Quantum electromagnetic fields in the presence of adielectric microsphere

Abstract

A general quantization scheme of the electromagnetic (EM) fields,including the near-field, of a dielectric material is presented.By expanding the classical EM fields in terms of orthonormalmodes, the Lagrangian is expressed as a sum of the Lagrangiansof the independent harmonic oscillators. For a dielectricsphere, as an example, two possible modes satisfying the boundaryconditions, as well as the orthonormality conditions, areexplicitly obtained: spherical modes and Mie modes. Using thesemodes, the EM fields are quantized and the transformationbetween two one-photon states having different modes is alsodiscussed. We apply our results to calculate the expectationvalue of the quantum momentum-density operator for a one-photonstate and show that it is equivalent to the classical Poyntingvector. Our rigorous results may be useful in the study of thequantum optical properties of the EM near-fields of adielectric microsphere of sub-wavelength size.