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We report on a novel kind of accelerating beams that follow parabolic paths in free space. This accelerating peculiar polygon beam (APPB) not only exhibits autofocusing property, but also possesses two types of accelerating intensity maxima, i.e., the cusp and spot structure. We also provide a detailed insight into the theoretical origin and characteristics of this spatially accelerating beam through catastrophe theory. Moreover, experimental results confirm the peculiar features presented in the theoretical findings, and the APPB is further verified to exhibit self-healing property during propagation with either obstructed cusp or spot-like main lobe reconstructing after a certain distance. This accelerating beam with multiple types of main lobes has never been achieved previously, and we anticipate that APPB will facilitate the applications in the areas of biomedical manipulation and optofludics.
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