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Third-generation synchrotron radiation light sources provide partially coherent radiation. Certain topics in the optical design of beam lines and monochromators can be studied with electric-field distributions only rather than with intensity distributions of the light source. The radiation characteristics of a dipole or an undulator can be represented by a two-dimensional complex-electric-field distribution at a distance of several meters from the insertion device center. A method is described that transforms this field distribution through a complete beam line (consisting of mirrors and gratings) to the sample plane. Diffraction at apertures can be modeled and Stokes parameters can be evaluated. The derivation of the equations was performed with the algebraic code REDUCE. A comparison with the brightness formalism is provided. Key words: Synchrotron radiation, ray tracing, Stokes parameters, diffraction.

One can derive fourth-order optical ray deviations for a mirror or grating from the optical path function by using the analytical code REDUCE. Some of the aberrations are discussed for normal-incidence and grazing-incidence monochromators. The nonlinear transformation of light from the source to the image plane of a mirror or grating is represented by a transformation matrix. The optical properties of a combination of several optical elements are given by the product of the matrices of the individual optical elements. The matrix elements can be interpreted as optical aberrations. A program has been written that optimizes a complete beam line with respect to various optical aberrations by minimizing an appropriate cost function that is built from a weighted sum of the matrix elements. Key words: Optical aberrations, ray tracing, phase-space transformation.