Considering the amplified spontaneous emission, the saturation effect and the energy distributions of the incident pump and seed lasers, a physical model is established to describe the kinetic process and the output performance of a four sided diode pumped alkali vapor laser amplifier. According to the experimental parameters of a single-side pumped configuration with a diffuse type hollow cylinder cavity, energy distributions in the cell and influences of several important factors are simulated and analyzed. The model is validated by comparing the simulation result with the experimental data, which shows the model can provide an effective way for designing an efficient diode four-side symmetrically pumped alkali vapor laser amplifier.
© 2015 Optical Society of America
Diode-pumped alkali vapor lasers (DPALs) have gained much attention and developed quite fast during the past decade. As a new class of laser, DPAL combines the major advantages of solid and gas lasers, such as high output power and high quantum efficiency, good thermal performance, narrow linewidth, compact size and so on , making it a potential candidate for future high power and high efficiency lasers. Till now, series experiments have demonstrated high output power and good beam quality of DPALs [2–7].
One way to increase the output power of alkali lasers is to use MOPA (master oscillator power amplifier), which is often implemented in high power solid-state laser systems. Up to now, some MOPA experiments on DPALs have been made in longitudinally pumped configuration [8, 9], and a corresponding model was set up by Pan et al , which has agreed well with experimental results.
Another more scalable pump scheme is the side pumped configuration, which has been applied in DPALs MOPA system , and the related models with rectangular cell were given by Parkhomenko and Yang et al [12, 13]. However, our model for a four-side pumped alkali vapor amplifier not only takes the amplified spontaneous emission (ASE), which accounts for a considerable proportion when the seed power is low, and the saturation effect when the seed intensity begins to approach the saturation intensity for the laser medium (that means the seed power is quite high) into consideration, but also uses the intensity distributions of the pump laser and the seed laser, and the numerical approaches are different from single-or-double-side pumped alkali vapor amplifier.
2. Modeling and numerical approaches
2.1. Modeling of a four-side pumped alkali vapor amplifier
The schematic diagram of a four-side pumped alkali vapor MOPA system is shown in Fig. 1. As most of the experiments, the amplifier cell is designed into a hollow cylinder , while the following calculation method for it can also be applied to the rectangular cell . The cell contains a homogeneous mix of alkali vapor and buffer gases at operation temperature. The pump lights enter into the cell through slits in the side, and its intensity along z-axis can be assumed homogeneous. The seed laser enters into the cell from x-y plane and propagates along z-axis to be amplified.
Different methods are used in the transverse (x, y) and longitudinal (z) dimensions as in . In the transverse dimension, a two-dimension division of the cross-sectional geometry of the amplifier cell pumped by LDAs with four symmetrical sides is made and shown in Fig. 2. The coupled LDAs are assumed to be in Gaussian profiles. We assume that the pump lights from four different directions are focused at the center axis of the vapor cell. R is the vapor cell radius, and are the seed and the pump beam waists, respectively.
In the longitudinal dimension, the model proposed by [10, 14] is used with some modifications to adjust the side pumped amplifier. The rate equations that govern the distribution of population densities in a volume element with dimensions of are
2.2. Numerical approaches
An iterative algorithm which we have proposed to solve for the four-side symmetrically pumped configuration is described as follow.
(a) Every element has four pump beams (, , and ) from four different directions. First, we only considerate the transverse population distribution for the double-side orthogonally pumped configuration, that isFig. 3. At position s (the black bold lines in Fig. 3, ), the expression of the pump beam waist is
In Fig. 3(a), the pump power should be obtained by different equations according to its position on black bold lines.
At the gray points
(b) For the pump light propagating along the y-axis (the black bold lines parallel to the x-axis), the incident spectrally resolved pump power at sides are given by
The pump powers at (x, s) are
According to , we give the expression of as16] with consideration of contributions from each hyperfine transition. represents the saturation power of the amplifier which can be obtained by the formula in , is the power distribution of seed laser, which is modeled as having a two-dimension Gaussian spectral profile.
The longitudinal amplified spontaneous emission rate is given by 
Because of the transverse population variation, it is difficult to know the transverse ASE effect among different volume elements, for simplification we only consider the longitudinal ASE effect.
The propagation equations of and are given by
(c) For the pump light propagating along the x-axis, all the items with or in step (b) should be changed into or , such as . And we can obtain the laser powers and population densities on the black bold lines parallel to the y-axis.
(d) For four-side symmetrically pumped configuration, the population densities in steady state should also be symmetrically distributed. Here, the symmetrically distributed population density is assumed to beEq. (19), we can obtain the unabsorbed pump power at the other side of the gain medium.
In steady state, the unabsorbed pump power for both forward and backward propagating pump lights should be equal, that is , where . Here we assume the amplifier to be in steady state, and the total pump power at one side should be
(e) By step (d), we can compare the new population density with , if they basically equivalent (<0.1%) to each other, then we get the final solution. If not, repeat the prior steps (b)-(d) to continue the iterative process for a final solution, note that the superscript of is a cycle index of the algorithm.
A flowchart for explaining the process of the iterative algorithm described above is diagramed in Fig. 4.
