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One-dimensional quasicrystal structures composed of Ⅲ-V semiconductor GaAs/AlGaAs multilayers in deterministic Thue-Morse (TM) sequences have been grown by using gas-source molecular beam epitaxy to investigate both the structural and the photonic bandgap properties. The x-ray measurements show that this aperiodic system exhibits obvious periodic spatial correlations, from which the precise thickness of the constitutive layers could be determined. Transmission and reflection measurements experimentally demonstrated plenty of photonic bandgaps with traditional or fractal features existing in those quasicrystal structures, which are in good agreement with the transfer matrix simulations. The diversity of this TM system makes it a good candidate for photonic device applications.