Conventional electric field integral equation based theory is susceptible to the spurious internal resonance problem when the characteristic modes of closed perfectly conducting objects are computed iteratively. In this paper, we present a combined field integral equation based theory to remove the difficulty of internal resonances in characteristic mode analysis. The electric and magnetic field integral operators are shown to share a common set of non-trivial characteristic pairs (values and modes), leading to a generalized eigenvalue problem which is immune to the internal resonance corruption. Numerical results are presented to validate the proposed formulation. This work may offer efficient solutions to characteristic mode analysis which involves electrically large closed surfaces.

Highlights 1. The enhanced photoactivity on Bi/BiOCl catalyst was studied 2. Bi/BiOCl photocatalyst was prepared in TEM in situ 3. The SPR of Bi nanoparticles was confirmed by mono-EELS and simulation 4. The direct Eg value of single Bi nanoparticle was determined by mono-EELS 5. The charge transfer characteristic between Bi/BiOCl was investigated

A generalized modal expansion theory for investigating arbitrary 3-D bounded and unbounded electromagnetic fields is presented. When an inhomogeneity is enclosed with impenetrable boundaries, the field excited by arbitrary sources is expanded with a complete set of eigenmodes, which are classified into trapped modes and radiation modes. As the boundaries tend to infinity, trapped modes remain unchanged, while radiation modes form a continuum. To illustrate the theory, several real-life structures are investigated with a conformal finite-difference technique in the frequency domain. Perfectly matched layers (PMLs) are imposed at finite extent to emulate the unbounded problems. Numerical examples show that, only a few system modes are prominent in expanding an excited field, leading to a reduced modal picture which provides a quick guidance as well as useful physical insight for engineering design and optimization of electromagnetic devices and components.

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