We introduce a fresh class of photonic band-gap materials, curvilinear-lattice photonic crystals, whose distinctive feature is that their individual scatterers are arranged in a curvilinear lattice. We show that adhering to some restrictions in the acceptable lattice transformations, one can achieve omnidirectional photonic band gaps for an entire subclass of such structures. We demonstrate, designing an efficient arbitrary-angle waveguide bend, that curvilinear-lattice photonic crystals can be employed for creation of original types of nanophotonic devices.

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