1. Introduction

The task of growing active elements based on chalcogenide crystals doped with Fe²⁺ ions and developing high-power and efficient mid-IR lasers has long been the subject of intensive research [1–10]. The main application areas of Fe²⁺:ZnSe and Fe²⁺:ZnS lasers include environmental remote sensing, laser communication systems, spectroscopic studies, medicine, and generation of high-power femtosecond pulses [10]. The highest energy characteristics were achieved with single-crystal samples under cryogenic cooling of the active element. To date, the energy of such lasers exceeds 10 J for Fe²⁺:ZnSe and 3 J for Fe²⁺:ZnS lasers [9]. At room temperature, the energy and efficiency of Fe²⁺:ZnSe and Fe²⁺:ZnS lasers have for a long time been limited by the absence of high-energy laser sources in the spectral range of 2.6–3 µm, having the pulse duration shorter than 300 ns [2,3]. However, the use of nonchain chemical HF lasers for optical pumping of Fe²⁺:ZnSe and Fe²⁺:ZnS crystals significantly increased the energy and efficiency of Fe²⁺:ZnSe and Fe²⁺:ZnS lasers operating at room temperature [11–15]. At present, the technology for creating high-energy nonchain chemical HF(DF) lasers with high average power is quite mature [16–20], and these devices are involved in various applications [21–23]. The highest values of the output energy and pulse repetition rate for Fe²⁺:ZnSe at room temperature were achieved with polycrystalline Fe²⁺:ZnSe samples pumped by radiation of a nonchain chemical HF laser [24,25]. The pulse repetition rate was as high as 200 Hz [24]. In [12,25], the energy of the Fe²⁺:ZnSe laser radiation generated in the bulk of the active medium with a transverse size of ∼15 mm (pump spot diameter) and a thickness of 4 mm exceeded 1.5 J in the spectral region of 4–5 µm, with the pulse duration being ∼150 ns. Thus, the pulsed power emitted from one cubic centimeter of the Fe²⁺:ZnSe crystal medium exceeded 15 MW at room temperature [12]. At a pulse repetition rate of 20 Hz, the average power of the Fe²⁺:ZnSe laser at room temperature was 20 W in [25], and the energy yield per unit volume of the crystal (without using forced cooling systems) was ∼30 W/cm³, with the efficiency of pump energy conversion being higher than 40%. However, in the case of low-doped single crystals, a further increase in the energy of Fe²⁺:ZnSe and Fe²⁺:ZnS lasers pumped by high-power chemical HF lasers is limited by the processes of the active element damage [26,27]. Alternatively, for highly doped wide-aperture polycrystalline plates, the limitations are due to the development of parasitic lasing [28], which is similar to the situation for disk lasers [29]. In [30], to solve this problem, a technology for growing layered Fe²⁺:ZnSe structures was proposed and tested. The use of a layered doping approach facilitated a reduction in the lasing threshold for the active element at room temperature. This provided the Fe²⁺:ZnSe laser with a high differential efficiency. However, the size of the pump region and the laser output energy were not increased significantly in this case. Moreover, the optical strength of these samples [30] was considerably lower than in [12]. However, detailed studies of the effect of the doping profile on the optical strength of the ZnSe crystal have not been carried out in [30]. In fact, the creation of high-quality active elements for Fe²⁺:ZnSe and Fe²⁺:ZnS lasers, having large transverse dimensions, requires significant investments, so replacing the direct experiments with adequate simulations is a relevant task [5]. Considering high-power pulsed pumping and with laser operation at a high pulse repetition rate, it is essential to gain the information on the thermoelastic stresses and optical distortions arising in the active element of the laser [31,32]. In [33], thermal loads and optical distortions in an elongated Fe²⁺:ZnSe crystal were analyzed at quasi-continuous optical pumping, and the doping profile was assumed to be uniform and axisymmetric in the calculations. However, for scaling the characteristics of a Fe²⁺:ZnSe laser, the disk geometry of the active element is more promising [12,31].

The present study was aimed at the simulation of thermoelastic stresses and analysis of optical distortions arising in Fe²⁺:ZnSe crystals with arbitrary doping profiles under repetitively pulsed pumping. To model the thermoelastic stresses and optical density perturbations in the laser crystal, the finite element method was used [34].

