The theory of partially coherent image formation in dark-field and phase-contrast microscopy is presented. Explicit expressions and three-dimensional plots of the transmission cross-coefficient for different imaging modes are given. These include central and oblique internal dark field, annular dark field, phase contrast, differential phase contrast using semicircular or quadrant condenser pupils, and differential interference contrast. Explicit expressions are given for the image intensity for pure-phase objects consisting of a single-object spatial frequency or combinations of object frequencies.
Sheppard et al. (Sun,) studied this question.