An ampere-like law for displacement vector field and near-field optical microscopy

We make up a novel and simple theory for near field and near-field optical microscopy (NOM). Our theory is composed of two parts. The first is a formulation to calculate the scattered near field of light by a small dielectric. We state that a wavenumber-vector-independent picture appears in the theory of the near field, and we find that this is expressed by an Ampere-like law for the displacement vector field. The second part is a formulation of field intensity for far field observation and near field observation from a unified point of view. We suggest a theoretical formula for the field intensity corresponding to the NOM image and demonstrate how to understand the relation between the near field and this image.