Maximum Power-Transfer Coefficient between Two Confocal Apertures*

Abstract

This paper discusses a modification of a maximum power-transfer theorem whose essential features were developed by Alan F. Kay and by Giorgio V. Borgiotti. It is shown that a C2 rotation symmetry with respect to the optic axis is a sufficient symmetry restriction for the shape of each aperture, and that the phase as well as the amplitude of the illuminating field can be included in a statement of the theorem if it is applied to confocal instead of flat surfaces. The modified statement of the theorem is that a maximum power-transfer coefficient between apertures in two confocal surfaces, whose shapes have C2 symmetry, is obtained when the illumination of one surface is identical to that of a confocal resonator having the same geometry and operating in the lowest diffraction-loss eigenmode, and that the power-transfer coefficient, T, is then related to the full-pass diffraction loss, D, by T=(1−D)12. It is further shown that the eigenfunctions of the confocal-resonator equation are identical to those of the extremum power-transfer condition, and that these functions form a complete orthogonal set. The actual solution for surfaces of rectangular shape is compared with results obtained for illumination with a gaussian-amplitude distribution. It turns out that attempts to minimize power radiated into sidelobes by using gaussian-amplitude distributions have been very close to the optimum solution of this problem.