3. Simulation results and discussion
In comparison with the single-side pumped Cs laser MOPA system whose entire cell is held inside a custom cylindrical diffuse reflector to assure homogenous pumping , we use its experimental parameters described as follows: the dimensions of the cylindrical glass cell filled with metallic cesium and 500 Torr ethane (measured at 20°C) are . The seed laser power is 5W with a 5mm diameter. The experimental amplifier was transversely pumped by 15 laser diode arrays with a total power of 280 W and a linewidth (FWHM) of 10GHz, therefore we assume that the pump power at each side of the four-side pumped configuration is 70W. The amplifier is operated at a temperature of 103°C. The lineshapes of the D2 atomic absorption line and the pump light are assumed to be in Voigt and Gaussian profiles with almost overlapped center wavelengths.
By Eqs. (1)-(20), the pump energy distributions inside the laser cell for a four-side pumped alkali vapor laser with different pump beam waist can be theoretically obtained as shown in Figs. 5 and 6, respectively.
Multi-side symmetrically pumped configuration with a larger pump beam waist possesses a more homogeneous pump energy distribution. We can see that the pump intensity reaches maximum in the center of the cell, and its distribution is similar to that of the four-side pumped solid-state laser .
The laser energy distribution in the x-y plane at the output end are also obtained and shown in Figs. 7 and 8.
When and both the pump power and the seed power are high enough, we can obtain a cross-shaped laser spot, while when , we may see a quasi-square spot, which can be used for some special research, such as the diffraction and interference experiments.
In following subsections, the influence of several important factors on the amplifier laser power are simulated, and the results can provide a useful guide to design an alkali vapor laser MOPA system.
3.1 Influence of the temperature
Temperature is a very important factor in designing an alkali vapor amplifier because the operation temperature will decide the number density of the alkali atoms , and the temperature variations in the active region can result in carbon contaminations [18, 19].
Using the parameters described above, dependence of the output power on the temperature is shown in Fig. 9. We can see that when T < 110°, the simulated result is consistent with the experimental result , and an optimal operation temperature of 110°C exists for a maximal amplifier power of 34W, both are higher than those of a single-side pumped configuration. When the temperature is low, the alkali atom concentration increases with the temperature which enhances the pump absorption fraction and the output power of the amplifier to the maximum. After that the output power falls quickly due to the further increase of alkali vapor density resulting in reabsorption of the seed lasers and the increase in pump threshold.
3.2 Influence of the seed laser and the pump powers
For comparison we used a total pump power of 280 W (70 W at each side) at the temperature of 103°C, a pump beam waist of 1mm and a seed beam waist of 2.5mm as experiment in  to calculate the dependence of the laser power of amplifier on the input seed power from the master oscillator, which is shown in Fig. 10. It can be seen that the simulation agrees well with the experimental result in . When the seed laser is weak, ASE will dominate the stored pump power, since the transverse ASE effect among different volume elements is out of consideration, the simulated result is higher than the experimental result. When the seed laser is strong, the ASE is effectively suppressed by the saturation effect, and the amplifier powers turn to be roughly equivalent.
Results of the amplifier output power as function of the seed power with different pump power at the same temperature are presented in Fig. 11, which shows that the output power increases with enhancing the seed power and slowly become saturated.
3.3 Influence of the length of the cell
Figure 12 presented the dependences of the amplifier output power on the cell length at an optimal temperature of 110°C when Psl = 5W with different pump power and other parameters described above. We can see that in the design of the dimensions of the alkali vapor cell, as other parameters are set and the length of the diode pumps is changed to match the length of the cell, for each pump power there is an optimal length Lo where the output power is maximal. When the pump power at each side Pp = 50W, Lo = 2.5cm, when Pp = 70W, Lo = 3.5cm, when Pp = 100W, Lo = 4.5cm, and when Pp = 200W, Lo = 7cm. Higher pump power means longer cell length, but too long cell will make the amplifier laser reabsorbed by the extra gain medium.
3.4 Influence of the mode overlap factor
The mode overlap factor defined in Eq. (14) is an important factor to influence operation of the four-side pumped alkali vapor amplifier. With an amplifier cell length of 3cm, a temperature of 110°C and a seed power of 5W, simulation results of the amplifier laser power as functions of the mode overlap factor at different pump power density are shown in Fig. 13. In the calculation process, we maintain the ratio of the seed beam waist to the cell radius unchanged. We can see from Fig. 13 that for each pump intensity the output power increases with the increase of the factor, and reaches a maximum when the factor equals unity.
In this article, a detailed physical model is set up to describe the kinetic and the laser amplification process of a diode four-side symmetrically pumped alkali vapor laser MOPA system, which includes the amplified spontaneous emission, the saturation effect, the energy distributions of the incident pump laser and the seed laser, and new iterative process different from the single-or–double-side pumped configuration. According to the experimental parameters in literature, the simulation result was calculated and compared with the single-side pumped configuration, which used a custom cylindrical diffuse reflector to ensure the homogeneous distribution of the pump energy, good agreement indirectly verifies the validity of the model. Energy distributions in the cell and influences of some important factors including the temperature, powers of the seed laser and pump light, length of the cell and mode overlap factor were simulated and analyzed. 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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