Quasinormal-Mode Expansion of the Scattering Matrix

Abstract

It is well known that the quasinormal modes (or resonant states) of photonic structures can be associated with the poles of the scattering matrix of the system in the complex-frequency plane. In this work, the inverse problem, i.e., the reconstruction of the scattering matrix from the knowledge of the quasinormal modes, is addressed. We develop a general and scalable quasinormal-mode expansion of the scattering matrix, requiring only the complex eigenfrequencies and the far-field properties of the eigenmodes. The theory is validated by applying it to illustrative nanophotonic systems with multiple overlapping electromagnetic modes. The examples demonstrate that our theory provides an accurate first-principle prediction of the scattering properties, without the need for postulating ad-hoc nonresonant channels.Scattering matrices have been playing a ubiquitous role in physics since the early history of quantum field theory [1]. Nowadays, scattering-matrix techniques represent an irreplaceable tool for scientists working in nuclear physics [2], electronic transport [3], or classically chaotic systems [4], just to mention some of the several fields of application. Scattering matrices also enjoy a well deserved popularity in electromagnetic modeling, ranging from microwave devices [5] to nanophotonics applications, such as scattering and transmission from nanostructured objects [6][7][8].Most of the systems that are usually investigated with scattering-matrix techniques display a highly structured resonant response as a function of the excitation frequency (or energy), with the resonances in the spectrum being directly related to the poles of the analytical continuation of the scattering matrix in the complexfrequency plane [9,10]. For electromagnetic systems, such poles correspond to quasinormal modes (also called resonant states), i.e., complex-frequency solutions of Maxwell's equations with outgoing-wave boundary conditions [11][12][13][14][15]. In a sense, quasinormal modes represent the bare skeleton around which the frequency-dependent response of the system is built. The interplay among different electromagnetic modes has proven to be crucial for explaining several intriguing phenomena, such as Fano resonances in optical systems [16], scattering dark states [17,18], and the optical analog of electromagnetically induced transparency and superscattering [19], and for designing new optical materials, such as optical metasurfaces for wavefront shaping [20]. For these reasons, it is desirable and extremely interesting to be able to reconstruct ab initio the entire scattering matrix of a system from the knowledge of its quasinormal modes. Not only would such quasinormal-mode expansion contribute to the understanding of complicated spectral features in terms of interference and superposition of resonant states, but it would also offer practical advantages from the numerical point of view, since a full eigenmode calculation is generally faster and more comprehensive than a large number of single-frequency simulations.A...

How to calculate the pole expansion of the optical scattering matrix from the resonant states

“…[35], the third test system is beyond its scope due to the large Ohmic losses. Its absorbance reaches values of nearly 40 %, as seen in Fig.…”

“…Note that the background term can be reduced to the scattering matrix of homogeneous and isotropic space for some highly symmetric geometries, as it is assumed in the examples of Ref. [35], but this is not necessarily the case.…”

“…As in Ref. [35], where S BG is called the direct-coupling matrix, we do not focus on the calculation of the background term, but derive expressions for the residue matrices R n with elements R n,NN . Applying Eq.…”

“…In contrast to previous works, the residues of the pole contributions in the scattering matrix are calculated directly from the resonant field distributions by projecting them onto basis functions of free space [36]. The derivation is based on selecting basis functions that satisfy appropriate orthogonality relations, and separating the total field into the background and scattered field [6,32,35,37]. The formulation is neither limited with respect to the number of resonant states nor restricted to certain geometries.…”

“…In a similar manner, Yang et al show how to derive the outgoing channels for a known electromagnetic near field [34], but both Perrin's and Yang's approaches do not provide the pole expansion of the scattering matrix, which directly relates incoming and outgoing channels. Alpeggiani et al [35] presented a formulation that is valid for a large number of resonant states. Based on symmetry considerations, a system of equations is derived that is solved in a least-square sense to calculate the residues for the pole expansion of the scattering matrix.…”