Comprehensive analysis of kinetic and fluid dynamic processes in flowing-gas diode-pumped alkali vapor amplifiers is reported. Taking into account effects of the temperature, the amplified spontaneous emission, the saturation power, the excitation of the alkali atoms to high electronic levels and the ionization, a detailed physical model is established to simulate the output performance of flowing-gas diode-pumped alkali vapor amplifiers. Influences of the flow velocity and the pump power on the amplified power are calculated and analyzed. Comparisons between single and double amplifier, longitudinal and transverse flow are made. Results show that end-pumped cascaded amplifier can provide higher output power under the same total pump power and the cell length, while output powers achieved by single- and double-end pumped, double-side pumped amplifiers with longitudinal or transverse flow have a complicated but valuable relation. Thus the model is extremely helpful for designing high-power flowing-gas diode-pumped alkali vapor amplifiers.
© 2015 Optical Society of America
Being regarded as one of most possible paths to high energy laser, diode-pumped alkali vapor lasers (DPALs) have attracted much attention and been extensively studied during the past dozen years. Combining the positive characteristics of solid and gas lasers, such as high quantum efficiency, good thermal performance, narrow linewidth, compact size and so on , DPALs have the potential to achieve high power in a high quality beam that is very desirable for various important applications in science, technology and national security areas . Till now, powers produced by DPALs have shown a remarkable increase from mW to kW, corresponding to multiple orders of magnitude. These increased power levels, with yet higher powers anticipated, have demonstrated the high efficiency of DPALs and the potential for power scaling [2–8].
Using MOPA (master oscillator power amplifier) is an important way to increase the output power of alkali vapor lasers, and preserves both the spectral and spatial beam qualities of the seed laser at the same time. A chain of such amplifiers can provide considerable increase of alkali lasers power . Up to now, some MOPA experiments on DPALs have been made [9–11] and their corresponding models were set up [12–15], which agreed well with experimental results.
Operating efficient high-power DPALs with MOPA system is hindered by the processes of heating of the gas mixture, photoexcitation, energy pooling and ionization of the alkali atoms. Using the flowing way of gas mixture can avoid such processes and replenish the lost neutral alkali atoms, which was taken into account by some theoretical models [1,16–22]. However, in spite of these excellent models, researches conducted on high-power flowing-gas DPALs using MOPA system are devoid.
Our model for flowing-gas alkali vapor amplifiers not only takes into consideration of the temperature rise, the amplified spontaneous emission (ASE) and the saturation effect, but also the processes of the alkali atoms excitation to high electronic levels, the ionization of these levels, and the electron-ion recombination. Chemical reactions are not included in the present model since, as explained below, they are negligible in the mixture . The model is able to describe both static and flowing-gas amplifiers. Detailed analysis for avoiding losses of neutral alkali atoms and improving the laser output should be meaningful for realizing high-power diode-pumped alkali vapor amplifiers.
2. Description of the kinetic and fluid dynamic model
Schematic diagrams of the end-pumped single amplifier, the end-pumped double amplifier and the double-side pumped amplifier are shown in Fig. 1. The seed beam enters a cylindrical cell of radius R and length L [in the case of the longitudinal flow shown in Fig. 1(a) and 1(b)] or a rectangular cell of height H, width W and length L [in the case of the transverse flow shown in Fig. 1(c)] from the ends through window with transmission t. The pump beams also enter into the cell which consists of a mixture of alkali vapors and buffer gases flowing with velocity u. The walls of the amplifier cell are heated to a temperature of Tw. Both the gas temperature at the entrance to the cell and the alkali source temperature are assumed to be equal to Tw .
2.1. Kinetic processes and rate equations
The energy diagram which shows the transitions taken into account in this work for an alkali laser is presented in Fig. 2, including the standard three levels, the relevant excited S and D states and the ionization limit.
In each of the considered alkali systems, the , and states can be populated by photon excitation from the and states by means of either pump or seed light, and then ionized by the same pump or seed light . All important kinetic processes in diode-pumped alkali vapor amplifiers (For X = alkali atom and M = CH4 or C2H6) are listed Table 1.
The rate equations for the population densities of various species of the alkali atoms are described as follows:
The number densities of He and M in the amplifying region and near the wall also have the same relation. Just as in [12,14,15,23,24], it is assumed for simplicity that the pressure in the cell is constant. and the alkali saturated vapor density are independent of , the densities of different species and T in the amplifying region are spatially uniform.
2.2 Transition rates and amplified power
The expressions of the transitions rates are shown as follows.
The incident spectrally resolved pump power is given by
The expressions of pump absorption rate and laser emission rate are given by15,25], we give the expressions of and asEq. (9) are proportional to . is the mode overlap factor between the pump laser and the seed laser defined by . is the total seed laser power. Because the seed laser is not a small signal and and are unsaturated populations in the absence of the stimulated emission, amplification saturation effect has to be included in the model, represents the saturation power of the amplifier which can be obtained by the formula in 15].
