## 2 Methods
### Waveguide Coupler Device Fabrication  
We propose a holographic lithography apparatus composed of holographic optical elements (HOEs) and a method for fabricating HOEs. We also simulate the refractive index spatial distribution of a 2D coupler prepared using a photosensitive resist as the recording material. The 2D holographic grating is fabricated using the holographic lithography apparatus shown in Figure 6a; a collimated and expanded plane wave is incident on the diffraction screen with HOEs, and the diffracted light intensity forms a periodic spatial distribution in the optical recording medium on a planar waveguide, inducing a spatial distribution of refractive index. The relative positioning and arrangement of HOEs, as  
Figure 5: The principle of probabilistic splitting (Full-colored arrows represent the real light-rays traced in probabilistic splitting mode whereas the vaguely-colored counterpart is taken as the reference and actually not exist). shown in the Figure 6, with the square region representing the HOE and other regions made of purely absorbing materials, are designed to avoid the influence of stray light. According to the theoretical analysis of 2D hexagonal photonic crystals made using HOEs, the lattice constants:  
\[a=\frac{2\lambda_{2}}{3\sin\theta_{d}} \tag{6}\]  
\(\lambda_{2}\) refers to the wavelength of the light source used in vacuum to fabricate the 2D holographic grating and \(\theta_{d}\) represents the first-order diffraction angle of the light passing through a single HOE. \(n\) denotes the refractive index of the ordinary waveguide substrate, the value of which is 1.723. The light source for the display is a He-Ne laser with a wavelength of 632.8 nm, while a He-Cd laser with a wavelength of 442 nm is used to fabricate the 2D holographic grating on the photosensitive resist. According to equation \(a=4\lambda_{1}/3n\), the required first-order diffraction angle for the HOE is obtained as \(\theta_{d}=36.995^{\circ}\). The fabrication of the two-dimensional grating is accomplished using holographic techniques. The first step involves interference, and analyzing the intensity of the interference light is beneficial for accurately guiding the fabrication experiment. Figure 7 represents the three-beam interference; the wave vectors of the three-beam plane wave are:  
\[\left\{ \begin{array}{c}.\\ k_{1}=-\sin\varphi e_{x}+\cos\varphi e_{z}\\.\\ k_{2}=\frac{1}{2}\text{sin}\ \varphi e_{x}-\frac{\sqrt{3}}{2}\text{sin}\ \varphi e_{y}+\cos\varphi e_{z}\\.\\ k_{3}=\frac{1}{2}\text{sin}\ \varphi e_{x}+\frac{\sqrt{3}}{2}\text{sin}\ \varphi e_{y}+\cos\varphi e_{z} \\ \end{array} \right. \tag{7}\]  
The distribution of optical intensity of the recording medium is:  
\[\left\{\begin{array}{c}I(x,y)=|E_{01}|^{2}+|E_{02}|^{2}+|E_{03}|^{2}\\ +2E_{01}\cdot E_{02}\text{cos}\Big{(}-\frac{3}{2}\text{kcsin}\varphi+\frac{ \sqrt{3}}{2}\text{kysin}\varphi\Big{)}\\ +2E_{01}\cdot E_{03}\text{cos} \begin{array}{c}-\frac{\sqrt{3}}{2}\text{kcsin }\varphi-\frac{\sqrt{3}}{2}\text{kysin}\varphi\\ +2E_{02}\cdot E_{03}\text{cos}\Big{(}\sqrt{3}\text{kysin}\varphi\Big{)}\\ \end{array} \right. \tag{8}\]  
The angle between the three planar waves and the surface normal is denoted as \(\varphi\), which is also the primary diffraction angle \(\theta_{d}\) of HOEs. As shown in Figure 7, the amplitudes of the three planar waves are \(A_{1},A_{2}\), and \(A_{3}\), and their polarization angles observed in the relative co-ordinate system are \(\phi_{1},\phi_{2}\), and \(\phi_{3}\), respectively. Figure 6: (**a**) Two-dimensional grating holographic lithography equipment; (**b**) HOEs for holographic lithography. equation \(\Delta d=T_{d}[v_{1}-\Delta\text{vex}(-\alpha_{0}E)]\); since \(\alpha_{0}E<<1\), the equation can be reduced to \(\Delta d=T_{d}(\Delta v\alpha_{0}E+v_{2})\).