Propagating analysis of Airy beams via complex phase space and ray method

A complex phase space and ray method is proposed to present intuitive interpretation of unique intensity profiles, parabolic accelerating trajectories, and distinctive phase shifts of Airy beam and its exponentially decaying version (i.e., finite-energy Airy beam). In the complex phase space, finite-energy Airy beam manifests itself as a complex parabolic phase space curve (PSC) which represents a cluster of light rays with complex wave vectors. The complex ray cluster converges to a complex parabolic caustic curve in complex coordinate space. For infinite-energy Airy beam, phase space, PSC, light ray and caustic curve change to real values. In the paraxial condition, Airy beams can maintain parabolic form of PSC and keep constant ray density, which guarantees the non-diffraction property of Airy beam and approximate non-diffraction property of finite-energy Airy beam. From the evolution of vertexes of parabolic PSC along a parabolic trajectory in phase space, one can also give the parabolic caustic curve for Airy beam and the complex parabolic caustic curve for exponentially decaying Airy beam. Further, the special phase and decay factors in the propagating solution of complex amplitude of Airy beams can be directly derived from the phase shifts of light ray cluster along transverse and longitudinal directions. The proposed phase space and ray method can present intuitive comprehension to distinctive propagating characteristics of Airy beams including intensity and phase, without resort to solving wave equation or diffracting integral formula.