# Static Structures in Leaky Mode Waveguides +
## 2 Supplies and Methods  
The geometries for order overlap are shown in Figure 6a. We will use K-vector analysis to show the relationship between points \(P\) and \(Q\). After choosing parameters for a near-eye geometry including an internal grating period, \(\Lambda_{D}\), and propagation distance, \(d_{3}\), we will plot the high-order rays to confirm a 'no-overlap' condition. We start with the back-cast chirp focus point, \(P=(x_{0},z_{0})\). This parameter is a function of the leaky mode drop angle, \(\theta\), whose value will change along the chirp. We can use the angles, \(\theta_{0}\) and \(\theta_{1}\) of light rays exiting both ends of the SAW chirp to define two vectors:  
\[\overline{k_{0}}=k_{sub}\sin(\theta_{0})\hat{x}+k_{sub}\cos(\theta_{0})\hat{z}\text { and } \tag{1}\]  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]  
\[\text{where }k_{sub}=\frac{2\pi n_{ord}}{\lambda_{0}}. \tag{3}\]  
We take the substrate index of refraction as \(n_{ord}=2.2864\) for \(\lambda_{0}=633\) nm [13]. Conservation of transverse momentum preserves the value of the vector components parallel to the device interfaces. Therefore, in our analysis we keep track of the propagation constants:  
\[\beta_{0}=k_{sub}\sin\theta_{0} \tag{4}\]  
\[\beta_{1}=k_{sub}\sin\theta_{1} \tag{5}\]  
The analysis can be thought of as having multiple layers separated in depth. They are as follows: \(d_{1}\), the absolute distance from the chirp focus point to the SAW plane; \(d_{2}\), the absolute distance from the SAW plane to the laser-induced internal grating; \(d_{3}\), the absolute distance from the internal grating to the output grating; and \(d_{4}\), the absolute distance traveled by light through the air to the viewer's eye (see Figure 6a). The vectors \(\overline{k_{0}}\) and \(\overline{k_{1}}\) bracket a ray bundle that propagates through each device layer toward the eye. The cross section of the ray bundle at each of these layers is:  
\[\Delta x^{i}=x_{1}^{i}-x_{0}^{i}\text{ where }x_{1}^{i}=d_{1}\tan\theta_{1}, \text{ and }x_{0}^{i}=d_{1}\tan\theta_{0} \tag{6}\]  
\[\Delta x^{ii}=x_{1}^{ii}-x_{0}^{ii},\text{ }x_{1}^{ii}=(d_{1}+d_{2})\tan \theta_{1},\text{ and }x_{0}^{ii}=(d_{1}+d_{2})\tan\theta_{0} \tag{7}\]  
\[\Delta x^{iii}=x_{1}^{iii}-x_{0}^{iii},\text{ }x_{1}^{iii}=(d_{1}+d_{2}+d_{3}) \tan\theta_{1},\text{ and }x_{0}^{iii}=(d_{1}+d_{2}+d_{3})\tan\theta_{0} \tag{8}\]  
\[\Delta x^{iv}=x_{1}^{iv}-x_{0}^{iv},\text{ }x_{1}^{iv}=(d_{1}+d_{2}+d_{3}+d_{4}) \tan\theta_{1},\text{ and }x_{0}^{iv}=(d_{1}+d_{2}+d_{3}+d_{4})\tan\theta_{0} \tag{9}\]  
Figure 5: High-order holographic images and view zones: **(a)** first through a fourth-order image for a horizontal parallax only (HPO) hologram. This hologram, created in grayscale photoresist, is meant to simulate the output of a leaky mode modulator passing through thin grating. **(b)** Here, we see how the high-order holographic view zones separate or overlap. In order to calculate the final orientation of \(\overline{k_{0}}\) and \(\overline{k_{1}}\) after they have passed through both the internal grating and the output coupling grating for every order \(N\), we simply subtract \(K_{G}\) once (every exiting ray must be diffracted by the output grating) and subtract (for negative orders) \(K_{D}\), \(N\) times (see Figure 6b). The magnitudes of the parallel components for the resulting \(k\)-vectors are,  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]0  
\[\overline{k_{1}}=k_{sub}\sin(\theta_{1})\hat{x}+k_{sub}\cos(\theta_{1})\hat{z}, \tag{2}\]1  
where \(m\) determines the sign of the diffracted orders; \(m=-1\) in this paper.