A novel systematic approach to the design of directivity patterns of higher order differential microphones is proposed. The directivity patterns are obtained by optimizing a cost function which is a convex combination of a front-back energy ratio and uniformity within a frontal sector of interest. Most of the standard directivity patterns-omnidirectional, cardioid, subcardioid, hypercardioid, supercardioid-are particular solutions of this optimization problem with specific values of two free parameters: the angular width of the frontal sector and the convex combination factor. More general solutions of practical use are obtained by varying these two parameters. Many of these optimal directivity patterns are trigonometric polynomials with complex roots. A new differential array structure that enables the implementation of general higher order directivity patterns, with complex or real roots, is then proposed. The effectiveness of the proposed design framework and the implementation structure are illustrated by design examples, simulations, and measurements.