The photonic properties of a two-dimensional (2D) photonic aperiodic crystal based on the Thue–Morse (ThMo) substitutional sequence were investigated theoretically. In contrast to traditional photonic quasicrystals based on the Penrose tiling, these structures were obtained by removing the lattice points from a square arrangement, following the inflation rules emerging from the ThMo sequence. The resulting structure does not exhibit the typical translational symmetry of photonic crystals. In particular, it is well known that the ThMo sequence has a singular continuous Fourier transform. This property was transferred directly on the 2D ThMo photonic aperiodic crystal represented by an array of pillars in air. The electromagnetic field distribution can be described as a quasi-localized state, with characteristics lying between the localized states, corresponding to the defect state in a photonic crystal, and the Bloch states …