A novel method for designing photonic crystals with high orders of rotational symmetry using an inverse Fourier transform (IFT) method is presented. The IFT of an n-sided polygon is taken and the positions of the peaks are computed in order to obtain a set of discrete points in real space where the scattering centres are to be located. We show, by simulating the diffraction pattern, that although these points appear disordered they possess long range order, which also confirms that the arrangement of points has re-fold rotational symmetry. The designed structures can possess an arbitrary number of rotational symmetries, whilst retaining the sharp diffraction patterns characteristic of known crystal lattices which exhibit wide bandgaps. We present simulation results using the finite difference time domain method (FDTDM) for large non-repeating patterns of scatterers produced by this method. We also present results where around 50 points have been generated in a square unit cell and tiled to produce a lattice. These were simulated using both the finite element method (FEM) and the FDTDM, which were shown to agree. Our results demonstrate that the method is capable of producing crystal structures with wide bandgaps where the scattering centres are either non-repeating with no fundamental unit cell, or consist of a (large) number of points in a unit cell, which may then be tiled to form a lattice.
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