(*a*) *The weak-phase-object approximation.* For sufficiently thin objects, the effect of the object on the incident-beam amplitude may be represented by the transmission function (2.5.2.16) given by the phase-object approximation. If the fluctuations, , about the mean value of the projected potential are sufficiently small so that , it is possible to use the *weak-phase-object approximation* (WPOA) where is referred to the average value, . The assumption that only first-order terms in need be considered is the equivalent of a single-scattering, or kinematical, approximation applied to the two-dimensional function, the projected potential of (2.5.2.16). From (2.5.2.42), the image intensity (2.5.2.35) becomes where the spread function *s*(*xy*) is the Fourier transform of the imaginary part of *T*(*uv*), namely .

The optimum imaging condition is then found, following Scherzer (1949), by specifying that the defocus should be such that is close to unity for as large a range of as possible. This is so for a negative defocus such that decreases to a minimum of about before increasing to zero and higher as a result of the fourth-order term of (2.5.2.33) (see Fig. 2.5.2.3). This optimum, `Scherzer defocus' value is given by or

The resolution limit is then taken as corresponding to the value of when becomes zero, before it begins to oscillate rapidly with *U*. The resolution limit is then For example, for mm and Å (200 keV), Å.

Within the limits of the WPOA, the image intensity can be written simply for a number of other imaging modes in terms of the Fourier transforms and of the real and imaginary parts of the objective-lens transfer function , where **r** and **u** are two-dimensional vectors in real and reciprocal space, respectively.

For dark-field TEM images, obtained by introducing a central stop to block out the central beam in the diffraction pattern in the back-focal plane of the objective lens, Here, as in (2.5.2.42), should be taken to imply the difference from the mean potential value, .

For bright-field STEM imaging with a very small detector placed axially in the central beam of the diffraction pattern (2.5.2.39) on the detector plane, the intensity, from (2.5.2.41), is given by (2.5.2.43).

For a finite axially symmetric detector, described by , the image intensity is where is the Fourier transform of (Cowley & Au, 1978).

For STEM with an annular dark-field detector which collects all electrons scattered outside the central spot of the diffraction pattern in the detector plane, it can be shown that, to a good approximation (valid except near the resolution limit) Since is the intensity distribution of the electron probe incident on the specimen, (2.5.2.48) is equivalent to the incoherent imaging of the function .

Within the range of validity of the WPOA or, in general, whenever the zero beam of the diffraction pattern is very much stronger than any diffracted beam, the general expression (2.5.2.36) for the modifications of image intensities due to limited coherence may be conveniently approximated. The effect of integrating over the variables , may be represented by multiplying the transfer function *T* (*u*, *v*) by so-called `envelope functions' which involve the Fourier transforms of the functions and .

For example, if is approximated by a Gaussian of width (at of the maximum) centred at and is a circular aperture function the transfer function for coherent radiation is multiplied by where