## 2 Theoretical Calculation
### Calculation of Displacement and Offset  
Both the oeeo wave and the eooe wave undergo displacement in the \(y\) and \(z\) directions. Due to the limited size of the prism, when the incident angle is too large or the incident point is too close to the edge of the prism, the wave may not exit correctly from the rear surface of the prism group, which can significantly limit the system's aperture utilization and luminous flux. Therefore, in order to ensure the maximum utilization of the DWP aperture, the light should be emitted from the edge of the prism, as shown in Figure 3a. Figure 1: The schematic of the SBFTIS. Figure 2: The schematic of the spatio-tempo hybrid modulate. Figure 3b is an enlarged view of the DWP part in Figure 3a, showing in detail the propagations of the eooe and oeeo waves passing through the DWP, where \(O\) is the incident point; \(A\), \(B\), \(C\), \(D\), and \(E\) are the intersection points of the eooe wave at each interface; and \(A^{\prime}\), \(B^{\prime}\), \(C^{\prime}\), \(D^{\prime}\), and \(E^{\prime}\) are the intersection points of the oeeo wave. In our previous work [24], the equation of the wave normal was obtained using the wave normal tracing method, which can be used to calculate the intersection coordinates. To analyze the aperture of the system, it is necessary to calculate the horizontal and vertical displacements \(\Delta\) of the wave normals of eoee and oeeo, respectively. \(\Delta\) is defined as the difference between the y and z coordinates of \(E\) and \(E^{\prime}\) and point \(O\), and the expressions are:  
\[\Delta_{ye}=\sin\omega\tan r_{e1}(t-l_{e})+\sin\omega_{e}\tan r_{e2}(t+l_{e})+ s\sin\omega_{e}\tan r_{e2}, \tag{1}\]  
\[\Delta_{yo}=\sin\omega\tan r_{e1}(t-l_{o})+\sin\omega_{o}\tan r_{e2}(t+l_{e})+ s\sin\omega_{o}\tan r_{e2}, \tag{2}\]  
\[\Delta_{ze}=\cos\omega\tan r_{e1}(t-l_{e})+\cos\omega_{e}\tan r_{e2}(t+l_{e})+ s\cos\omega_{e}\tan r_{e2}, \tag{3}\]  
\[\Delta_{ze}=\cos\omega\tan r_{e1}(t-l_{o})+\cos\omega_{o}\tan r_{e2}(t+l_{e})+ s\cos\omega_{o}\tan r_{e2}, \tag{4}\]  
where  
\[l_{e}=\frac{t\cos\omega_{e2}\tan r_{e2}+s\cos\omega_{e2}\tan r_{e3}}{\cot\beta -\cos\omega_{e2}\tan r_{e2}}, \tag{5}\]  
\[l_{o}=\frac{t\cos\omega_{e2}\tan r_{e2}+s\cos\omega_{e2}\tan r_{e3}}{\cot\beta -\cos\omega_{e2}\tan r_{e2}}, \tag{6}\]  
In the above expression, \(t\) is the thickness of the WP\({}_{1}\) and WP\({}_{2}\); \(s\) is the thickness of the air gap; and \(\beta\) is the structural angle of the prisms. \(r_{e1}\) and \(r_{e1}\) are the refraction angles of eooe and oeeo waves in the left crystal of WP1 and the right crystal of WP2. \(r_{e2}\) and \(r_{e2}\) are the refraction angles of the two waves in the right crystal of WP\({}_{1}\) and the left crystal of WP\({}_{2}\); and \(\omega_{e1}\) and \(\omega_{e1}\) are the incident plane angles of the two waves at the same position. \(R_{e3}\) and \(r_{e3}\) are the refraction angles of the two waves in the air gap; and \(\omega_{e2}\) and \(\omega_{e2}\) are the incident surface angles of the two waves in the air gap. The expressions of all angles can be found in reference [24]. Figure 3: (**a**) Waves’ propagation and displacement in SBFTIS (**b**) Waves’ propagation and displacement in DWP (For calcite WP, projected on the OXZ plane). Equations (1)-(6) do not incorporate any positional coordinate information from the incident point. Hence, the displacement of the prism group is solely dependent on the incident direction of the incident light and the thickness of the air gap and is not related to the position of the incident point. The difference between the \(y\) and \(z\) coordinates of points \(E\) and \(E^{\prime}\) represents the offset amount (\(d\)) of the two rays in the \(y\) and \(z\) directions, and its expression is:  
\[d_{y}=\left|\Delta_{ye}-\Delta_{yo}\right|, \tag{7}\]  
Figure 2:0  
Figure 2:1  
where \(d_{total}\) is the transverse offset amount of the prism group, and its physical meaning is the distance between the outgoing two parallel light.