# A 1 \(\times\) 2 Two-Dimensional Slanted Grating Based on Double-Layer Cylindrical Structure
## 1 Introduction  
Diffraction gratings are the key elements in the optical devices and systems, such as spectrometers [1, 2], metrology scales [3, 4], wavelength division multiplexing, optical communication systems [5, 6], etc. In addition, diffraction gratings play an increasingly important role in various planar optical systems, such as near-eye display systems for virtual reality (VR) and augmented reality (AR) [7, 8]. The VR glasses that were originally invented used traditional optical elements. Free-form surfaces have achieved relative optimization in the aspects of evaluation dimensions of display effect, mass production, cost performance, and processing technology. However, due to the large volume of traditional optical elements and the bulky products, AR optical elements are evolving into optical waveguides. With the rapid development of micro-nano processing technology, diffractive optical waveguides have gradually attracted attention, and the products have gradually begun to be developed. Furthermore, 3D display technology is developing rapidly [9], and optical gratings are widely used in 3D display technology. Therefore, most mainstream AR manufacturers are choosing to explore innovative optical waveguide solutions to greatly reduce the size of glasses. Many authors studied the gratings for virtual reality (VR) and augmented reality (AR) applications in display technologies. [PERSON] et al. designed a waveguide structure with three polarizer gratings as the coupling elements to achieve large outgoing exit pupilsize [10]. [PERSON] et al. proposed a blazed grating structure which diffracts green light with an efficiency of 78.6% and red and blue light with an efficiency of 69.7% [11]. [PERSON] and [PERSON] et al. proposed a transparent near-eye display with an exit pupil expander (EPE) consisting of two multiplexed slanted gratings [12]. [PERSON] et al. designed a surface microstructure with a triple-carved-sub-grating, the period of which is 18.9 microns. It can be used in diffractive waveguide near-eye display devices [13]. HoloLens uses three layers of waveguides with slanted gratings to realize full-color display and two-dimensional exit pupil replication [14]. [PERSON] et al. proposed a tilted grating for optical communication [15]. [PERSON] et al. proposed a waveguide display design based on polarization-dependent grating. They use slanted gratings and rectangular gratings to extract TM light and TE light, respectively [16]. [PERSON] et al. presented a two-dimensional dielectric grating; it can achieve high absorption in monolayer graphene in visible and near-infrared frequencies [17]. [PERSON] et al. proposed a pair of stacked metal-dielectric grating structures which are polarization-independent spectrum selective absorbers. The absorber can maintain a high absorption rate around a wide range of incidence angles (0-33\({}^{\circ}\)) [18]. However, about 60% of light is wasted in the light coupling when a symmetrical structure is used [19]. Therefore, in order to achieve high diffraction efficiency, it is crucial to design an asymmetric diffraction grating structure. Most reported gratings achieving high diffraction efficiency at the \(-1\) order under normal incidence are 1D gratings, or involve complex combinations of multiple 1D slanted gratings. For one-dimensional gratings, optical waveguides can achieve only one-dimensional pupil expansion in one direction, with low integration. In order to induce propagation for the optical waveguide in two directions, one method is to use a turning grating group, such that the optical path turns form two directions of pupil dilation. The other is to use two-dimensional diffraction grating. The advantage of two-dimensional grating is that there are only two grating areas, which reduce the loss of light in the propagation.