system. The last, sinusoidal term of the contrast transfer function will determine the sign with which components of frequency u will enter contrast in the final image. If one takes into account only spherical aberration to third order and defocus, χ is rotationally symmetric about the optical axis of the microscope and thus only depends on the modulus u = |u|, given by where Cs is the spherical aberration coefficient, λ is the electron wavelength, and Δf is the defocus. In transmission electron microscopy, defocus can easily be controlled and measured to high precision. Thus one can easily alter the shape of the contrast transfer function by defocusing the sample. Contrary to optical applications, defocusing can increase the precision and interpretability of the micrographs. The aperture function cuts off beams scattered above a certain critical angle (given by the objective pole piece for ex), thus effectively limiting the attainable resolution. However it is the envelope function E(u) which usually dampens the signal of beams scattered at high angles, and imposes a maximum to the transmitted spatial frequency. This maximum determines the highest resolution attainable with a microscope and is known as the information limit. E(u) can be described as a product of single envelopes: due to Es(u): angular spread of the source Ec(u): chromatic aberration Ed(u): specimen drift Ev(u): specimen vibration ED(u): detector Specimen drift and vibration can be minimized in a stable environment. It is usually the spherical aberration Cs that limits spatial coherency and defines Es(u) and the chromatic aberration Cc, together with current and voltage instabilities that define the temporal coherency in Ec(u). These two envelopes determine the information limit by damping the signal transfer in Fourier space with increasing spatial frequency u where α is the semiangle of the pencil of rays illuminating the sample. Clearly, if the wave aberration ('here represented by Cs and Δf) vanished, this envelope function would be a constant one. In case of an uncorrected transmission electron microscope with fixed Cs, the damping due to this envelope function can be minimized by optimizing the defocus at which the image is recorded (Lichte defocus). The temporal envelope function can be expressed as . Here, δ is the focal spread with the chromatic aberration Cc as the parameter: The terms and represent instabilities in of the total current in the magnetic lenses and the acceleration voltage. is the energy spread of electrons emitted