_Article_  
**Quantification of the Transmission Properties of Anisotropic Metasurfaces Illuminated by Finite-Size Beams**  
**[PERSON] 1,2, [PERSON] 1,3 and [PERSON] 1,4**  
1 Departement d'Optique, Institut FEMTO-ST, Unite Mixte de Recherche (UMR) Centre National de la Recherche Scientifique (CNRS) 6174, Universite Bourgogne Franche-Comte, 25030 Besancon, France  
2 Departement de Physique, Ecole Normale Superieure de Kouba B.P. 92, 16050 Algiers, Algeria  
3 Department of Electrical and Computer Engineering, University of California San Diego,  
La Jolla, CA 92093-0407, USA  
*Correspondence: [EMAIL_ADDRESS]; Tel. : +33-3-81666421  
**Abstract:** The aim of this paper is to present an analytical method to quantitatively address the influence of a focusing illumination on the optical response properties of a metasurface illuminated by a finite-size beam. Most theoretical and numerical studies are performed by considering an infinite periodic structure illuminated by a plane wave. In practice, one deals with a finite-size illumination and structure. The combination of the angular spectrum expansion with a monomodal modal method is used to determine the beam size needed to acquire efficient properties of a metasurface that behaves as an anisotropic plate. Interesting results show that the beam-size can be as small as \(5\times 5\) periods to recover the results of a plane wave. Other results also show that the beam-size can be used as an extrinsic parameter to enhance the anisotropic metasurface performance and to adjust its expected properties finely (birefringence and/or transmission coefficient). These findings are important for the design of real (finite) structures and can be adapted for experimental conditions to achieve optimized results and take full advantage of the metamaterial properties. **Keywords:** metamaterials; transmission properties; quantification; theory  
More often than not, theoretical studies dealing with the design of a periodic metamaterial exhibiting unique properties (extraordinary or enhanced transmission, large anisotropy, enhanced nonlinearity, chirality, etc.) involve an infinite periodic structure illuminated by a plane wave [1, 2, 3, 4]. This approach is questionable when dealing with finite-size optical beams and/or finite-size structures. This becomes critical when the properties of the metamaterial are very sensitive to the illumination angle of incidence. This happens especially when surface plasmon resonances are involved. In the latter case, it was found that a structure larger than the propagation length of the surface plasmon is generally needed to recover the properties of an infinitely-sized structure [5]. [PERSON] et al. [6] studied the propagation of a finite-size beam through a 2D metamaterial (slabs). They showed that the group velocity direction can be non-collinear with the orthogonal to the iso-frequency curves of the dispersion diagram due to a collective contribution of the evanescent waves that are excited inside and at the edges of the beam. Unfortunately, there are only a few studies where extensive numerical simulations are performed using commercial software by considering finite-size structure and/or beams [7, 8, 9, 10]. These simulations are generally essential to model non-periodic metamaterials [11, 12]. However, there are simpler methods to treat the case of periodic structures. [PERSON] [13] studied the transmission of different polarization state beams through an array of coaxial apertures engraved into a perfectly conducting screen. An angular spectrum expansion was used in that study; however, the mean beam propagation direction was restricted to normal incidence, and only near-field transmittedbeam distributions were calculated. We propose to combine the angular spectrum expansion with a monomodal modal method to determine the transmission properties of a specific metasurface behaving as a half-wave or quarter-wave plate [14].