# Multiple Reflections and the Near-Field Effects on a Metamaterial Quarter-Wave Plate
## 1 Introduction  
The polarization degree of freedom has played a critical role in advanced sciences and technologies [1] involving weak to strong intensities of light. Recent research that considers the light polarization includes high-efficiency polarization locking [2], generalized polarization transformations [1], full-Stokes polarization perfect absorption [3], frequency tunable polarization conversion [4], and enhanced broadband nonlinear optical processes [5], to name a few. Metamaterials have garnered significant interest in the field due to their extraordinary properties and versatile applications, which stem from their ability to manipulate light in ways that are unattainable with conventional materials [6; 7]. One metamaterial-based optical device is the quarter-wave plate (QWP), which is an indispensable tool for polarization control and manipulation [8; 9; 10]. QWPs work by introducing a phase shift between the orthogonal components of an incident lightwave, effectively transforming linearly polarized light into circularly polarized light or vice versa. Recent advances in nanofabrication techniques and material science have enabled the construction of innovative metamaterial-based optical devices [11; 12]. This progress has given rise to the design and fabrication of ultra-thin, sub-wavelength QWPs [13; 14; 15; 16; 17; 18], emphasizing the ongoing trend towards device miniaturization and integration [19; 20; 21; 22; 23]. Such miniaturization and incorporation of metamaterials present exciting prospects in classical and quantum optical information systems (e.g., imaging, sensing, communications, and cryptography). This is made possible by the extensive degrees of freedom that metamaterials provide [26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36]. As the integration of metamaterials into optical systems undergoes size reductions and neighboring structures approach the near-field region, their interaction with electromagnetic fields becomes increasingly intricate. Controlling near-field interactions within the unit cells has enabled novel metamaterials and optical phenomena, such as classical analog of electromagnetically induced transparency [37], coupled plasmonic metamaterials [38],and, more recently, chiral perfect absorbers with strong near-field enhancement [3]. While metamaterials are recognized for their inherent near-field effects [39; 40; 41; 42; 43; 44; 45; 46; 47], the precise extent of this effective region and its consequential impact on adjacent structures remain a less explored domain. Unless accounted for within a metadevice, coupling among neighboring metamaterials or optical components can trigger unwanted energy transfer. Thus, to better understand the consequences of cascading metamaterials for future integration, a comprehensive analysis of how near-field effects [26; 27; 28; 48; 49] and multiple reflections [50; 51] affect the performance of the metamaterial is essential. This will ensure appropriate integration into optical designs and to understand potential complications arising from multiple reflections between devices. To understand some of the complications when integrating metamaterials into a single device, we present an analysis of a grating structure design for a metamaterial QWP, defining its effective length due to near-field effects and its performance in a cascaded system. The paper is organized as follows. In Section 2, we provide an overview of the theoretical framework utilized to design a grating structure-based QWP. We discuss the principles of birefringence in the context of grating structures and outline the key considerations for designing our metamaterial QWP. Furthermore, we introduce a specific design, detailing the materials and parameters employed. Then, we present the simulation results obtained using COMSOL Multiphysics software and analyze the performance of the designed grating structure as a QWP. In Section 3, we analyze the near-field performance of the grating structure, highlighting the effective length required for QWP operation.