Simple analytical solutions for the optical properties of multilayered confocal prolate spheroids in the quasistatic approximation

Abstract We present simple analytical solutions to Laplace’s equation for multilayered confocal prolate spheroids within the quasi-static approximation, tailored to configurations where the polarisation of the incident field aligns with the spheroid’s major axis. The field potential in each region is expressed as a superposition of two analytic functions, with amplitude coefficients conveniently determined through a compact matrix formulation derived from the boundary conditions. A key advancement of this work is the derivation of an explicit expression for the amplitude coefficient $$K_0$$ of the perturbed field at the surface of the multilayered spheroid, obtained from the ratio of two matrix determinants. This coefficient inherently incorporates all constituent material properties and geometrical parameters of the nanostructure. Building on this, we derive a simple and general expression for the effective polarisability $$\alpha $$ of the multilayered spheroidal nanoparticle, which serves as the foundation for calculating its optical characteristics, including scattering and absorption. In contrast to previous models, the proposed analytical expressions enable rapid and accurate evaluation of optical responses–such as scattering, absorption, field enhancement, and plasmonic resonance–across arbitrary multilayered spheroidal configurations. Numerical examples of two-, three-, and five-layer structures are provided to demonstrate the model’s accuracy and computational efficiency, underscoring its potential as a powerful analytical tool for the design and optimisation of complex plasmonic nanostructures.