The problem of the optimal synthesis of circularly symmetric aperture distributions able to radiate pencil beams fulfilling well-defined requirements on the far-field and/or the directivity is addressed and solved. In particular, constraints concerning both the sidelobe level (SLL) and the directivity within a coverage zone are dealt with while being careful not to incur into the so called super-directive sources or into aperture fields actually difficult to be realized. The case wherein a constraint is enforced on the dynamics of the excitations is also considered. Numerical examples concerning cases of practical relevance are also shown in support to the given theory.
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