[en] In this paper, we derive the thickness profiles of metallic transmission-grating bars which maximize either the power throughput into the m^th diffracted order or the ratio of the m^th-order diffracted power to the total output power (in the soft x-ray range). The derivation is performed for both general and symmetric bar shapes and for the two physically important cases of continuous gratings and gratings with integral bars. The solutions derived are shown to be valid for cases where the optical constants are generalized to be functions of position in a direction perpendicular to the grating bars. Examples of some optimum profiles for gold in the soft x-ray range are computed on the basis of the presented analysis and tabulated for convenient reference. 18 references