math intensity equation of light at surface

Such conditions are often fulfilled at terahertz frequencies, facilitating applications in electron acceleration 14, pulse compression 15, 16 and spectral reshaping 17, 18.Īt even higher frequencies in the optical domain, a further regime is encountered when the electron coherence time exceeds the optical period, or equivalently, the kinetic energy distribution of the electron beam is narrower than one photon energy. In contrast, when an electron’s interaction time with an oscillating field becomes comparable to the period, both its transverse and longitudinal momentum distributions may be altered as a function of the electron arrival time within the cycle. At sufficiently low frequencies, typically in the radio-frequency range, the electron beam samples the instantaneous potential, with just a marginal net change in electron energy. These approaches were previously extended to image time-dependent potentials, employing pulsed electron beams synchronized to an external excitation 9, 10, 11, 12, 13. In particular, Lorentz microscopy encompasses TEM variants designed to image phase shifts induced by nanoscale fields in electric and magnetic devices, primarily involving spin textures containing domain walls 5, vortices 6 and skyrmions 7, 8. Transforming this Aharanov-Bohm phase 1 to measurable intensities, contrast is obtained by defocused imaging, beam deflection (differential phase contrast), off-axis holography or the use of Zernike phase plates 2, 3, 4.

Atomic-scale and elemental variations of the Coulomb potential imprint a phase profile onto a transmitted electron wave function. The primary source of contrast in transmission electron microscopy (TEM) is elastic scattering at magnetic and electric potentials in an investigated specimen.

Math intensity equation of light at surface