# Static Structures in Leaky Mode Waveguides +
## 2 Supplies and Methods
Using coordinate geometry, we determine that \(u_{N}\) and \(v_{N}\) can be expressed as:  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]2  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]3  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]4  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]5  
We define the non-overlap condition for order when the view zones are \(x_{1,1}^{iv}<x_{0}^{iv}\) (i.e., the first \(m=-1\) diffraction of \(\overline{k_{1}}\) is below the \(m=0\) order of \(\overline{k_{0}}\) at the viewing distance). For the non-overlap condition to be satisfied at any distance, \(K_{D}\) must be greater than \(\Delta\beta=\beta_{1}-\beta_{0}\). To determine the distance, \(d_{4}\), at which the non-overlap condition is satisfied, we must find the intersection of two lines. We compose the first ray from the slope of the un-diffracted \(\overline{k_{0}}\) vector and the coordinates of the point, \([x_{0}^{iii},\)\(-(d_{1},d_{2},d_{3})]\). We compose the second ray from the slope of the negative first-order diffracted, \(\overline{k_{1}}\) vector and the coordinates of the point, \([x_{1,1}^{iii},\)\(-(d_{1},d_{2},d_{3})]\). We solve for the intersection of these two lines to obtain,  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]6  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]7  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]8  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]9  
Note that the slopes in this case have perpendicular \(k\) components traveling in the opposite direction of those in Equations (14) and (15) above. This is because these rays are going down rather than up. The coordinate, \(z_{int}\), gives the depth _within_ the material at which the zero and first-order views separate. This is a higher than necessary upper bound as the two orders still have additional time to separate as they propagate through distance, \(d_{4}\). However, this provides time for other orders, that may have stronger overlap, to separate and to form a set of view windows that is well-separated at the viewing plane. We chose the parameters listed in Table 1 to define our near-eye geometry. We chose a 6 mm (\(d_{2}=1\) mm + \(d_{3}=5\) mm) thick substrate with an internal grating at a 1 mm distance from the SAW layer with a period, \(\Lambda_{D}\), of approximately 4 \(\upmu\)m and an output coupling grating with a period, \(\Lambda_{G}\) of approximately 300 nm. These values correspond to \(K_{D}\approx 2*\Delta\beta\) and \(K_{G}\approx\beta_{0}\), respectively. An approximately \(10^{\circ}\) bias was assumed to give maximum and minimum leaky mode angles of \(\theta_{0}=70^{\circ}\) and \(\theta_{1}=77^{\circ}\). All of these fit within reasonable fabrication parameters for a two-substrate system. \[\text{where }k_{sub}=\frac{2\pi n_{ord}}{\lambda_{0}}. \tag{3}\]0 Multiple-order leaky mode concept: (**a**) leaky mode light diffracts into multiple orders within the substrate to form multiple image points at different angles and visible from different angles. (**b**) The K-vector analysis showing momentum change for both the internal grating and the out-coupling grating.