Characterization of Yb:YAG active slab media based on a layered structure with different doping A.Lapucci*a, M.Ciofini a, L.Espositob, P.Ferrarac,L.A.Gizzic, J.Hostašab,e, L.Labatec, A.Pirrid, G.Tocid, M.Vanninid a C.N.R. - National Research Council, Istituto Nazionale di Ottica (CNR-INO), Largo Enrico Fermi 6, I-50125 Firenze, Italy; b C.N.R. - National Research Council, Institute of Science and Technology for Ceramics (CNR-ISTEC), Via Granarolo 64, 48018 Faenza, Italy; c C.N.R. - National Research Council, Istituto Nazionale di Ottica (CNR-INO), Via G. Moruzzi, 1 - I-56124 Pisa, Italy; d C.N.R. - National Research Council, Institute of Applied Physics "Carrara" (CNR-IFAC), Via Madonna del Piano 10, I-50019 Sesto Fiorentino (FI) Italy; e Department of Glass and Ceramics, ICT Prague, Technická 5, 166 28 Prague, Czech Republic ABSTRACT Slabs with non-uniform doping distribution are studied with the aim of reducing thermal deformations in high-energy high-average-power Yb:YAG slab systems. We present a numerical analysis based on Finite Element Mesh (FEM) methods suitable to model non-uniform devices. The thermal variation of the refractive index, the end-faces deformations and the photo-elastic effect have been calculated in order to estimate the total thermal-lens effect. The stress distributions are also obtained. Some results of this numerical approach are compared to experimental thermal lens measurements in a simple geometry for both uniform and structured samples, in order to validate the numerical procedures. Finally we compare numerical simulations for different single- or double-sided pumping and cooling geometries. They show that structured slabs can reduce thermal gradients with respect to uniformly doped means with comparable absorption and geometry. This means a reduction of thermal lens effect and thus an increase of maximum allowed pump power loading. Previous literature reports some work made with structured slabs where higher doping was located in layers with lower pump radiation levels, in order to get a more uniform absorption. Interestingly our modeling indicates that reduced thermal effects are instead obtained when a higher doping is located close to the cooled surfaces. Keywords: Yb:YAG Lasers, High Energy Lasers, Slab lasers, Ceramic lasers 1. INTRODUCTION Ytterbium doped YAG materials are the chief candidates for the attainment of High Energy Diode Pumped Solid State Laser (DPSSL) systems working in high rep. rate pulsed regimes [1]. YAG ceramics have already proven their advantages over single crystals for high average power systems, both in terms of higher fracture limit levels and in terms of ease of production. The ceramic sintering process has also the advantage of allowing the production of “structured active media”, namely samples consisting of several volumes with different doping levels. Recently structured slabs have been obtained in YAG ceramics shaped from a stack of differently doped layers [2-4]. In this work the ceramic material is prepared by stacking differently Yb-doped layers through cold isostatic pressing. Calcination, reaction sintering, annealing and polishing processes complete the slab production.The details of the production process have been reported elsewhere [5]. The optimization of the doping distribution is strongly related to the pumping and cooling geometry of the active medium and a complex numerical analysis is needed to assess the advantages or disadvantages of a specific design in terms of thermal deformation and fracture limit of the active medium. In many classical cases, such as those of uniform cylindrical rods or uniform disk lasers, symmetry considerations help in simplifying the numerical approach. *[email protected]; phone +39.055.2308226; fax +39.055.2337755 High-Power, High-Energy, and High-Intensity Laser Technology; and Research Using Extreme Light: Entering New Frontiers with Petawatt-Class Lasers, edited by Joachim Hein, Georg Korn, Luis Oliveira Silva, Proc. of SPIE Vol. 8780, 87800J Proc. of SPIE Vol. 8780 87800J-1 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 04/01/2014 Terms of Use: http://spiedl.org/terms In this paper we present a thermo-mechanical modeling based on Finite Element Mesh (FEM) methods suitable to model non-uniform devices. We firstly show some results of this approach applied to a simple geometry that we can test experimentally. The measurements of thermal lens effective focal lengths (e.f.l.s) are reported for both uniform and structured samples in order to validate the numerical approach. Finally we compare numerical simulations for different single- or double-sided pumping and cooling geometries. They show that structured slabs can reduce thermal gradients with respect to the uniformly doped means with comparable absorption and geometry. This means a reduction of thermal lens effect and thus an increase of maximum allowed pump power loading. Previous literature reports structured slabs usually with higher doping located in layers where a lower pump radiation level is present, in order to get a more uniform power absorption [2,3]. Interestingly our modeling indicates that reduced thermal effects are instead obtained locating higher doping levels close to the cooled surfaces. 