Sparse and composite coherent lattices

A method is described that yields a series of $(D+1)$-element wave-vector sets giving rise to ($D=2$ or 3)-dimensional coherent sparse lattices of any desired Bravais symmetry and primitive cell shape, but of increasing period relative to the excitation wavelength. By applying lattice symmetry operations to any of these sets, composite lattices of $N>D+1$ waves are constructed, having increased spatial frequency content but unchanged crystal group symmetry and periodicity. Optical lattices of widely spaced excitation maxima of diffraction-limited confinement and controllable polarization can thereby be created, possibly useful for quantum optics, lithography, or multifocal microscopy.