Transmission of Light in Crystals with different homogeneity: Using Shannon Index in Photonic Media

Light transmission in inhomogeneous photonic media is strongly influenced by the distribution of the diffractive elements in the medium. Here it is shown theoretically that, in a pillar photonic crystal structure, light transmission and homogeneity of the pillar distribution are correlated by a simple linear law once the grade of homogeneity of the photonic structure is measured by the Shannon index, widely employed in statistics, ecology and information entropy. The statistical analysis shows that the transmission of light in such media depends linearly from their homogeneity: the more is homogeneous the structure, the more is the light transmitted. With the found linear relationship it is possible to predict the transmission of light in random photonic structures. The result can be useful for the study of electron transport in solids, since the similarity with light in photonic media, but also for the engineering of scattering layers for the entrapping of light to be coupled with photovoltaic devices.


Introduction
Propagation of electromagnetic waves in complex and structured photonic media is a topic of major importance for the comprehension of some general properties of transport phenomena and for a better understanding of the transport of electronic in solids owing to well-known analogies between electronic and photonic transport. [1][2][3] Complex dielectric structures show variations of the refractive index on a length scale comparable to the wavelength of light. In ordered structures, namely in photonic crystals, for a certain range of energies and certain wave vectors, light is not allowed to propagate through the medium [4][5][6]. This behaviour is very similar to the one of electrons in a semiconductor, where energy gaps arise owing to the periodic crystal potential at the atomic scale. Photonic crystals are present in nature or can be fabricated through a wide range of techniques, with the dielectric periodicity in one, two and three dimensions [7][8][9]. Nowadays, these materials are extensively studied since they find application in several fields, including photonics for low threshold laser action, high bending angle waveguide, super-prism effect, sensors and optical switches. [10][11][12][13][14][15] The optical properties of photonic crystals, as for example the transmission of light, can be predicted by several efficient mathematical methods [6,[16][17][18][19], but for less homogeneous structures these calculations become very cumbersome. For a better comprehension of the optical properties of such complicated systems, simple methods that are not time consuming can be very useful. Recently, concepts and methods widely used in statistics have been successfully applied to explain light transport phenomena in Lévy glasses [20].
In this work, we have analyzed the light transmission properties of two-dimensional photonic media, studied by the use of a finite element method, and we have demonstrated a simple scaling law between transmission of light over a wide range of wavelengths and the distribution of the diffractive elements in the photonic lattice, the grade of homogeneity of the structure being quantified by the Shannon index, commonly employed in statistics and information theory [20]. In particular, we have shown that the transmission of light increases linearly by increasing the Shannon index, i.e. by increasing the homogeneity of the distribution of pillars in the crystals.

Outline of the Method
We consider a well-known structure of two-dimensional photonic crystal: a square lattice of dielectric circular pillars [6]. The diameter of the column d is set to 75 nm and the lattice constant a is 300 nm (Fig. 1), the pillars are made of Titanium dioxide (n T = 2.45) and the matrix where the pillars are embedded is Silicon dioxide (n S = 1.46). Note that, for such a geometrical setting n T d ~ n S (a-d) is satisfied [6]. Starting from this regular structure, keeping constant the number of pillars throughout the crystals, a number of less regular crystals have been synthesized by concentrating more and more pillars in certain unit cells of the original crystal. In this way, some unit cells of the crystal have no pillars. Figure 2 shows a few representative realizations of the different crystals configurations. For clarity, the different crystals are named corresponding to the number of pillars per cell (1, 2, 3, …, 16). The pillar distributions in the different photonic structures are set up in order to have a selected homogeneity, that we describe herein with the so called Shannon index. In several fields of science, the diversity is correlated to a Shannon index [21,22]. The Shannon-Wiener H' index is a diversity index used in statistics, defined as (1) where p j is the proportion of the j-fold species and s is the number of the species. This index is widely used in statistics and ecology as an evenness measurement and in physics in the field of information theory [21][22][23][24]. Dividing H' by log(s) we have normalized the index constraining it within the range (0,1). We used the normalized Shannon index (i.e. 0 ≤ H' ≤ 1) as a measurement of the homogeneity of the transmission medium: in this study p j indicates the proportion of pillars belonging to the j-fold cell and s the total number of cells. In an analytical way, H' is the highest when the medium has perfect uniform structure and is the lowest when the medium is completely scalar (in our case the most aggregated it is when all the pillars are in one cell, a limit structure that we have not used in our experiment since it is not possible, in our two dimensional model, to put more than sixteen pillars in a cell). To set up the crystals, we consider a 12x12 photonic crystal cells and we start getting one pillar inside each cell to build the perfect uniform structure: a regular lattice of Titanium dioxide, i.e. crystal 1 in Figure 2. This first structure is the square lattice of dielectric pillars [6], the crystal with the maximum evenness [22] in which H' is equal to 1. At second step, we build a less uniform crystal by aggregating two pillars in each cell, thus a certain number of cells becomes empty.
Crystal 2 in Fig. 2 gives a possible realization as an illustrative example. At third step we concentrate three pillars in each cell leaving an increasing number of cells without pillars. We continue the concentration of diffractive elements with this method for the following crystals until to reach the most aggregated structure we can make, by taking into account the maximum number of pillars which can be contained in a cell respecting the optical distance between pillars (in our case 16). Thus, the aggregation method makes crystals in which all no empty cells have the same number of pillars, i.e. in Crystal 1 every no empty cell has one pillar, in Crystal 2 every cell has 2 pillars, in Crystal 3 every no empty cell has 3 pillars, etc... For each crystal n, ten different realizations have been considered by allocating in a random fashion the (12×12)/n clusters among the available 12×12 cells of the original crystal. In this way, the ten realizations of each crystal n have the same Shannon index. As an example, the different distributions for Crystal 2 are depicted in Figure S1 in the Supporting Information.
Since for a crystal where the pillars aggregate in clusters in some unit cells (and the other unit cells become consequently empty) the Shannon index decreases, we have obtained a set of crystals, from the most uniform to the most aggregated, in inverse proportion to H'.
For the calculation of the light transmission of the photonic structures through the finite element method, we assumed a TM-polarized field and used the scalar equation for the transverse electric field component E Z (2) where n is the refractive index distribution and k 0 is the free space wave number [6,25]. As input field, a plane wave with wave vector k directed along the x axis has been assumed. Scattering boundary conditions in the y direction has been used.

Results and Discussion
We have calculated, by the finite element method [6,25], the transmission of light in a broad range of wavelength (450 -1400 nm) for different crystals, starting from a perfect uniform one, in which the evenness and the Shannon index is at the maximum, to the most aggregated structures.
For each crystal the Shannon index, calculated according to Eq. 1, is reported in Table 1    For all the crystals, the fractional power transmission, averaged over the entire spectral range, has been computed and correlated to the Shannon indices (see Table 1). In Figure

Conclusions
In conclusion, in this in silico experiment we have found that the evenness of the photonic crystal trapping enhancement in photovoltaic devices [27] and for the study of light scattering in diffusive media for diagnostic imaging [28]. Figure S1. Different distributions for Crystal 2