In this chapter we examine the spectral properties of operators commonly encountered in electromagnetics. We emphasize the role of eigenvalues and eigenfunctions in spectral theory since these quantities play an important role in many applications, as both mathematical and physical entities. The primary goal of this chapter is to present elements from the spectral theory of operators on infinite-dimensional spaces. Operators on finitedimensional spaces and their associated matrix representations are also briefly covered, as this material forms an appropriate starting point for discussion.

Spectral Theory of Linear Operators

The modes of silicon meta-atoms are investigated, motivated by their use as building blocks of Huygens' metasurfaces. A model based on these modes is presented, giving a clear physical explanation of all features in the extinction spectrum. Counter-intuitively, this can show negative contributions to extinction, which are shown to arise from the interference between non-orthogonal modes. The direct and interference contributions to extinction are determined, showing that conservation of energy is preserved. The Huygens' condition of matched electric and magnetic dipole moments leads to strong forward scattering and suppressed back scattering. It is shown that higher order modes with appropriate symmetry generalise this condition, leading to multiple bands of directional scattering. The presented results are obtained using a robust approach to find the modes of nano-photonic scatterers, commonly referred to as quasi-normal modes. By utilising an integral formulation of Maxwell's equations, this work avoids the problem of normalising diverging far-fields, which other approaches require. The model and presented results are implemented in open-source code.

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Interference between the modes of an all-dielectric meta-atom

The Singularity Expansion Method (SEM) was derived as a means of interpreting/estimating responses measured during electromagnetic testing of aerospace systems. These responses appeared to consist of a superposition of exponentially damped sinusoidal oscillations. It was shown that the electromagnetic response of a finite-sized, perfectly conducting object to a delta-function incident plane wave is a meromorphic function of the complex frequency. The physical interpretation, computational advantages and fundamental problems associated with using the poles (natural frequencies) of the meromorphic function to construct the transient responses of objects are reviewed. Areas of future investigations, both for the purpose of improving the mathematical foundations and the computational tools are discussed.

Lennart Marin

Papers

Major results and unresolved issues in singularity expansion method