Far-field illumination of targets by annular apertures

This paper considers the maximum performance of annular apertures. Pupil functions are found that place the largest possible fraction of energy on circular, far-field targets, thus establishing the truly diffraction-limited performance of annuli. Properly contoured pupil functions can put significantly more energy on targets than do the uniform pupil functions that presently are used as the measure of diffraction-limited performance in laser technology. For example, an annulus with a ratio of inside to outside diameter of 0.1, with the optimum function, places 12% more energy on a target that is one Airy disk in size than does a uniform pupil function. The optimal propagation of energy from annulus to target is found to be described by the same integral equation that describes the hole-coupled resonator. Design and evaluation of annular devices should be based on these optimal pupil functions rather than on uniform illumination.