Optimization and stability of the contrast transfer function in aberration-corrected electron microscopy

The Contrast Transfer Function (CTF) describes the manner in which the electron microscope modifies the object exit wave function as a result of objective lens aberrations. For optimum resolution in C₃-corrected microscopes it is well established that a small negative value of C₃, offset by positive values of C₅ and defocus C₁ results in the most optimal instrument resolution, and optimization of the CTF has been the subject of several studies. Here we describe a simple design procedure for the CTF that results in a most even transfer of information below the resolution limit. We address not only the resolution of the instrument, but also the stability of the CTF in the presence of small disturbances in C₁ and C₃. We show that resolution can be traded for stability in a rational and transparent fashion. These topics are discussed quantitatively for both weak-phase and strong-phase (or amplitude) objects. The results apply equally to instruments at high electron energy (TEM) and at very low electron energy (LEEM), as the basic optical properties of the imaging lenses are essentially identical.