# Considerations of Curvature for a Near-Eye Holo-Video Display
## 3 Circular Screen  
### Purpose and Initial Concept  
Our goal in creating a curved display is to increase the FOV while sacrificing the maximum aperture size allowed. Specifically, if the display has a radius of curvature (ROC) equal to its distance from the eye, then the FOV will automatically be \(180^{\circ}\) (or at least the angle subtended by the actual curved device). This behavior is demonstrated visually in Figure 5a. Figure 4: (**a**) The virtual point is limited by the aperture \(2a\) and distance drawn. The angle allowed by these parameters depends on the maximum deflection angle of the device. (**b**) The maximum length perceived by the eye at any distance is found by dragging the aperture along the active region until the back-traced rays fall just barely on the eye. The angle from the eye to the virtual points defines the FOV. Figure 5: (**a**) For a curved display with the eye placed at the radius of the curvature, as the virtual point is radially drawn at a fixed distance from the eye, the normal to the device will always fall on the eye, and the point is guaranteed to be seen by the eye. Thus, the FOV of the device is \(180^{\circ}\). (**b**) The closest  
drawable point at distance, \(d\), is determined by the radius of curvature. As the curvature increases, the angle between the normals will offset the \(\theta_{MAX}=2^{\circ}\) deflection on each side. This limits the closest point possible faster than in the flat case. Both the flat and curved devices use the same underlying technology, and the flat case is already a proven method [7; 8; 9; 10]. Curved or circular LiNbO\({}_{3}\) has been shown to propagate SAWs at similar levels to those achieved in flat cases [24]. Furthermore, we show that, at the radius of curvatures and interaction lengths available, the curved device operates in a very similar fashion to the flat case. The major consideration for this setup is the allowable aperture for drawing the virtual point. In contrast to the flat-screen case, where aperture and allowable draw distance has a linear relationship, the curved display adds a complication; that is, as the display curves, so does the \(\theta_{MAX}=2^{\circ}\) off the normal (see Figure 5b). This means that, in order to achieve the maximum deflection possible, and therefore the closest point possible, the aperture size must be reduced. Figure 6a shows the relationship between aperture size and closest drawable point for a curved display of this type. Note that, as the arclength used increases (for a fixed ROC), the closest drawable points approaches infinity. The display can no longer converge to a point at any distance, since, with the distance along the curved surface increased, and therefore the angular offset between the endpoints increased, the \(\theta_{MAX}=2^{\circ}\) of deflection is no longer sufficient to converge to a point. Again, refer to Figure 5b to understand this limitation. Figure 6: **(a)** As the arclength (\(2a\) in Figure 7) increases, the ability of the device to focus on a point near the eye is reduced. This eventually reaches a point where, due to the curvature of the device, the \(\theta_{MAX}=2^{\circ}\) afforded by the device is insufficient to draw any point. **(b)** Here, we consider different deflection angles, \(2\cdot\theta_{MAX}\). As expected, increasing the deflection angle allows us to use a larger arclength. Figure 7: The various angles are labeled as \"\(\theta_{-}\)\". Half of the aperture is labeled “\(a\)”, and the secant under the arc is “\(l\)”. The maximum deflection from the normal is required at either end of the shaped beam and is defined, for this paper, as \(\theta_{MAX}=2^{\circ}\). The figure is not drawn to scale, in order to improve viewability. In order to facilitate the design for this near-eye display, we choose to limit the allowable aperture to one size for all points drawn (for a given ROC). In practice, a variable aperture size would allow for greater control over points drawn further in space, but this would complicate the design outside the scope of a paper.