Normalization approach for scattering modes in classical and quantum electrodynamics

Abstract

We propose a novel scheme to normalize scattering modes of the electromagnetic field. By relying on analytical solutions for Maxwell's equations in the homogenous medium outside the scatterer, we derive normalization conditions that only depend on the electromagnetic field on the surface of a sphere containing the scatterer. We pay special attention to the important cases of plane wave illumination and illumination with a multipolar field, for which an explicit and easy to use normalization condition is derived. We demonstrate the versatility of our method by normalizing scattering modes of some selected metallic and dielectric scatterers of different geometries in the context of different application scenarios. Since every quantum mechanical treatment of light-matter interaction requires the proper normalization of electromagnetic fields, we deem our proposed normalization scheme broadly applicable independent of the scatterer involved.

Quantum description of radiative decay in optical cavities

“…• Calculate the light-matter coupling constant from the properly normalized field strength (according to [64]), the cavity parameters and the emitter's transition dipole moment.…”

“…For the rest of this work we drop the mode index k from ω k , making the dispersion relation implicit. The electromagnetic field modes are of the form [64] …”

Second Quantization of Scattering Modes of Absorptive Photonic Nanostructures

“…(ω, |x| > R) = b ∈ R. This last condition guarantees the existence of a set of normalizable and orthogonal scattering modes [7]. In the Coulomb gauge and in the absence of external charges, the classical Lagrangian of such a system can be written as…”

“…it satisfies Silver-Müller radiation conditions. In [7] it is demonstrated that the scattering modes are complete, orthogonal and normalized with respect to the scalar product…”

“…The quantitative quantum description of optical systems, therefore, often requires a large amount of heuristic modeling, phenomenological inference and/or experimental input. In two earlier contributions we tried to overcome this problem by first achieving proper field normalization of scattering modes [7] as a necessary step towards quantizing the field and then demonstrating how discrete eigenmodes can emerge from the resulting quantum field theory in systems with optical resonances [8]. In the process we found means to determine light-matter coupling paramters quantitatively, discussed the emergence of dissipative dynamics due to radiation losses and established an input-output formalism linking cavity and scattering dynamics.…”