All fourteen Bravais lattices can be fabricated by triple exposure of two-beam interference fringes

Abstract

A simple method for fabricating three-dimensional photonic crystals with all fourteen Bravais lattice structures and arbitrary lattice constant is proposed. The lattice structure is defined by three primitive translation vectors. The vectors form three triangles on the non-parallel three planes, which shape a tetrahedron, and a united two tetrahedrons forms a parallelepiped, which corresponds to a primitive cell of the lattice. All atoms lie at the vertexes of the parallelepiped where these three planes intersect. This means that all crystal lattices can be described by three sets of periodically aligned planes. The periodically aligned planes, or walls, can easily be fabricated by recording two-beam interference fringes in the recording material with a recording threshold such as a photoresist. Therefore, triple exposure of the twobeam interference fringes can create arbitrary crystal lattice structure. The optical arrangement of the practical interferometer is simplified when the angles between the normal to the recording plane and the three planes are equal. In this arrangement the triple exposure is followed subsequently by simply rotating the recording material around the normal to the recording plane. To create a lattice with arbitrary lattice plane with respect to a recording plane, two axes rotating stage must be used to rotate and tilt the recording plane. This method can create all fourteen Bravais lattices with arbitrary lattice constant while the conventional four-beam interference method creates only a limited number of lattice constants. Mathematical proof and numerical analysis results of this method are also shown.