Method of smoothing photoluminescence spectra

Abstract

The smoothing method for experimental photoluminescence spectra, which uses Tikhonov regularization method, is proposed. Smoothing problem statement is performed as a statement of the minimization problem based on the least squares criterion. The functional to be minimized is a compound of two terms. The first one is the basic term, which determines the quality of data approximation provided by smooth function. The second one is the regularization term, which sets a constraint on the energy by smooth function derivative of a given order. A solution of the smoothing problem is obtained in the closed form analytically. The solution can be implemented by a simple array of matrix operations; one of those is an operation of matrix inversion. The necessity of matrix inversion imposes constraints on the order of derivative used in the proposed method and on noise level. It is related to the increase of either the order of derivative at the given noise level in data or the noise level at the given order of derivative, that deteriorates the conditionality of a matrix to be inverted and limits the scope of the proposed method, accordingly. Efficiency comparison of the proposed smoothing method, the Savitzky-Golay method, and the moving average method was performed by the numerical simulation of two models of photoluminescence spectra. Their advantages, disadvantages, and also the scopes of applicability are noted. The best values of operating parameters of these methods for the different noise levels in data were obtained. The conditions, under which the potential capabilities of the proposed smoothing method excel the potential possibilities of Savitzky-Golay method, were indicated. The results of smoothing the experimental photoluminescence spectra obtained for ZnO:Mn nanocrystals and ZnS:Mn crystals are provided.