Determining Upconversion Parameters and Invertibility of Nonlinear Dynamical Systems
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Determining Upconversion Parameters and Invertibility of Nonlinear Dynamical Systems Makhin Thitsa and Thanh Q. Ta Department of Electrical and Computer Engineering, Mercer University Macon, GA 31207, U.S.A. ABSTRACT Determining upconversion parameters is of high interest in laser material development. For many materials these parameters cannot be directly measured by experimental methods. These upconversion coefficients appear as unknown parameters in the laser rate equations, which are a system of coupled nonlinear differential equations that are used to model the dynamics of population densities in different energy levels. In this paper we propose the well-established system theoretic tools pertaining to the system inversion to be applied in this case. The unknown parameters can be considered as the inputs and the fluorescence signals can be considered as the outputs of the dynamical system. Therefore the determination of the unknown upconversion rates in the system equations from the output data is a classical system inversion problem. In this paper we demonstrate how to compute the unknown coefficients in the rate equations from the experimental emission data utilizing this method. INTRODUCTION In developing new laser materials, it is essential to understand the various energy transfer processes that take place in the material. One of the most important and studied energy transfer processes is the upconversion process, in which, the emission photon energy exceeds the excited photon energy by 10 - 100 kT, where k is the Boltzmann’s constant and T is the temperature in Kelvin. These processes are often referred to as energy transferred upconversions (ETUs) and they are prevalent in ion doped laser crystals such as rare earth doped glasses. While the energy transfer upconversion rates play a crucial role in the laser performance, they are in general, not directly measurable. The dynamics of energy transfer processes are usually analyzed by laser rate equations, which are a system of coupled nonlinear differential equations describing the temporal evolution of the population densities in the energy manifolds due to various interaction processes. The upconversion rates appear as the unknown parameters in the laser rate equations. Currently in the literature they are determined by searching for the set of parameters that will fit the experimentally measured fluorescence data. Specifically, the nonlinear differential equations are solved numerically for each set of guessed parameter values and the set that reasonably fits the experimental data is taken to be the actual set of parameters [1]. When no single set of parameters is found to fit the experimental data, the rate equation model is concluded to be invalid and proper modification to the model is sought [2]. In this paper we propose the wellestablished system theoretic tools pertaining to the system inversion to be applied in this case. The unknown parameters can be considered as the inputs and the fluorescence signals can be considered as the outputs of the dyna
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