The general formulation of double refraction or internal double reflection with different directions of propagation vector and wave vector in a uniaxial crystal is analyzed in terms of the Huygens principle. Then double refraction and double reflection along the sequential interfaces in a crystal are discussed. With the relations between the output of the former interface and the input of the latter, double refraction and/or double internal reflection in a crystal along two and multiple successive interfaces can be calculated. On this basis, a configuration of electrooptic 1×N optical switch integrated in a single slab of LiNbO3 is suggested, which consists of an array of electrooptic polarization modulators between the interfaces for total internal reflection. Due to double refraction effect, the ordinary ray and the extraordinary ray have different reflections. From different switching states of modulators, internal double reflection along the sequential interfaces of a crystal can lead to different directions of the output beam. The suggested 1×N optical switch is simple and compact in construction, low in loss and insensitive to environment. An experimental 1×4 switch is demonstrated, too.