2. NUMERICAL F.E.M. ANALYSIS 2.1 Three dimensional numerical simulation approach Several papers on solid state lasers dealt with the thermal analysis in the past [6-8]. They all end with the conclusion that the optical path difference at different transverse positions is related to three contributions. The refractive index variations with temperature, the surface bending due to thermal expansion and the stress induced birefringence due to the photo-elastic effect. ⎛ d dn ⎞ ⎛ 3 d ∂n ⎞ OPD(r) = ⎜ ⋅ΔT (r, z)dz⎟ + n ⋅Δu(r) + ⎜ ⋅ε (r)dz⎟ (1) ⎜ ∫ dT ⎟ 0 ⎜ ∑∫ ∂ε ij ⎟ ⎝ 0 ⎠ ⎝ i, j=1 0 ij ⎠ Where d is the sample thickness , n0 is the refractive index without deformations, Δu(r) is the surface deformation at radial position r, and εij is the strain tensor. In what follows z is the direction parallel to the cylindrical axis and to the “optical rays propagation” . Variables (x,y) or (r,Φ) describe a cartesian or polar coordinate system in the planes transverse to propagation. Our approach is based on the adoption of a commercial F.E.M. (Finite Element Mesh Calculation Method) software package [9] to obtain the thermal and mechanical behavior of our structured samples, given the heat loading distribution and thermal boundary conditions. The software package output consists in a scalar temperature variation field ΔT=T- TREF , the geometrical displacements of each node of the mesh and, finally, the 6 components of the stress tensor (σ i and τ ij ). Strains can be easily calculated from the stress and temperatures fields using the conventional linear elasticity equations: 1 ⎧ ⎡⎤ εσνσσαrr=−++−()φ z()TT REF ⎪ E ⎣⎦ ⎪ ⎪ 1 . ⎡⎤ ⎨εσνσσαφφ=−++−⎣⎦()()r zTT REF ⎪ E 1 ⎪ ⎡⎤ ⎪εσνσσαz =−++−z()φ r()TT REF ⎩ E ⎣⎦ . (2) ⎧+⎛⎞1 ν γ rrrφ ==22⎜⎟τεφφ ⎪ E ⎪ ⎝⎠ ⎪ ⎛⎞1+ν ⎨γ rz==22⎜⎟τε rz rz ⎪ ⎝⎠E ⎪ ⎛⎞1+ν ⎪γ φz ==22⎜⎟τεφφzz ⎩ ⎝⎠E Where, considering our host material (YAG), we have that E=2.8 1011 Pa is the Young modulus, ν = 0.28 the Poisson -6 -1 coefficient , α = 7.9 10 °C the thermal expansion coefficient and TREF = 20°C the reference temperature, that is the temperature of the cooling system. Proc. of SPIE Vol. 8780 87800J-2 Downloaded From: http://proceedings.spiedigitallibrary.org/ on 04/01/2014 Terms of Use: http://spiedl.org/terms The first two contributions to the OPD are directly calculated from the output of the thermo-mechanical software package. To determine the third contribution we need the relation between the refractive index (for the different polarization directions) and the strain tensor. This relation is in general expressed by a variation of the dielectric nd impermeability 2 rank tensor Bij [7] and is given by the set of equations: ΔBij = pijkl εkl (3) 3 n0 * Δni = − ⋅ ΔB (4) 2 ii th Where pijkl is the 4 rank elasto-optical tensor, ni are the refractive indices along the principal axes of the stress-perturbed * indicatrix [6] and ΔBij is the diagonalized dielectric impermeability variation tensor. Fortunately equation (3) reduces to a much simpler formulation in case of a cubic crystal with [111] orientation [7]. Moreover Khazanov [8] demonstrated that the YAG ceramic behavior is similar to that of the [111] oriented single crystal, given the averaging effect of the different grains orientations. This fact was also confirmed by experimental birefringence measurements in Nd:YAG ceramics [10]. Under these conditions the elasto-optical tensor simplifies to the form: pp11εεε 1++ 12() 2 3 p 44 ε 6 p 44 ε 5 (5) Δ=Bx()', yz ', ' p44εεεεε 6 p 11 2 ++ p 12() 1 3 p 44 4 pppp44ε 5 44εεεε 4 11 3++ 12() 1 2 with p11= - 0.0290 , p12= 0.0091 , p44= - 0.0615. Now it is important to note that previous literature has always looked for an analytical solution for the strains and thus it has mainly treated “rod like” cases [6] (also referred to as “plain strain approximation” [7]) or “disk like” cases [11] (“plain stress approximation”) in which a two-dimensional approach can be adopted either because of the reduced thickness or because of a translationally invariant cylindrical symmetry. In order to maintain the features of our structured samples (see Fig.1), translational invariance along z cannot be claimed and a complete 3D approach is to be adopted. We thus performed the B-matrix elements calculation from the output of our software package, on two selected “ray paths” (parallel to the z axis), one on the axis of the cylindrical pumped volume and one on the lateral surface of that same volume. Transverse refractive indices nr(k) and nΦ(k) (with respect to the z-propagating waves) have been obtained through a diagonalization of the complete B matrix on each node (k) of the model mesh belonging to the two selected paths.