How to properly fit the dielectric permittivity and resonances of optical materials?

Resonances strongly impact the optical response of materials and optical cavities. They occur in optical materials due to the electronic inter or intra band transitions, and in optical cavities where they are governed by the shape and composition of the cavity. Characterizing resonances has attracted a strong attention from a wide scientific community since the knowledge of resonance properties allows for predicting the optical response of the system in the whole spectrum. In this work, we show how to extract resonances without a priori knowledge on the associated physical system. The method relies on the spectral analysis of the optical response of the system in the framework of the scattering matrix formalism. A coefficient of the S-matrix corresponds to a specific channel from which the system releases the accumulated energy to the outside. Each scattering spectrum of an S-matrix element can be expanded onto a set of functions that depend on complex singularities corresponding to complex frequencies for which the scattering coefficient becomes infinite. This method is called the singularity expansion method –SEM [1-3]. In the framework of the SEM, we have developed a method that fits the spectrum of a given scattering channel by retrieving the number and position of singularities and their associated residues in the complex frequency plane. This method is very general and can be applied to any linear physical system. The singular expansion method must apply to the dielectric permittivity that is equivalent to a transfer function in the constitutive relations [4]. We recently showed how a simple change of variables allows recasting the singular expansion of the dielectric permittivity into a formulation that generalizes the Drude Lorentz –DL- model. This generalized Drude-Lorentz model is derived without any a priori knowledge on intrinsic resonances in the material, but the number and location of the singularities retrieved from the fitting with experimental data of the dielectric permittivity unveils reliable and fruitful information on the intrinsic resonances of optical materials. This approach applies to any kind of material such as dielectrics, 2D materials or metals [4]. Metasurfaces, diffractive optics and optical cavities are well described by the S-matrix formalism. We applied this approach to analyse the reflection and transmission spectra of metallic and dispersive metasurfaces. This analysis reveals the singularities of the scattering channels. We recently introduced a quality function that characterizes the strength of resonances over the whole spectrum with respect to the singularities and zeros of the scattering channel [5]. References 1 C. Baum, E. Rothwell, K.-M. Chen, D. Nyquist, “The singularity expansion method and its application to target identication,” Proceedings of the IEEE 79, 1481-1492 (1991) [2] V. Grigoriev, A. Tahri, S. Varault, B. Rolly, B. Stout, J. Wenger, N. Bonod, “Optimization of resonant effects in nanostructures via Weierstrass factorization,” Phys. Rev. A 88, 011803(R) (2013) [3] I. Ben Soltane, R. Colom, F. Dierrick, B. Stout, N. Bonod, “Multiple-Order Singularity Expansion Method,” New J. Phys. 25, 103022 (2023) [4] I. Ben Soltane, F. Dierick, B. Stout, N. Bonod, “Generalized Drude-Lorentz Model Complying with the Singularity Expansion Method,” Advanced Optical Materials 12, 2400093 (2024). [5] I.B. Soltane, N. Bonod, “Extracting Complete Resonance Characteristics From the Phase of Physical Signals,” arXiv:2412.10157 (2025)