Airy transform of an electron Landau state in a longitudinal constant magnetic field

In this article, a relationship between the electron beams in a longitudinal constant magnetic field and light beams in a parabolic refractive index distribution medium is revealed and demonstrated analytically. We propose a new method to work out the properties of the singularity in that analytical expression and have a discussion on some universal features of electron beams in a longitudinal constant magnetic field. Based on the theories above, we introduce a new type of structured electron beams, which is the Airy transform of electron Landau states (ATLSs). We not only study the dynamic of ATLSs and the magnetic field analytically and numerically but also carry out the movement of centroid and the dual auto-focusing property of ATLSs. Moreover, we have a discussion on the influence on ATLSs of Airy control parameters and the parameters of electron Landau states.