Anomalous behaviour of light reflection in crystals with different homogeneity Michele Bellingeri a , Francesco Scotognella b,⇑ a Dipartimento di Scienze Ambientali, Università di Parma, Parco Area delle Scienze, 33/A 43100 Parma, Italy b Dipartimento di Fisica, Politecnico di Milano, piazza Leonardo da Vinci 32, 20133 Milano, Italy article info Article history: Received 4 December 2010 Received in revised form 17 February 2011 Accepted 21 February 2011 Available online 17 March 2011 Keywords: Photonic crystals Shannon–Wiener index Structure-property relationship abstract The light reflection as a function of the sample length has been studied for an ideal two-dimensional pho- tonic crystal and for a two-dimensional photonic structure with smaller homogeneity with respect to the photonic crystal. We have found that, although the number of the scattering elements is constant for the two structures, the behaviour of the light reflection increases linearly with the sample length in the less homogeneous photonic structure, while it is strongly sub-linear in the photonic crystals. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction In the last two decades great attention has been devoted to the study of the light transmission in photonic structures. In such structures, for a certain range of energies and certain wave vectors, light is not allowed to propagate through the medium [1–3]. This behaviour is very similar to the one of electrons in a semiconduc- tor, where energy gaps arise owing to the periodic crystal potential at the atomic scale. Photonic structures can possess a periodical modulation of the dielectric constant. These structures are called photonic crystals and they are present in nature or can be fabri- cated through a wide range of techniques, with the dielectric peri- odicity in one, two and three dimensions [4–7]. Several efficient mathematical methods can predict the light transmission in pho- tonic crystals [8–11] and these instruments can be useful for differ- ent applications, such as the fabrication of distributed feedback lasers [12]. Instead, these calculations become very cumbersome for aperiodic and random structures. Recently, concepts and meth- ods widely used in statistics have been successfully applied to ex- plain light transport phenomena in materials where the local density of scattering elements is position-dependent [13–15]. To efficiently predict the optical properties of such complicated sys- tems, also as a function of the sample length, the implementation of simple and not time consuming methods can be very useful. In this work, we have studied the light reflection as a function of the sample length in a non-trivial engineered two-dimensional photonic structure. This structure is less homogeneous with re- spect to a perfectly ordered structure, i.e. a 2D photonic crystal [16]. We have observed that the less homogeneous structure shows a linear behaviour of the average reflection, over a wide range of wavelengths, as a function of the sample length, while an ordered photonic crystal, with the same number of scattering centres, shows a strong sub-linear behaviour. 2. Outline of the method For this study, we first consider an ideal two-dimensional pho- tonic crystal [3]. This photonic structure is a square lattice of dielectric circular pillars, where the pillars have a diameter d of 75 nm and are made of Titanium dioxide. The lattice constant a of the crystal is 300 nm and the matrix where the pillars are embedded is Silicon dioxide. The refractive indexes of TiO 2 and SiO 2 are n T = 2.45 and n S = 1.46, respectively. Note that, for such a geometrical setting n T d  n S (a–d) is satisfied [3]. We consider 12  12 cell photonic crystal, to have a size of 3.6  3.6 lm (Fig. 1, crystal PC1). In order to analyse the light transmission as a function of the sample length, we have built the structures de- picted in Fig. 1, where PC2 is PC1 repeated two times and PC3 is PC1 repeated three times. We have realised structures from PC1 up to PC7, where PC7 is PC1 repeated sevenfold. The other photonic structure we have used for this study is al- ready reported in Ref [16]. Briefly, to design this crystal, we have assigned pillars in cells by a fitness model [16,17]. We have used this model in order to realise a crystal space with skewed clusters size without benchmark distribution. Thus, we have obtained a random crystal in which the clusters size distribution (i.e. pillars for cells distribution) is skewed. The whole structure has the same size 3.6  3.6 lm of PC1 and is depicted in Fig. 2 (R1 diagram). Also for this structure, R2 is the structure R1 repeated two times, R3 is 0925-3467/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.optmat.2011.02.026 ⇑ Corresponding author. E-mail address: francesco.scotognella@polimi.it (F. Scotognella). Optical Materials 33 (2011) 1258–1261 Contents lists available at ScienceDirect Optical Materials journal homepage: www.elsevier.com/locate/optmat