Abstract
Ordered and chaotic superlattices have been identified in Nature that give rise to a variety of colours reflected by the skin of various organisms. In particular, organisms such as silvery fish possess superlattices that reflect a broad range of light from the visible to the UV. Such superlattices have previously been identified as ‘chaotic’, but we propose that apparent ‘chaotic’ natural structures, which have been previously modelled as completely random structures, should have an underlying fractal geometry. Fractal geometry, often described as the geometry of Nature, can be used to mimic structures found in Nature, but deterministic fractals produce structures that are too ‘perfect’ to appear natural. Introducing variability into fractals produces structures that appear more natural. We suggest that the ‘chaotic’ (purely random) superlattices identified in Nature are more accurately modelled by multi-generator fractals. Furthermore, we introduce fractal random Cantor bars as a candidate for generating both ordered and ‘chaotic’ superlattices, such as the ones found in silvery fish. A genetic algorithm is used to evolve optimal fractal random Cantor bars with multiple generators targeting several desired optical functions in the mid-infrared and the near-infrared. We present optimized superlattices demonstrating broadband reflection as well as single and multiple pass bands in the near-infrared regime.
Highlights
Fractal geometry, which was introduced by Mandelbrot in the 1970s, has been called the ‘geometry of Nature’ because it helps us describe complex structures found in Nature that are often irregular, wiggly, self-similar on multiple length scales, and difficult to represent using conventional Euclidean geometry [1]
While multi-generator fractal trees were well suited to represent the arrangement of antenna elements in an array, a more natural choice of a fractal geometry for representing superlattices would be the Cantor bar, which is composed of one-dimensional line segments
The multi-generator fractal Cantor superlattice synthesis technique presented here was developed with the objective of imitating the broadband mirror behaviour of the silvery fish skin
Summary
Fractal geometry, which was introduced by Mandelbrot in the 1970s, has been called the ‘geometry of Nature’ because it helps us describe complex structures found in Nature that are often irregular, wiggly, self-similar on multiple length scales, and difficult to represent using conventional Euclidean geometry [1]. While this model is able to reproduce the broadband reflectivity of the organism skin, in some cases, it ignores thicker layers of cytoplasm which are present within the skin, such as demonstrated by the ribbonfish [8] Both these ordered and chaotic regions indicate that there is a potential design space that can be explored using fractal geometry. While multi-generator fractal trees were well suited to represent the arrangement of antenna elements in an array, a more natural choice of a fractal geometry for representing superlattices would be the Cantor bar, which is composed of one-dimensional line segments This class of fractals has previously been studied as a basis for producing superlattices [18], where deterministic Cantor bar superlattices were found to have a characteristic, ordered structure and scattering spectra with many peaks and nulls in the reflection. A GA is used in order to evolve optimal fractal random Cantor superlattices with varying degrees of structure to possess specific filter functions, including a broadband reflectivity and single or multiple passbands
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