2. Numerical model

Thermoelastic stresses arising upon optical pumping were simulated for an active element in the form of a ZnSe disk with various Fe²⁺ doping profiles. The diameter of the active disk element was 20 mm, with its thickness being 6 mm. The pumping conditions were assumed to be the same as in [30], i.e., the energy of a radiation pulse with λ≈3 µm (150 ns in duration) was absorbed in a layer containing Fe²⁺ ions. The calculations were performed under the assumption that the energy of the pump pulse was uniformly distributed over a circular spot with a diameter of 15 mm, centered on the optical axis. In these calculations, to estimate the effect of the doping profile on the maximum thermoelastic stresses arising in the active element, modeling was performed without taking into account the crystal transmittance dynamics, and the pump energy was assumed to be totally absorbed in the doped layer, with its heating to temperature T_max=T₀+E_abs/(m·c_m). Here Т_max is the maximum temperature established in the absorbing layer after the absorption of the pump pulse energy E_abs; m is the mass of the doped layer, c_m is the specific heat of the crystal. The intentional disregard of the thermal conductivity effect on the temperature distribution in the absorbing layer is justified by the fact that the pump pulse duration is rather short (t_las=150 ns), and the main goal of the study is to identify the role of the doping profile in the thermo-optical distortions in the active element.

In the calculations of the thermoelastic stresses, two profiles of the ZnSe crystal doping were considered: 1) iron atoms were uniformly distributed in a single layer with a thickness of 1 mm, located under the undoped ZnSe layer (i.e., same conditions as in experiments [30]); 2) four inner layers of the crystal were doped with iron atoms, with square doped areas arranged according to the mask shown in Fig. 1. In the figure, the mask for four consecutive layers with periodic doping regions is shown. The entire active element is doped using such mask; the thickness of the doped region is also 1 mm. As seen in Fig. 1 (top view), the doped regions consist of squares, with each layer having a specific color. The doped squares with side L are separated by undoped areas (these are uncolored areas). Linear size L is taken based on the condition L ≤ d_p/$\sqrt 2$.

 

Fig. 1. Schematic representation of the mask used for doping with Fe²⁺ in each of four layers in the active element of the ZnSe:Fe laser.

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Here d_p is the maximum possible size (critical size) of the pump spot at a given radiation density, which does not lead to the development of parasitic lasing.

Thus, with doping method 2, the areas absorbing the pump radiation are divided in each layer into several regions; between these regions, the crystal is undoped. When the four layers shown in Fig. 1 are placed one under the other in the direction of the optical axis, the formed absorption region appears to be the same as in the case of the uniform doping in a single layer. However, in the active element doped by method 2, the critical size of the irradiation spot can be increased, with the pump density being the same, since the width of the doping region is reduced in each layer and the total doping area in the entire element is the same due to the spatial separation of the doping regions. A simple theoretical estimate shows that in the case of four doped layers, the energy of the Fe²⁺:ZnSe laser can be increased by a factor of 2. Figure 2 and Fig. 3 schematically show the structure and arrangement of the absorbing regions for a quarter of a disk element doped by methods 1 and 2, respectively. The following pumping conditions are simulated here. The pump spot is a circle with a diameter of 15 mm, the thickness of the heated region is 1 mm, and the size of the pieces is chosen as L=2 mm. All material constants necessary for the numerical simulations are taken from [5,35]; the mechanical and thermophysical properties of Fe²⁺:ZnSe and ZnSe are assumed to be identical. Thus, for the Fe²⁺:ZnSe crystal, we take specific density ρ=5.27 g·cm⁻³; elastic limit σ_t=55.1 MPa; Young’s modulus Е=67.2 GPa; Poisson’s ratio ν=0.28, coefficient of thermal expansion α=7.8·10⁻⁶ K⁻¹, and specific heat с_m=0.339 J·g⁻¹·K⁻¹.

 

Fig. 2. a) schematic representation of the ZnSe crystal region doped with Fe atoms by method 1; b) nodes of finite elements, the pumped area is colored.

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Fig. 3. a) schematic representation of the ZnSe crystal region doped with Fe atoms by method 2; b) nodes of finite elements, the pumped area is colored.

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To define the stress-strain state in the three-dimensional model, Hooke’s law for an elastic medium is used:

When building the model, we use three-dimensional prismatic 20-node isoparametric finite elements with 27 integration points in the Gaussian quadrature, shown in Fig. 4.

 

Fig. 4. Schematic representation of a 20-node finite element used in the numerical simulations.