The amplified spontaneous emission rate is given by .
The rates of relaxation , spontaneous emission , and , quenching , photoexcitation , energy pooling , photoionization , penning ionization and recombination , are given by . For a whole amplified cell, note that the photoexcitation and photoionization rates should be changed into exponential absorption form:Eq. (13) of .
When X = Cs, estimates in  show that the rate of the three-body recombination, , and of the two-photon ionization, , calculated using the experimental values of the rate constants are negligibly small. Following [17,22] we also assumed that chemical reactions of the excited alkali atoms due to T are negligible. At their rate constant calculated by  is , far smaller than the other rate constants, and their products due to the temperature variations  will be removed quickly by the flowing gases. Therefore, these processes were not taken into account in the computations.
2.3 Fluid dynamic processes and thermal balance equations
The heat removal from the amplifying region to the walls can be calculated by17,21]. denotes the number density of the total mixed gases in the cell and presents the Avogadro constant. The Nusselt number is given by1]:1,17,21]
For the first term in the right-hand-side of Eq. (17) describes the convective heat transfer due to the gas flow whereas the second term corresponds to the thermal conductivity through the lateral surface of the cylindrical amplifying region, which is negligibly small in comparison with the first convective term even for , meaning that the output power is mainly affected by the first term. Since L is far larger than , we can know from Eqs. (17) and (19) that for the same flow velocity, temperature and pressure at the laser cell inlet, the convective heat and hence the number densities for the transverse flow, are larger than those for the longitudinal flow.
The heat release due to relaxation between the fine-structure levels and their quenching can be calculated byEq. (23) corresponds to the heat release due to relaxation between the fine-structure levels of the alkali atoms, the first term in the square brackets to the quenching of these levels and the second term to the ion-electron recombination described in detail in . The heat release due to relaxation and quenching of the higher levels , and is estimated to be negligibly small and is not taken into account [17,21].
T is found from the energy balance assuming that the heat removal is equal to the heat release :
Using Eqs. (1-24) we can calculate the population densities of the eight levels, the temperature T in the amplifying region and the output power .
2.4 Numerical approaches
It is worth mentioning that there are too many unknown variables for the program to find the explicit solutions. Therefore, we can give an initial value of T for Eqs. (1-16) to calculate the population densities and hence the transition rates. Then by substituting the transition rates into Eqs. (17-23) to solve Eq. (24) we can obtain a new temperature T for the amplifying region, compare it with the old one, if they basically equivalent to each other (<0.1%), then we can get the final solution of the temperature, as well as the population densities and the output power. If not, substitute this new temperature into Eqs. (1-16) to continue the iterative process for a final solution.
In the longitudinal dimension of a double-end pumped amplifier, the gain medium is divided into small volume segments (). In each segment, the temperature in the amplifying region is assumed to be constant for longitudinal flow with high flow velocity and transverse flow. First, let the pump light from one end pass half of the gain medium to obtain the population distribution, then let the pump light go on passing the rest of the gain medium through this distribution, we can obtain the unabsorbed pump power at the other end of the gain medium. The unabsorbed pump powers for both forward and backward propagating pump lights are assumed be equal, so we get the total pump power at one end, use it to calculate the population distribution and compare them with the old ones, if they basically equivalent (<0.1%), then we get the final solution. If not, repeat the prior steps, continue the iterative process and finally get the solution.
Since the simulation results of [17,20,21] with the main assumption on the uniform densities of different species and temperature are in good agreement with the measurements in static and flowing-gas end-pumped alkali vapor lasers, our model for end-pumped amplifiers does not take into account the accurate density and temperature distribution, but the one for side-pumped amplifiers does.
For double-side pumped configuration, a two-dimension division of the cross-sectional geometry of the amplifier cell pumped by laser diode arrays is made, each divided volume element has dimensions of (for the cell dimensions of ). Combining Eqs. (1-24) and the iterative algorithm proposed by [14,15] in the transverse dimension, we can obtain the side-pumped amplifier power.
3. Results and discussion
To test the model we first applied it to the diode-pumped Cs vapor amplifiers with broadband pumping (0.7nm). We calculated the power for this case and found out that for the laser parameters indicated below and high flow velocity there is no significant difference in the calculated power between single and double amplifier, end-pumped and side-pumped configurations, longitudinal and transverse flow.
The cylindrical Cs vapor cell is filled with ethane and helium with a molar ratio of 1/3.5, this ratio was assumed for total pressures up to 4.5 atm . = 0.97 and t = 0.98. For end-pumped configuration the cross section of the pump beam has a circular shape and a 3.5 mm radius with the same cross section of the seed beam in order to achieve the highest mode overlap factor. For side-pumped configuration the rectangular cell has the same height and width of 1 cm with a seed beam waist of 3.5 mm and an optimal ratio of pump beam and seed beam waists of 3/7 calculated by . For all cases presented here the seed power is assumed to be 20 W and kept as constant.
