Advances in Optical Technologies

Volume 2017 (2017), Article ID 9256053, 15 pages

https://doi.org/10.1155/2017/9256053

## Finite Element Analysis of Thermal Effects in Diode End-Pumped Solid-State Lasers

^{1}Physics Department, Damascus University, Damascus, Syria^{2}Faculty of Informatics and Communications, Arab International University, Daraa, Syria

Correspondence should be addressed to Moustafa Sayem El-Daher

Received 23 November 2016; Revised 6 February 2017; Accepted 2 March 2017; Published 10 April 2017

Academic Editor: Augusto Beléndez

Copyright © 2017 Moustafa Sayem El-Daher. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Thermal effects are the main obstacle to getting high power and good beam quality in diode end-pumped solid-state lasers. In this work, a theoretical investigation of thermal effects in single and dual end-pumped solid-state lasers is carried out using finite element analysis (FEA) for a selected number of widely used laser producing materials, namely, Nd:YAG, Yb:YAG, and Nd:KGW. Crystals with different dimensions are also investigated both in single and in dual end-pumped configuration. Finally, the effect of using composite crystals on thermal lensing is investigated. An experiment to measure the thermal focal length for two different crystals was carried out and a comparison with FEA computed focal length of the thermal lens is made. In all cases studied in this work, results show clear effects of thermal lensing with some differences depending on crystal type, pump power, and size.

#### 1. Introduction

Obtaining high power output in diode pumped solid-state (DPSS) lasers with good beam quality is limited by the thermal effects in the laser crystal which induce thermal lensing and losses due to depolarization and cracks in the laser crystal [1–3].

For these purposes, a longitudinal pumping configuration was developed which is called end-pumped solid-state lasers which give high quality output beam and efficiency in comparison to side pumped lasers. For this goal, a fiber-coupled diode laser is used to pump the laser crystal longitudinally, and the end of the fiber is coupled to microlenses to adjust the pumping beam shape to match the mode of the laser resonator to make use of all the output power of the diode laser [4]. Using fiber-coupled diode lasers provides a uniform pumping beam but the drawback is the focusing of high power on a small area of the face of the laser crystal which results in generating a large temperature gradient between the crystal center and the outside surface which gives thermal lens and stresses and birefringence [5]. In this work, we carried out FEA calculations for a selected number of laser producing crystals with different dimensions for a number of sizes and pumping parameters; we discussed the results and compared some of them to the actual experimental setup which gives us a better understanding of the numerical results and their deviation from the real measurement.

#### 2. Thermooptic Effects

The difference between the energy of pumping photons and the energy of laser’s photons in optically pumped solid lasers using diode lasers and quantum defects, which are called quantum defect heating, is the main reason for the heat generated in the crystal lattice of the lasing medium in addition to heat resulting from transfer of non-laser-emitting upper levels to ground levels [6].

The amount of heat generated inside the laser rod resulting from the absorption process of pumping light is dissipated by a heat sink on the surface of the laser rod cylindrical length; this is a condition of steady state assuming uniform internal heat generation. We can get such a setup using symmetrical pumping beam. End-pumping using a diode laser coupled with an optical fiber gives such beam distribution. Assume homogeneous cooling on the surface of the laser rod, which means the temperature at each point along the axis of the crystal is fixed and the thermal conductivity coefficient is independent of temperature [7].

Under these assumptions, we can write a differential equation for thermal transfer in a cylindrical crystal as follows [1, 6–8]: is the temperature in kelvins as a function of radial distance and position along the -axis, is the thermal conductivity (W/m·K), and is heat in unit volume.

Thermal distribution within a laser crystal is a function of absorbed power density, which in turn takes the form of distributed light pumping at any vertical section on the axis of the laser crystal and parallel to the pumping beam. On the one hand, the intensity of light pumping decreases along the -axis, subject to the law of absorption; can be given in a number of forms depending on the beam shape, for example, flat hat, top hat, Gaussian, or super Gaussian [9].

Boundary conditions are set based on the assumption that the side surface of the laser crystal is in direct contact with the heat radiator, generally made from a piece of copper in contact with a chiller. The first boundary condition is the continuity of thermal flow through these surfaces [7] and the second boundary condition is , where is the temperature set by the chiller.

The change in the refractive index can be separated into two terms: the first part depends on the temperature distribution and the second part depends on the strain [6]. So, we can writewhere is the total index of refraction, is the part of index of refraction related to temperature, is the part of index of refraction related to stress, and is the material index of refraction.

#### 3. Theoretical Calculations and Results

In this work, we carried out a finite element analysis study for three common types of solid-state laser crystals, namely, Nd:YAG (neodymium-doped yttrium aluminum garnet), Nd:Y_{3}Al_{5}O_{12}, Yb:YAG (ytterbium-doped yttrium aluminum garnet), and Nd:KGW (neodymium-doped potassium-gadolinium tungstate crystals (Nd:KGd(WO_{4})_{2}). We studied thermal and stress properties within crystals choosing different dimensions for crystals and different pump power values as shown in Table 1(a).