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In the finite element method, displacements $\{\textrm{f} \}= {[\textrm{N} ]_\textrm{e}}{\{\mathrm{\delta} \}^\textrm{e}}$ in the finite element region are calculated using the shape functions ${[\textrm{N} ]_\textrm{e}} = [{{\textrm{N}_1}({\textrm{x},\textrm{y},\textrm{z}} ),{\textrm{N}_2}({\textrm{x},\textrm{y},\textrm{z}} ),{\textrm{N}_3}({\textrm{x},\textrm{y},\textrm{z}} ), \ldots ,\; {\textrm{N}_{20}}({\textrm{x},\textrm{y},\textrm{z}} )} ]$ and node displacements ${\{\mathrm{\delta} \}^\textrm{e}}. $ Here the relationship between deformations and node displacements is given as

The problem is solved in the region of elastic strains where the Mises condition ${\mathrm{\sigma}_\textrm{i}} < {\mathrm{\sigma}_\textrm{T}}$ [34] is valid. Here ${\mathrm{\sigma}_\textrm{T}}$ is the elastic limit of the material, and ${\mathrm{\sigma}_\textrm{i}} = \sqrt {[{{{({{\mathrm{\sigma}_\textrm{x}} - {\mathrm{\sigma}_\textrm{y}}} )}^2} + {{({{\mathrm{\sigma}_\textrm{y}} - {\mathrm{\sigma}_\textrm{z}}} )}^2} + {{({{\mathrm{\sigma}_\textrm{z}} - {\mathrm{\sigma}_\textrm{x}}} )}^2}} ]/2 + 3[{{\mathrm{\tau}_{\textrm{xy}}}^2 + {\mathrm{\tau}_{\textrm{yz}}}^2 + {\mathrm{\tau}_{\textrm{zx}}}^2} ]} $ are the stress intensities.

3. Simulation results and related discussion

The simulations for the pulsed pump mode showed that the highest distortions occur when the product of the pump power and the absorption coefficient is maximum. The highest stresses arise at the boundaries of the hottest and coldest regions in the material. In experiments [30], the crystal damage was observed precisely at these boundaries. For crystals with different doping profiles, Figs. 5 and 6 show the calculated values of the intensity of thermoelastic stresses (specified for the indicated range of values) arising in the crystal upon absorption of a pump pulse. It is seen in these figures that the maximum levels of thermoelastic stresses arising at the boundaries of hot and cold regions turn out to be close for the two doping methods. However, the total volume of the highly stressed material appears to be significantly smaller in the case of doping method 2. This is due to the geometry of the stressed region. In the case of the doping method 2, regions with an unperturbed material are close to the heated ones.

 

Fig. 5. Intensity of thermoelastic stresses (σ_i) arising after absorption of a pump pulse in the crystal doped by method 1 (regions corresponding to the indicated range of values are shown).

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Fig. 6. Intensity of thermoelastic stresses (σ_i) arising after absorption of a pump pulse in the crystal doped by method 2 (regions corresponding to the indicated range of values are shown).

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Therefore, internal stresses can be relieved not only along the optical axis but also in the transverse direction. It should be noted that the heat transfer is intentionally disregarded here. If we take into account the heat transfer between the heated regions, the advantages of doping method 2 will become even more obvious since, in this case, the heated regions cool down in all three directions, whereas in the case of doping method 1, the heat flow from the central region is one-dimensional and is directed along the optical axis. Figure 7 shows the profiles of linear displacements for the boundary of the crystal upper surface, which are due to bulk thermoelastic deformations arising in the crystals doped by method 1 and method 2. The surfaces are constructed such that the coordinates of the smallest calculated deformations match. It is seen in this figure that the deformations, and, consequently, the optical distortions are several times higher in the case of doping by method 1 than the corresponding values for the active elements with a piecewise discontinuous doping profile. This is due to the fact that with the piecewise discontinuous doping profile, as mentioned above, excessive material stresses are relieved at the free boundaries. In contrast, in the case of the uniform doping, the stresses are localized on the optical axis, and a strong optical lens is formed there.

 

Fig. 7. Linear displacements of the boundary of the crystal upper surface due to bulk thermoelastic deformations. The upper surface corresponds to the deformation of the active element doped uniformly in one layer; the bottom surface is obtained for the piecewise discontinuous doping profile.