Lowering the temperature properly will be helpful for weakening the thermal effects  and hence increasing the output power. Thus we first simulated the dependence of the calculated and T on the longitudinal flow velocity u with and as shown in Fig. 3. For small u, dramatically grows with u, however, at larger u the growth rate decreases and saturates approaching the maximum value for the given . At small the amplified power saturates at low . At higher pump power () a higher is need  for efficient operation.
T decreases with increasing u from ∼700 K at to ∼400 K at . A slight increase in T with increasing u from 0 to is caused by an increase in the absorption of the pump beam due to reduction of losses of neutral alkali atoms .
When , an amplification of 7 is achieved for this case which was close to the experimental results in [9,11].
Output power and population densities as functions of the pump intensity of an end-pumped amplifier is shown in Fig. 4 with and other parameters as described above. It is seen that the increase of and changes of densities () are substantial due to increase of the pump intensity and stronger gas heating by the pumping beam. After that they reach the saturated values, the continued increase of the density of ions show that the influence of photoexcitation, energy pooling and ionization cannot be simply ignored at rather high pump intensity.
Multiple amplifier approach is an important and efficient way to further achieve higher power and studying it will become an inevitable trend. Figure 5 shows the different output powers of end-pumped single and double amplifiers for two values of pump power.
The pump power and cell length of first stage of the cascaded two-stage amplifier are and while the second are and that meet conditions of , and , where and are the pump power and cell length of the single amplifier, respectively. We can see from Fig. 5 that for the same total pump power and cell length, the final output power of double amplifier is always higher than that of single amplifier. A maximum value of is obtained when for both of the pump powers (1 kW and 2 kW), which is practical for designing the setup of double amplifier.
The maximum optical to optical conversion efficiency of this double amplifier was ~20% with respect to total pump power of 1 kW, more than 50% larger than those of the single amplifier with a pump power of 1 kW and double amplifier with a total pump power of 2 kW.
Because the pumped intensity along z-axis is assumed to be homogeneous for side-pumped configuration, the only considerable difference between single and double amplifier is the losses of laser energy caused by the windows between the two stages. Thus the cascaded case of side-pumped configuration is not analyzed here.
Figure 6 shows the calculated values of as function of for amplifiers in end- and side-pumped configurations. The calculations were performed for longitudinal and transverse flows with and . At , the values of the output power calculated for different configurations and two mutually perpendicular flow directions have the following relation: , where the subscripts and t indicate single-end pumped, double-end pumped and double-side pumped configuration, longitudinal and transverse flow, respectively. The output powers are very close to each other for different flow directions with this sufficiently large u. At higher , . The calculated of double-side pumped configuration, which can provide a uniform distribution of pump power along the z-axis, are much larger than those of double-end pump configuration, while the latter are much larger than those of single-end pumped configuration. Figures 6(b) and 6(c) display the difference of output powers of longitudinal and transvers flow more clearly at a large range of pump power. The main reason for the higher output power achieved for transverse flow with larger flow cross section and shorter flow path in the same configuration is the much more efficient cooling caused by faster replacement of the hot active volume gas, which results in lower temperature of the mixtures in the amplifying region. For example, at and the average temperatures over the amplifying volume of the single-end pumped amplifier for the longitudinal and transverse flows are 456 K and 388 K, respectively. High flow velocity can reduce the temperature rise and make the output power of longitudinal flow close to the one of transverse flow as shown in Fig. 6(d).
Taking into account the rise of temperature T, the amplified spontaneous emission, the saturation effect, the excitation of the alkali atoms to high electronic levels, and their losses due to ionization in the gain medium, modeling the performances of flowing-gas diode-pumped alkali vapor amplifiers are reported. Using these models, coupled equations for laser kinetics, laser optics, and gas flow were solved and different numerical approaches for the three kinds of configurations were proposed for Cs DPAL MOPA systems
For flowing-gas amplifiers with longitudinal flow, the model predicts a substantial increase of with increasing u. At moderate (), the processes of excitation of atoms to high levels, ionization, and rise of temperature strongly affect the amplified power at low flow velocity , but their influence is weak at larger u. Thus the maximum values of can be substantially increased by optimization of the flowing-gas amplifier parameters.
Cascaded double-stage end-pumped amplifier can achieve larger amplification than the single-stage one under equivalent total pump power and cell length. In particular, double amplifier with the same cell length of the two stages will achieve the highest output power.
Comparison of the longitudinal and transverse flow is made. For low pump power both models predict very close values of output power. However, at higher pump power, when the absorption on the pump transition is saturated, the rise of the temperature strongly affects the output power, while high flow velocity can avoid this decrease.
Amplifiers in different configurations with different flow directions have different capabilities to amplify the seed power. Especially, double-side pumped configuration with transverse flow under the same condition can obtain the maximal amplified power. Thus the model can provide an effective way to design an efficient alkali vapor laser MOPA system.
This work was supported by the Zhejiang Provincial Natural Science Foundation under Grant No. LY14A04005.
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