Quantum theory of light in a dispersive structured linear dielectric: a macroscopic Hamiltonian tutorial treatment

Abstract

ABSTRACT These notes – intended to be self-contained and tutorial – present a direct, macroscopic approach to quantizing light inside a linear-response dielectric material when both spectral dispersion and spatial non-uniformity are present, but the spectral region of interest is optically transparent so that explicit treatment of the underlying physics of the medium is not needed. The approach is based on the macroscopic Maxwell equations and a corresponding Hamiltonian, without the use of Lagrangians or dynamical models for the medium, and uses a standard mode-based quantization method. The treatment covers energy density and flux in a dispersive dielectric; the inverse permittivity formalism; a new derivation of the mode normalization condition; a direct proof of mode nonorthogonality; examples of quantized field expressions for the general case and various special cases; the relationship between group velocity and energy flux; the band approximation and the continuum limit; and quantum optical treatment of waveguide modes.