  Citation: Ciarella, L.; Tognazzi, A.; Mangini, F.; De Angelis, C.; Pattelli, L.; Frezza, F. Finite-Size and Illumination Conditions Effects in All-Dielectric Metasurfaces. Electronics 2022, 11, 1017. https:// doi.org/10.3390/electronics11071017 Academic Editor: Longqing Cong Received: 14 February 2022 Accepted: 21 March 2022 Published: 24 March 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). electronics Article Finite-Size and Illumination Conditions Effects in All-Dielectric Metasurfaces Luca Ciarella 1,† , Andrea Tognazzi 2,3, * ,† , Fabio Mangini 4,† , Costantino De Angelis 3,4,5 , Lorenzo Pattelli 6,7,† and Fabrizio Frezza 1,5 1 Dipartimento di Ingegneria dell’Informazione, Elettronica e Telecomunicazioni, Sapienza Università di Roma, Via Eudossiana 18, 00184 Roma, Italy; ciarella.1656100@studenti.uniroma1.it (L.C.); fabrizio.frezza@uniroma1.it (F.F.) 2 Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze ed. 8, 90128 Palermo, Italy 3 Istituto Nazionale di Ottica-Consiglio Nazionale delle Ricerche (INO-CNR), Via Branze 45, 25123 Brescia, Italy; costantino.deangelis@unibs.it 4 Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Brescia,Via Branze 38, 25123 Brescia, Italy; fabio.mangini@unibs.it 5 Consorzio Nazionale Interuniversitario per le Telecomunicazioni (CNIT), Viale G.P. Usberti 181/A Sede Scientifica di Ingegneria-Palazzina 3, 43124 Parma, Italy 6 Laboratorio Europeo di Spettroscopia Nonlineare (LENS), Via Nello Carrara 1, 50019 Sesto Fiorentino, Italy; pattelli@lens.unifi.it 7 Istituto Nazionale di Ricerca Metrologica (INRiM), Strada delle Cacce 91, 10135 Torino, Italy * Correspondence: andrea.tognazzi@unipa.it † These authors contributed equally to this work. Abstract: Dielectric metasurfaces have emerged as a promising alternative to their plasmonic coun- terparts due to lower ohmic losses, which hinder sensing applications and nonlinear frequency conversion, and their larger flexibility to shape the emission pattern in the visible regime. To date, the computational cost of full-wave numerical simulations has forced the exploitation of the Floquet theorem, which implies infinitely periodic structures, in designing such devices. In this work, we show the potential pitfalls of this approach when considering finite-size metasurfaces and beam-like illumination conditions, in contrast to the typical infinite plane-wave illumination compatible with the Floquet theorem. Keywords: all-dielectric metasurfaces; multipolar decomposition; T-matrix; BIC 1. Introduction All-dielectric metasurfaces have recently emerged as a valid and promising alternative to their plasmonic counterparts. Their lower dissipative losses and thermal heating at opti- cal frequencies allow working at higher pump power for nonlinear frequency generation and the performance of non-invasive sensing of biological samples [1–12]. Moreover, in all- dielectric resonators, electric and magnetic resonances can be used to engineer near and far fields, thus providing more degrees of freedom in the design of optical devices [1,13–16]. In this framework, III-V compounds have been used for second-harmonic generation, due to their non-centrosymmetric structure and their high second-order nonlinear coeffi- cients [2–5,17–23]. Strong third-harmonic generation has been demonstrated instead with Si-based platforms [24,25]. Lately, great efforts have been made to improve the design of optical nanoresonators, either in the limit of single isolated nanoantennas or in the opposite scenario where perfect periodicity is assumed and thus the finite-size real-world structure is approximated by an infinite structure [26–28]. The two opposite situations are required in order to limit the numerical burden in the modelling when resorting to the full-wave solution of Maxwell’s equations. Despite the fact that this has given very helpful guidelines, these approaches fail in the accurate design of a device, because they Electronics 2022, 11, 1017. https://doi.org/10.3390/electronics11071017 https://www.mdpi.com/journal/electronics