Photonic crystals that are aperiodic or quasi-crystalline in nature have been the focus of research due to their complex spatial distributions, resulting in high order rotational symmetries. Recently we proposed aperiodic patterns that were rotationally symmetric while being random in the radial direction. The structures are designed by segmenting the circular design space, randomly populating one segment, and repeating that segment about a center of rotation. Studying the symmetries and geometrical attributes of aperiodic structures is typically performed in reciprocal Fourier space by examining the distribution of the Fourier coefficients. This allows the translational symmetry to be directly extracted and the rotational nature to be interpreted. Instead we propose comparing the typical Fourier analysis with the use of a Fourier-Bessel space. The Fourier-Bessel approach expands the dielectric layout in cylindrical coordinates using exponential and Bessel functions as the angular and radial basis functions. The coefficients obtained in this fashion directly provide the rotational symmetries that are present. This work will examine both the Fourier and Fourier-Bessel distributions of the proposed structures as well as other quasi-crystals in order to explore the strengths and weaknesses of both techniques.

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Keywords Fourier expansion, Fourier-Bessel expansion, Photonic quasi-crystals
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Conference Photonic and Phononic Properties of Engineered Nanostructures III
Newman, S.R., & Gauthier, R. (2013). Photonic quasi-crystals in fourier and fourier-bessel space. Presented at the Photonic and Phononic Properties of Engineered Nanostructures III. doi:10.1117/12.2005621