Classification of photonic crystals with Bravais lattice structure fabricated by four-beam interference

Abstract

The number of independent lattice constants of three-dimensional photonic crystals, which can be fabricated by four-beam interference, is analyzed for 14 Braivas lattices. The equation of maximum intensity point condition for interference fringe among four plane waves is the same as that between the lattice vector and the reciprocal lattice vector in the solid-state physics. This relation gives us the way to derive the wave number vectors for incident four plane waves to fabricate any desired three-dimensional photonic crystal structures. It is analyzed that the effective combination of incident wave number vectors is 16 for each of 14 Bravais lattices. Lattice constant is numerically analyzed for 16 combinations of wave number vectors for each 14 Bravais lattices. The resultant 16 lattice constants are not necessarily independent due to the symmetry of lattice. It is found that the maximum and minimum numbers for lattice constants are 16 for Triclinic and Face-centered orthorhombic lattices, and 1 for Primitive orthorhombic, Primitive tetragonal and Primitive cubic lattices. The others are 10 for Centered monoclinic, 9 for Body-centered orthorhombic, 6 for Body-centered tetragonal, 5 for Face-centered cubic, Body-centered cubic and Trigonal, and 2 for Primitive monoclinic, Centered orthorhombic and Hexagonal lattices. As a result, total number of 81 photonic crystals with different lattice structure or different lattice constant can be fabricated by using four-beam interference with a fixed wavelength.