## 2 Methods
### Diffraction Grating Ray Tracing
The order where the intersection point with the dotted line cannot be found on the semicircle is the evanescent order, which is represented by the wave vector \(\vec{k}_{Enumecent}\) in Figure 1.  
When the incident wavevector and the grating are not in the same plane, the diffraction of the grating is referred to as \"conical diffraction\" [22; 23; 24]. In this case, using the circular diagram of incident wavevector \(k\) shown in Figure 1 is no longer convenient to describe the deviation of light caused by the grating. Therefore, a spherical diagram of wavevector \(k\) is used to describe the deviation of light caused by the grating under oblique incidence. The tracing process of the oblique incidence with a one-dimensional grating follows the same procedure as that for normal incidence, except that the grating vector \(\vec{K}\) and the incident wavevector \(\vec{k}_{inc}\) are not in the same plane. In the same manner, a spherical diagram of the incident wavevector \(\vec{k}_{inc}\) is plotted to describe the deviation of light caused by the grating, as shown in Figure 2.  
According to the spherical diagram of wavevector \(k\), the curve formed by the reflection order \(\stackrel{{\rightarrow}}{{k}}_{-1}\stackrel{{ \rightarrow}}{{-}}\stackrel{{\rightarrow}}{{-}}\stackrel{{ \rightarrow}}{{+}}_{3}\) is exactly a half-cone, and the curve formed by the transmission order \(\stackrel{{\rightarrow}}{{k}}_{-2}\stackrel{{ \rightarrow}}{{k}}_{+4}\) is also a half-cone. Due to the difference in refractive index between the transmission and reflection regions, the radius of the half-cone is different in each case. Combining the cases of normal incidence and oblique incidence, in a one-dimensional grating, due to the spatial invariance of the planar interface, the tangential component of the diffracted wavevector differs from the tangential component of the incident wavevector by an integer multiple of the tangential component of the grating vector of the modulation grating. The generalized one-dimensional grating diffraction equation for volume holographic gratings can be written as:  
\[n_{1}\sin\theta_{m}\ =n_{0}\sin\theta_{mc}+m\frac{\lambda}{\Lambda}\sin\phi \tag{2}\]  
\(\phi\) represents the angle between the grating vector and the normal to the planar interface. The addition of one degree of freedom in the two-dimensional grating results in an increase in the number of grating vectors from one to two, necessary to describe the far-field diffraction behavior of the grating. Figure 1: Circular diagram of incident wavevector \(k\) on a one-dimensional grating (Blues represent grating vectors. Reds represent diffractive vectors). Figure 2: Spherical diagram of wavevector \(k\) for “conical” diffraction of a one-dimensional grating (Blues represent grating vectors. Reds represent diffractive vectors). As depicted in Figure 3, \(\overset{\rightarrow}{T}_{1}\) and \(\overset{\rightarrow}{T}_{2}\) represent the primitive translation vectors of a two-dimensional hexagonal lattice, and their corresponding primitive reciprocal lattice vectors \(\overset{\rightarrow}{K}_{\alpha}\) and \(\overset{\rightarrow}{K}_{\beta}\) can be obtained through the reciprocal lattice relation. When the diffraction grating exhibits two-dimensional periodicity, its grating vectors are essentially the primitive reciprocal lattice vectors of its periodic lattice in reciprocal space. There exists a relationship among the tangential component of the diffracted wavevector, the tangential component of the primitive reciprocal lattice vector in reciprocal space, and the tangential component of the incident wavevector in the plane interface. \[\left\{