An application of the Hilbert transform in electron microscopy: I. Bright-field microscopy

The application of Hilbert transforms to the evaluation of electron microscope images is examined. In particular, the use of complementary semicircular apertures in the objective lens back focal plane to determine from intensity measurements the complex image wavefunction is discussed. Bright-field optics is considered where the primary electron beam is allowed to interfere with the scattered electron beam. It is shown that in this modified bright-field situation, using semicircular apertures rather than the normal circular objective aperture, it is possible to determine for complementary positions of the apertures a Hilbert transform relation between the real and imaginary parts of the image wavefunction. The initial estimate for the complex wavefunction may be corrected for nonlinear terms occurring in the equations for the image intensity. In this way it is possible to correct the image wavefunction for nonlinear terms contributing as much as 50% to the modulation of the image intensity; and an image contrast as high as 70%, well above that encountered in the conventional microscopy of biological materials, does not seem to invalidate the solution for the image wavefunction. The application of the Hilbert transform to electron microscopy is illustrated with computed images comprising an array of single atoms, and the effects of objective lens aberrations are included. The solutions for the complex wavefunction are valid in all cases examined and the correction for nonlinear terms in the image intensity may be made in 1-5 iterations, the lower figure corresponding to the contrast normally achieved in the transmission electron microscopy of biological materials. Additionally, the stability of the solution for the image wavefunction is examined when the effects of photographic noise, objective aperture misalignment and inelastic background are included in the image intensity profiles.