title: Arrangements of four beams for any Bravais lattice.
abstract: A single geometric model based on a new concept of a reciprocal primitive pyramid (RPP) in reciprocal space is proposed for investigation of relationships between any three-dimensional (3D) lattice and arrangements of four beams (AFBs) that produce the lattice. A ternary linear equation set, described for the one-to-one correspondence between a RPP and AFB, can readily reveal all AFBs for the same lattice (AFBSLs). Quantitative AFBs for bcc and fcc real lattices are illustrated to show that various AFBSLs can modulate the properties of a photonic bandgap (PBG) both by tuning the lattice constant and by changing the lattice-point shape. This fact may yield the appropriate AFB for a complete 3D PBG with the desired center wavelength. The nonuniqueness of AFBSLs can provide abundant choices for persons who plan interference experiments, especially for holographic fabrication of 3D photonic crystals (PCs).