In this paper, the results of a theoretical study of the radiation characteristics of semicircular, circular, and rectangular surface sources have been presented when these surface sources radiate uniformly over their surfaces and obey Lambert’s cosine law. Equations for all three types of sources have been derived giving the total flux falling on an elementary receiving area when this elementary receiving area has arbitrary coordinates and its surface normal arbitrary direction cosines. The classical method of surface integration has been used in each case. These equations are very general in form so that a large number of the equations already published for these three sources become special cases of these equations for which either the coordinates or the direction cosines of the surface normal of the elementary receiving area have particular values. These equations are equally applicable to problems for which the previously designated elementary receiving area becomes the source and the semicircular, circular, and rectangular areas the receiving areas. A simple translation and rotation of coordinates also makes it possible to consider the equally important problems for which the sources are permitted to have arbitrary coordinates and surface normals arbitrary direction cosines. With these equations in this form, it becomes possible to calculate the total flux falling on the elementary receiving area in the presence of an array of semicircular, circular, and rectangular surface sources having arbitrary coordinates and surface normals with arbitrary direction cosines. One numerical example is included for each of these three sources showing how these equations may be used to determine the total flux falling on the elementary receiving area when it is limited to particular planes.
A. I. MAHAN and W. F. MALMBORG, "Radiation Characteristics of Semicircular, Circular, and Rectangular Surface Sources," J. Opt. Soc. Am. 44, 644-646 (1954)