## 1 Introduction  
At present, diffractive waveguide based AR-HMD is favored by the market and consumers for its compact structure, low color crosstalk, high transparency, and low pupil swing distortion. The diffraction waveguide display system uses a microdisplay as the image source, collimates the light from the image source using a collimation system, couples the collimated light into the waveguide transmission medium through a coupling-in element, and, finally, transmits the image to the human eye through a coupling-out element. The coupling element directly influences the field of view (FOV), eyebox, and imaging brightness of an AR-HMD in a diffractive waveguide system. In practice, the coupling elements of diffractive waveguides are holographic gratings, surface relief grating (SRG), polarization volume grating (PVG), and super surface [1; 2; 3; 4]. For example, Sony LMX-001 smart glasses (SONY, Tokyo, Japan) use reflective holographic grating, Digilens (Digilens, Sunnyvale, CA, America) uses volume holographic grating, and Hololens 2 (Microsoft, Redmond, WA, America) and Magic Leap One (Magic Leap, Plantation, FL, America) both use surface relief grating [5; 6; 7]. Eyebox is a key parameter in the design of HMD devices; the larger the geometry of the eyebox, the greater the range of motion of the human eye and the better the wearing experience for the user. Waveguide eyebox enlargement is achieved with the help of EPE. In order to reduce the impact of the input optical system on the overall size and weight of the HMD, it is necessary to expand the pupil in two directions. The basic structure of 2D-EPE has been theoretically determined [8], and Microsoft's Hololens 1 adopts an \"L-shaped\" architecture [9; 10], in which the grating periods of the coupled-in and coupled-out grating and the deflected grating are not parallel in the waveguide plane, enabling pupil expansion in two directions. WaveOptics uses a 2D grating as the coupling grating, which has two degrees of freedom in the grating vector, combining the functions of both the deflection grating and the coupling grating in the \"L-shaped\" architecture, while the waveguide size can be smaller, with less diffraction loss and better manufacturing tolerances, contributing to improved imaging quality and lower manufacturing costs [11]. However, the current waveguide devices using the above two 2D-EPE schemes usually suffer from low optical efficiency and severe reverse light leakage. Improving the diffraction performance of coupled gratings is a direct solution to the above problems. New optical elements composed of PB-phase liquid crystals possess high flexibility, and the Display Technology Research Center of Southeast University has realized EPE structures by PVG gratings [12; 13]. In recent years, researchers have implemented EPE for diffractive waveguide systems through joint simulations with multiple software: Gsolver5.2 and TracePro73, Rsoft 2020 and LightTools8.7.0 for geometry-micro-nano data sharing through static files, and Zemax OpticStudio2023R1 and Lumerical2023R1 for data interaction through dynamic links, and also through field-tracing simulation software Virtualab Fusion2023.1 for the integrated design and analysis of the system [14; 15; 16; 17]. A research group at Southeast University implemented a one-dimensional PVG by customizing the diffraction function in Zemax OpticStudio as a coupled grating for far-field diffraction characterization and used this as the basis for the optimal design of diffractive waveguide EPE [18]. Research teams from Beijing Institute University of Technology and Changchun University of Technology have proposed strategies and algorithms for designing and optimizing EPE [19; 20; 21], respectively. For static data sharing schemes, the efficiency of the optimized design depends on the software interpolation and search algorithms.