Revival and Expansion of the Theory of Coherent Lattices

An effective way to design structured coherent wave interference patterns that builds on the theory of coherent lattices, is presented. The technique combines prime number factorization in the complex plane with moir\'e theory to provide a robust way to design structured patterns with variable spacing of intensity maxima. In addition, the proposed theoretical framework facilitates an elegant computation of previously unexplored high-order superlattices both for the periodic and quasiperiodic case. A number of beam configurations highlighting prime examples of patterns for lattices with three-, four-, and fivefold symmetry are verified in a multibeam interference experiment.