Effects of pumping profiles on the temperature distributions in double- end pumped solid-state lasers: A comparison study

A R T I C L E I N F O Handling editor: Ivan A. Hashim


H I G H L I G H T S A B S T R A C T
 Understanding the thermal behavior of dual end-pumped solid-state lasers.  The exponent of the beam profile has a significant effect on the temperature distributions.  Monitoring the thermal behavior keeps the performance of the lasers before being subject to damage.  A theoretical basis for laser cavity design of dual-end-pumped cylindrical laser rods was provided.
In this work, the effects of pumping profiles on the temperature distributions in solid-state lasers pumped by laser diode from two different end faces were studied. With different pump spot radii of 0.4 0.6 , various exponent factors were examined to evaluate the behavior of temperature distributions due to Gaussian and super-Gaussian profiles. For a Gaussian pumping (n=2), the maximum temperature difference on each rod face was calculated to be 295.5 K and 226 K, respectively, at a total pump power of 92.8 W (46.4 W per face). While at the same pumping power, the temperature decreased as the n exponent increased. A good match was obtained between the proposed model and the previous works listed in the literature.

Introduction
Understanding the induced temperature distribution over the laser active medium is important. This is due to the need for a tiny focused pump beam volume in end-pumped solid-state lasers, resulting in relatively higher pump deposition density and thus a high density of thermal loading. Such a result may lead to a regression in both beam quality and laser performance [1].
Several previous works studied the thermal model and thermal effects of solid-state lasers in different geometries [2][3][4][5][6][7][8][9][10][11]. Full temperature distribution analytical evaluations in end-pumped solid-state lasers under all profiles types were presented in Ref. [12]. A numerical investigation of thermal effects in single and double end-pumped solid-state lasers was achieved using the finite element method (FEM) for many commonly used lasers media crystals [13][14]. The FEM incorporated in LASCAD software was also used to evaluate thermal focal length and temperature distribution in a CW Nd: YAG laser rod for a double end-pumped laser media under Gaussian and super-Gaussian pumping beam profiles [15]. By taking into account thermal conductivity as a function of temperature, the analytical thermal model for the double-end laser diode pumped cylindrical laser rod was derived by solving the heat equation. Also, the temperature distribution and the thermal stresses were estimated under top-hat and Gaussian pump beam profiles by Refs [16][17][18]. With complicated boundary conditions utilizing COMSOL Multiphysics simulation software, the thermal model of a slab crystal was investigated. A new theoretical method to slab laser crystal (Tm: YLF) was derived for the first time by Ref. [19]. 344 In this work, an analytical steady-state-thermal model was derived to evaluate temperature distributions in cylindrical shape solid-state lasers in dual-end-pumped geometry with Gaussian and Super-Gaussian pump beam profiles. The derived model provides a theoretical basis for laser cavity design of dual-end-pumped cylindrical laser rods.

Thermal modeling
The proposed 2D configuration of double-end pumping Nd: YAG laser rod is shown in Fig. 1. With an unequal pumping radius focused on the center of the laser rod from two different pumping sources, the heat was solved analytically using Kirchhoff's transformation method. . , , 0 1 Where Q(r,z) denotes the heat source density, this is proportional to pump power density. The temperature-dependence thermal conductivity is denoted by K(T), which is provided by [12]: 2 where m, K0, and T0 are the best-fit to measured data provided by [14]. For a Super-Gaussian double-end pumping, Q(r,z) can be presented as: where W1,2 are 1/e 2 radii intensity profiles of the two pump beams, L is the laser's rod length, α is the pump coefficient of the absorption, and Q1,2 are normalization constants that can be calculated using the following relation: Therefore, the two temperature solutions for eq. (7) in two regions (i.e. the pumped region (0 ≤ r ≤ w) and the unpumped region (w < r ≤ b)) are: At the unpumped region where (w ≤ r ≤ b)) the temperature given by:

Results and Discussion
To assess our proposed model, we make a comparison with an earlier analytical work presented by Refs. [16][17]. The heat loading and the radial temperature distribution had been modeled using Nd: YAG as an active medium. Table 1 lists the parameters represented in this analysis.
Under double-end Super-Gaussian pumping profile with different pump spot radii, Figure 2 shows the influence of n th order on the behavior of the generated heat load. It can be seen that as the n th exponent increases, Q(r,z) is decreased on both rod end faces, and the shape of the pump profile becomes more flattened. It was also observed that the heat deposition is increased as the spot radius decreased, as indicated in Figure 2 (a and b). Under the Gaussian pump profile where n=2 and for the same laser parameters, the obtained results perfectly match a previous work presented by Ref. [17]. Figure 3 (a and b) depict the temperature distribution between the rod center and the rod surface. With different pump spot radii . 0.4, 0.6 . For the same laser parameters listed by Ref. [17], the maximum temperature of 295.5 K and 225.9 K were obtained on both rod end faces. For n much greater than two, which tend to a top-hat profile, the temperatures gradient between the center and surface of the laser rod were shown in figures 3c,d. The obtained results are perfectly matched for the same parameters listed by Ref. [16]. The various effects of the beam spot radius and the exponent factor on the steady-state axial temperature distribution of the gain medium are shown in Figure 4 (a-d). It was observed that along the rod axis, the temperature decreased exponentially due to the incident power's exponentially absorbed.
Also, Figure 5a-e shows the contour mapping of the temperature distribution on each face of the super-Gaussian pumped region for different n's factors, namely (2,8,16, and 32) respectively, which indicate that the maximum temperature in laser media is reduced as the n th exponent increased.

Conclusions
In conclusion, the effects of pumping profiles on the thermal effects in double-end-pumped solid-state lasers were modeled analytically using Kirchhoff's integral transform method. By taking into account different pump radii focused on the end faces of the laser rod with various n th exponent factors, the behavior of the temperature distributions was obtained, and a good matching was obtained with previous works listed in the literature. Furthermore, the results show that as the exponent of the Gaussian pumping beam profile increased, the maximum temperature difference in the laser rod decreased exponentially since the beam became flatter than the Gaussian profile.

Author contribution
All authors contributed equally to this work.

Funding
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability statement
The data that support the findings of this study are available on request from the corresponding author.