Effective phase correction function for high-resolution exit wave reconstruction by a three-dimensional Fourier filtering method

The phase correction function used in the three-dimensional Fourier filtering method (3D-FFM) for compensating lens aberrations was investigated to reconstruct a high-resolution exit wave of a sample. An appropriate function, which hardly suffered from imperfect illumination conditions, was determined by comparing two types of phase correction functions with numerical calculations and experiments using through-focus images of an amorphous thin film and a [1 1 0]-oriented Si single crystal taken under tilted illumination or partially coherent illumination. Theoretical calculations indicated that a function in terms of w (an axial Fourier component), available uniquely in the 3D Fourier space, compensated for the phase shift due to the spherical aberration more precisely than did a conventional function in terms of g (the two-dimensional (2D) planar Fourier components). Experimentally, exit waves reconstructed using the w-function showed sample structures at ∼20% higher resolution than those reconstructed using the g-function. Image contrast simulations proved that the w-function had a significant advantage over the g-function: the former canceled out the effect of illumination divergence, resulting in a high-resolution exit wave. These results demonstrated that exit waves, which are uniquely realized in the 3D-FFM, should be reconstructed using the w-type phase correction function.