Quantum electromagnetic fields in the presence of a dielectric microsphere

A general quantization scheme of the electromagnetic (EM) fields, including the nearfield, of a dielectric material is presented. By expanding the classical EM fields in terms of orthonormal modes, the Lagrangian is expressed as a sum of the Lagrangians of the independent harmonic oscillators. For a dielectric sphere, as an example, two possible modes satisfying the boundary conditions, as well as the orthonormality conditions, are explicitly obtained: spherical modes and Mie modes. Using these modes, the EM fields are quantized and the transformation between two one-photon states having different modes is also discussed. We apply our results to calculate the expectation value of the quantum momentum-density operator for a one-photon state and show that it is equivalent to the classical Poynting vector. Our rigorous results may be useful in the study of the quantum optical properties of the EM near-fields of a dielectric microsphere of sub-wavelength size.