Theory of Electron Beam Moiré

When a specimen surface carrying a high-frequency line grating is examined under a scanning electron microscope (SEM), moiré fringes are observed at several different magnifications. The fringes are characterized by their spatial frequency, orientation, and contrast. These features of the moiré pattern depend on the spatial frequency mismatch between the specimen grating and the raster scan lines, the diameter of the electron beam, and the detailed topography of the lines on the specimen. A mathematical model of e-beam moiré is developed that expresses the spatial dependence of the SEM image brightness as a product of the local intensity of the scanning beam and the local scattering function from the specimen grating. Equations are derived that give the spatial frequency of the moiré fringes as functions of the microscope settings and the spatial frequency of the specimen grating. The model also describes the contrast of several different types of moiré fringes that are observed at different magnifications. We analyze the formation of these different fringe patterns, and divide them into different categories including natural fringes, fringes of multiplication, fringes of division, and fringes of rotation.