer gases in the jth cylindrical annulus as given by [12, 23–25]

K (T_{j}) = \frac{P_{H e}}{P_{H e} + P_{C_{2} H_{6}}} K_{H e} (T_{j}) + \frac{P_{C_{2} H_{6}}}{P_{H e} + P_{C_{2} H_{6}}} K_{C_{2} H_{6}} (T_{j}),

where

K_{H e} (T_{j})

and

K_{C_{2} H_{6}} (T_{j})

are respectively the molar heat capacity of helium and ethane [8], A_j stands for the contact area between the jth and the (j + 1)th cylindrical annuli as given by

A = 2 π r_{j} L .

Appointing the total heat transferred out from a vapor cell as P_Thermal , the relationship between P_Thermal and the total generated heat

\sum_{i = 1}^{N} Q_{i}

as well as the flowed heat

\sum_{i = 1}^{N} F_{i}

can be described as

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

Equation (15) indicates that the total generated heat is equal to the sum of the heat removed by the flowing mixed gases and the heat conducted to the cell wall from the gas-state medium. Such a process is schematically illustrated in Fig. 2.The temperature distribution and the output characteristics can be simultaneously evaluated by using Eqs. (2)-(15) and relations shown in Fig. 2.

Fig. 2 Diagram for representing the relationship of generated heat, transferred heat and flowed heat in the cross-section of a vapor cell.

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3. Results and discussions

By using the theoretical model introduced in Section 2, we calculate the thermal features of a cesium cell and the physical characteristics of a Cs-DPAL. In the computation, the temperature of the cell wall T_w is 400 K, the waist radius of the pump Gaussian beam is 1000 μm, and the other parameters are the same as those in Ref. 11.

Figure 3 shows the radial temperature distribution with the velocity of mixed gases of 0, 1, 5, and 10 m/s, respectively, while the pump power is set to 1000 W. The horizontal axis indicates the distance to the center of a vapor cell and the vertical axis of the graph denotes the temperature. The temperature distribution has an obvious gradient in the cross section of a vapor cell for the pump power of 1000 W and the temperature peaks appear at the cell center for every curve. It can be found that the absorbed pump power has led to a more obvious temperature rise if the mixed gases are not flown. The maximum temperature rise even reaches about 550 K for the static medium. However, such a temperature gradient can be effectively decreased with a flowing approach. The faster the flowing velocity is, the lower the temperature rise becomes. The results demonstrate the essentiality and effectiveness of the flowing-gas procedure for an even-higher DPAL system.

Fig. 3 Temperature distributions in the cross-section of a cell with different velocities of the mixed gases when the pump power is 1000 W.

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The population distributions with different velocity of the mixed gases are shown in Fig. 4 with the pump power of 1000 W. The velocity of the flowing-gas is 0, 1, 5, and 10 m/s, respectively. It is seen that the cesium number density n₀ increases with the radial position, which means that the temperature gradient in the cell cross-section must influence the distribution of the gain medium. The inflection points in n₁ and n₂ curves are located in the lasing boundary line. With increase of the flowing velocity of the mixed gases, the cesium number density n₀ obviously increases in both the lasing region and the no-lasing region. One can suppose that the absorbed capability inside the cell can be greatly improved by the flowing process.

Fig. 4 Population distributions with different velocities of the mixed gases when the pump power is 1000 W.

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Figure 5(a) shows the variation trend of the output characteristics of a Cs-DPAL with different flowing velocities of mixed gases when the pump power is 1000 W. The output power first rapidly raises and then flattens out with increase of the flowing-gas velocity. If the flowing velocity is up to 10 m/s, the output power can be increased as about two times of a static DPAL system. The reason can be explained to the enhancement of the absorbed power inside a vapor cell when using the flowing procedure. In Fig. 5(b), we can observe that the generated heat is totally transferred to the outside in a static cell, but, in a flowing-gas cell, the generated heat is mainly taken away from the cell by the flowing gas if the speed is larger than 4 m/s. For a DPAL system with the output of tens of thousands watts, the flowing velocity should be raised as high as several tens meter per second or even more.

Fig. 5 Output power (a) and heat features (b) of a Cs-DPAL system versus the flowing velocity of mixed gases with the pump power of 1000 W.

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4. Conclusions

In this paper, we introduce a theoretical scheme to analyze the temperature distribution of the flowing-gas cell in a Cs-DPAL system by simultaneously considering the laser kinetics, heat transfer and fluid dynamic. The population distributions at the cross-section of a vapor cell and the output characteristics have been systematically evaluated with the different flowing velocities. From the calculation results, we can understand that the flowing-gas procedure is effective to dramatically eliminate the temperature gradient and increase the output power. The theoretical analyses would be helpful to construct a feasible high-powered DPAL with good beam quality in the future.

Acknowledgments

We are very grateful to Dr. Boris D. Barmashenko in Ben-Gurion University of the Negev for his meaningful helps in calculating the saturated alkali number densities inside a static alkali vapor cell.

References and links

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Algorithm for evaluation of temperature distribution of a vapor cell in a diode-pumped alkali laser system (part II)

Abstract

1. Introduction

2. Theoretical analyses

2.1. Analyses of laser kinetics

2.2. Analyses of fluid dynamic and heat transfer

3. Results and discussions

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (5)

Equations (15)

Optics Express