Nonparaxial Accelerating Electron Beams

We investigate nonparaxial accelerating electron beams theoretically in two and three dimensions. Starting from the Klein–Gordon equation, we obtain the Helmholtz equation for electron beams. We demonstrate that the electron beams can accelerate along semi-circular, parabolic, and semi-elliptic trajectories. The shape of the trajectory is determined by the input beam, which can be constructed by using phase masks that reflect the shape of the relevant special functions: half-Bessel, Weber, or half-Mathieu. The corresponding self-healing and ballistic-like effects of the nonparaxial accelerating beams are also demonstrated. The depth of the focus of the electron beam can be adjusted by the order of the function that is included in the input. Our investigation enriches the accelerating electron beam family, and provides new choices for improving the resolution of transmission electron microscope images.