Complete solutions of finite Airy beams in free space and graded index media with fourier analysis

In order to analyze the far-field propagation characteristics of the new non-diffractive Airy beams theoretically and deeply on the condition of finite energy, the Fourier analysis method was utilized. The analytical solution to paraxial wave equation which described one-dimensional Airy beams' propagation dynamics in free space was given. Furthermore, a new analytical solution to the modified paraxial wave equation which described one-dimensional Airy beams' linear propagation dynamics in graded index media was also given, which was different from the known one. The far-field propagation characteristics of one-dimensional and two-dimensional Airy beams in free space were studied by utilizing the acquired analytical solution. In addition, the influence of different parameters on non-diffractive propagation and transverse acceleration characteristics of Airy beams were studied in detail. The non-diffractive propagation distances would become longer if the decay factor became smaller when the arbitrary transverse scale kept invariant. Moreover, the Airy beams would accelerate faster if the arbitrary transverse scale became smaller when the decay factor kept invariant. The linear propagation characteristics of one-dimensional Airy beams in the graded index media was also analyzed. Interesting phenomena were observed,and it was found the trajectory of the Airy beams could be modulated arbitrarily if the parameters of the media were selected properly.