Additional boundary condition for electric quadrupolar continua derived from Maxwell's differential equations

Electric quadrupolar continua satisfying a physically reasonable constitutive relation supports both an evanescent and a propagating eigenmode. Thus, three interface boundary conditions, two plus an “additional boundary condition” (ABC), are required to obtain a unique solution to a plane wave incident from free space upon an electric quadrupolar half‐space. By generalizing the constitutive relation to hold within the transition layer between the free space and the quadrupolar continuum, we derive these three boundary conditions directly from Maxwell's differential equations. The three boundary conditions are used to determine the unique solution to the boundary value problem of an electric quadrupolar slab. Numerical computations show that for long wavelengths, two previous boundary conditions, derived under the assumption that the electric quadrupolarization contains negligible effective delta functions in the transition layer, produce an accurate solution by neglecting the evanescent eigenmode, that is, by assuming it decays within the transition layer. It appears that the general method used to derive the electric quadrupolar ABC can be applied to obtain the boundary conditions for any other realizable constitutive relation in a Maxwellian multipole continuum.