Entropy Applied to Morphological Analysis and Modelisation of Nanomaterial Optical Properties

— The normalized configuration entropy, based on the theory of information, when applied to the image of heterogeneous media, points out a characteristic length of the disorder, `opt, at which we calculate the optical properties. The models we propose make a partition of the image between percolated and non percolated cells of size `opt, in which we calculate effective dielectric functions. Two models are then developed performing respectively a coherent and a non coherent treatment in the calculation of the optical properties of the whole medium. The coherent model gives a good account of the metallic grain resonance and of the infrared behavior of both reflectance and transmittance of granular gold films, close to the percolation threshold, domain where the effective medium theories fail. Résumé. — L’entropie de configuration normalisée, basée sur la théorie de l’information et appliquée à l’image d’un milieu hétérogène, permet de mettre en évidence une longueur caractéristique du désordre, `opt, à laquelle nous calculons les propriétés optiques. Les modèles que nous proposons effectuent une partition de l’image entre cellules de taille `opt, percolées et non percolées, dans lesquelles nous calculons une fonction diélectrique effective. Deux modèles ont ainsi été développés, réalisant respectivement un traitement cohérent et incohérent lors du calcul des propriétés optiques du milieu global. Le modèle cohérent rend bien compte de la résonance de grains métalliques et du comportement infrarouge de la réflexion et de la transmission de films d’or granulaires, aux alentours du seuil de percolation, hors du domaine de validité des théories de milieu effectif. Introduction The morphology of a medium can be analyzed as a quantity of information, as defined by Shannon [1] and Brillouin [2]. This quantity is a characteristic of the disorder of an image, related to a given statistical state of configuration, and to an entropy. We have adapted these information and entropy concepts to the image analysis of random two phase media [3, 4]. () Author for correspondence () Unité associée au CNRS D0781 c © Les Éditions de Physique 1997 550 JOURNAL DE PHYSIQUE III N◦3 This analysis leads to a length, characteristic of the image, at which we choose to calculate the optical properties of the material (Sect. 1). In Section 2, we propose two models of the optical properties, based on the same principle of the image partition [5]. The first model uses a self consistent calculation, while the second allows a local calculation of the optical properties. We finally compare the predictions of our models to the predictions of classical models (Bruggeman, Maxwell Garnett, renormalization) and to optical measurements on gold granular films prepared by thermal evaporation. 1. Normalized Configuration Entropy 1.1. Definition. — The binary image is composed of black pixels (gray level: 0), considered as active, and white pixels (gray level: 1). It is analyzed through a sliding window of variable size `. We define the relative frequency pk(`) of occurrence of a cell of size ` enclosing k black pixels in the image as: pk(`) = Nk(`) N(`) (1) where Nk(`) is the number of cells of size ` containing k black pixels, and N(`) is the total number of cells of size `. The number of possible configurations is: ` + 1. The entropy is defined as the sum over k of all information quantities of each event k weighted with their own weight pk(`): H(`) = ` ∑ k=0 pk(`) ln[pk(`)]. (2) In order to compare the different entropies calculated for various sizes of the cell of analysis, we need to normalize the entropy, by dividing it by the theoretical maximum entropy (H max). H max is determined by using the Lagrange Multiplier Method, with the single constraint: ` ∑ k=0 pk(`) = 1. We obtain H max = ln(` 2 + 1), which is the entropy corresponding to the equally probable distribution (pk = 1/(` 2 + 1) for all k values). We thus define the normalized configuration entropy: