Reconstructing the scattering matrix of photonic systems from quasinormal modes

Abstract

The scattering matrix is a fundamental tool to quantitatively describe the properties of resonant systems. In particular, it enables the understanding of many photonic devices of current interest, such as photonic metasurfaces and nanostructured optical scatterers. In this contribution, we show that the scattering matrix of a photonic system is completely determined by its quasinormal modes, i.e., the self-sustaining electromagnetic excitations at a complex frequency. On the basis of temporal coupled-mode theory, we derive an expression for the expansion of the scattering matrix on quasinormal modes, which is directly applicable to an arbitrary number of modes and input/output channels. Our theory does not require any ad-hoc assumptions, such as the fitting of an additonal nonresonant background. We validate and discuss the theoretical formalism with some illustrative examples. This demonstrates that the theory represents a powerful and predictive tool for calculating the highly structured spectra of resonant nanophotonic systems, and, at the same time, a key for unravelling the physical mechanisms at the heart of such intricate spectral structures. Keywords: scattering matrix, quasinormal modes, resonant states, theoretical photonics, photonic crystals. INTRODUCTIONThe scattering matrix constitutes an important tool in multiple fields of science, such as nuclear physics, electronic transport, or classical electrodynamics [1,2]. In particular, it plays an essential role in helping us to understand intriguing optical phenomena, such as Fano resonances in optical systems [3] or scattering dark states [4], and to tailor the properties of novel photonic devices, like photonic metasurfaces [5] or hybrid plasmonicphotonic nanoresonators [6].These electromagnetic systems typically display a highly structured spectral response, with multiple resonances in the spectrum. The resonances are associated with quasinormal modes (also called resonant states), i.e., complex-frequency solutions of Maxwell's equations with outgoing-wave boundary conditions [7][8][9]. In order to decipher, understand, and engineer the behaviour of complex optical systems, it is vital to unravel the connection between the scattering matrix and the underlying quasinormal modes.In this contribution, we develop a rigorous and general theory for the expansion of the scattering matrix of electromagnetic systems on quasinormal modes. The method is based on the framework of temporal coupledmode theory for optical resonators [3,10]. Such framework has been effectively used to study the transmission of gratings and photonic-crystal structures [3,[10][11][12] and the scattering cross section of nanoparticles [4]. However, in these applications, coupled-mode theory has been usually restricted to only one or two modes of the optical system, with the residual spectral response being taken into account with a slowly varying frequency-dependent background, fitted from independent simulation data [3,10,11]. Our theory provides a direct and general relation ...