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.

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