Airy accelerating beams have aroused a great deal of interest because of their non-diffracting and self-accelerating properties, which have many potential applications such as in optical micromanipulations, nonlinear optics and vacuum electron acceleration. A key issue in research of the acceleration beam and its applications is how to generate this specific type of beams with high energy efficiency. In this paper, the spatial oscillation properties of the complex amplitude of an accelerating beam are analyzed and a theory describing the accelerating properties of the beam based on its local spatial frequency is proposed. The local spatial frequency of a general Airy beam is calculated through the zero-point coordinates of the Airy function, and an analytical formula accurately describing the local spatial frequency distribution is given. The relationship between the local spatial frequency and the accelerating trajectory of the beam is also given, based on which a simple algorithm for finding the pure-phase expression of an acceleration beam from its given accelerating trajectory is presented. Finally an analytical expression of the pure-phase function of an acceleration beam for generating a circle-arced trajectory is found out, based on which a pure-phase diffractive optical element is designed and demonstrated successfully.

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