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Figure 8 shows the dependences of the maximum crystal deformation in the vertical direction on the heating temperature of the doped regions of the crystal, calculated for the case of uniform doping and for the sample with nonuniform doping. It is seen that when a crystal with uniform doping is heated, substantially higher crystal deformations arise in the entire investigated temperature range compared to the case of the piecewise discontinuous doping profile. Naturally, large optical distortions in the laser crystal lead to a significant deterioration in the quality of the laser beam. Therefore, the piecewise discontinuous doping profile is preferable.

 

Fig. 8. Dependences of the maximum crystal deformation δ_m in the vertical direction on the heating temperature of the doped regions of the crystal ΔT: curve 1 – for a layer with uniform doping; curve 2 – for the sample doped by method 2.

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In [33], to reduce the thermal lens effect in the active element of the Fe²⁺:ZnSe laser during crystal pumping, it was proposed to use an axially symmetric doping profile with a minimum on the optical axis as well as low concentrations of iron ions (∼10¹⁶ cm⁻³) with a uniform distribution of Fe²⁺ over the crystal length. With this approach, active elements should be made from bulk Fe²⁺:ZnSe single crystals doped with iron during their preparation [2]. However, this technology [2,9] is quite complicated, expensive, and at present, it provides homogeneous crystals with transverse dimensions no larger than 6 cm. Also, under the conditions of repetitively pulsed pumping, bulk crystals are much more difficult to cool than thin disks grown by the technology described in [4–6]. When the active element is rather long in the direction of the optical axis, crystal damage due to self-focusing effects may occur [27]. In the case of high-power repetitively pulsed optical pumping, the disk geometry of the active element has distinct advantages and is more promising [29]. Therefore, the combination of technologies [4,30] with the doping performed according to the piecewise discontinuous profile described in the present study appears to be favorable. We have considered only four layers in this paper; however, their number can be 4k (k = 1, 2, …) with decreasing the transverse size of the elementary doping region by k times. This approach potentially allows the production of rod-type polycrystalline active elements free from imperfections, which are specific to the elements made from single crystals [27,33]. It should be noted that the calculations were performed for ZnSe, and heating was assumed to be due to the absorption of pump radiation in the region doped with Fe²⁺ ions. Therefore, qualitatively the same results will be obtained in the calculations for the active elements of Fe²⁺:ZnS, Cr²⁺:ZnS, and Cr²⁺:ZnSe lasers [36].

In conclusion, let us make a point about how Fe²⁺:ZnSe laser crystals with an inhomogeneous doping profile can be fabricated. In fact, if laser ceramics are used, the recipe will be quite obvious. In the case of polycrystalline laser elements, the algorithm can be as follows. A protective neutral coating layer is applied to a prepared layer of the CVD-grown ZnSe crystal. Then, square areas are etched according to the mask shown in Fig. 1, and after that iron atoms are deposited in these areas. Next, the coating is removed and the crystal is annealed to allow the diffusion of iron atoms into ZnSe; to accelerate this process, the methods described in [8] can be applied. After that, the surface is polished and a layer of pure ZnSe is grown, as described in [37]. Then, the process is repeated with a shift of the mask for doping the second layer and so on up to the fourth layer. The technology for preparing a doped layer under undoped ZnSe was first used to create a Cr²⁺:ZnSe laser [37] and is now available for a Fe²⁺:ZnSe laser [30].

4. Conclusion

Based on the developed mathematical model involving the finite element method [34], qualitative and quantitative analyses of the effect of the doping method on the behavior of thermoelastic stresses and deformations arising during pulsed pumping were carried out for the active element of the Fe²⁺:ZnSe laser. It was found that with the same energy absorbed in the crystal, selection of a special (nonuniform, both along the optical axis and in the transverse direction) doping profile of the ZnSe crystal provides a means to significantly reduce the optical density distortions and also suppress the development of parasitic lasing due to reduction in the effective gain length in each doped crystal layer. Thus, the numerical simulations show that the method for doping the active element of a Fe²⁺:ZnSe laser proposed in the study can significantly increase the energy characteristics of Fe²⁺:ZnSe lasers and improve the quality of their radiation. In fact, crystals with a nonuniform doping profile can also be used as wide-aperture saturable absorbers and in tandem with various solid-state lasers [38,39]. However, inhomogeneous doping would be of maximum value when creating high-energy Fe²⁺:ZnSe lasers with energies above 10 J and pulse power exceeding 1 MW. For example, in the case when high-energy HF lasers are used for pumping the crystal [40,41]. The proposed doping method is expected to be helpful when creating high-power disk laser systems, which suffer from an acute problem consisting in the development of parasitic lasing in the direction transverse to the optical axis [29,42].