## 2 Methods
### Direct Projection of the DV-EIA onto the CMA for the DV 3-D Display
#### 2.3.1 Parallel-Ray Approximation Method  
In the field of the array antenna, the parallel-ray approximation (PRA) method has been used in computing the frequency response of the array antenna rather than the ray-tracing method [27; 28; 29; 30]. Here, the array antenna is built with a 2-D periodic array structure. Since this array antenna looks very similar to the convex mirror array (CMA), the PRA method can be applied for the performance analysis of the proposed CMA-DPII system under the far-field condition.  
Thus, in this paper, the PRA method was employed for the ray-optical analysis of the proposed system. Figure 6 shows an optical geometry for analyzing the proposed system based on the PRA method. In Figure 6, \(P\) and \(K\) represent the pitch of an elemental convex mirror and the maximum number of convex mirrors along the \(x\)-direction. The center of the \(0\)th convex mirror was set to be the origin of the vertical \(x\)-coordinate and a point source whose vertical coordinate was the same as that of the \(0\)th convex mirror. \(L\) denotes the distance between the point source and CMA. Here, two red and blue-color rays coming from the point source were assumed to arrive at the upper and bottom edges of the \(K\)th convex mirror, respectively, where \(R\) and \(\theta_{Red}\) denote the optical path length of the red-color ray and its incidence angle to the \(K\)th convex mirror, while \(r\) and \(\theta_{Blue}\) denote the optical path length of the blue-color ray and its incidence angle to the \(K\)th convex mirror, respectively. In addition, \(\theta_{Center}\) represents the angle of the ray incoming to the center of the \(K\)th convex mirror.  
Figure 5: Optical configuration of the direct-projection integral imaging system for the DV 3-D display.  
The proposed CMA-DPII-based DV 3-D display system can be made under the far-field condition, which enables the use of the PRA method for the analysis of its operational performance. Under the far-field condition of \(P/L<0.01\), the optical path lengths of the red and blue-color rays from the point source to the \(K\)th convex mirror, \(R\) and \(r\), as well as the incidence angles of the red and blue-color ray to the \(K\)th convex mirror, \(\theta_{Rad}\) and \(\theta_{Blue}\), can be approximated to be equal. With this relationship of \(\theta_{Rad}\)\(\approx\)\(\theta_{Blue}\) red and blue-color rays can be assumed to be parallel rays with the same propagation angles to the \(K\)th convex mirror. Thus, the propagation angles of all those rays in a bundle can be given by \(\theta_{Center}\approx\theta_{Rad}\approx\theta_{Blue}\), where \(\theta_{Center}\) represents the incidence angle of the center ray. In other words, the incidence angles of rays to the \(K\)th convex mirror can be modeled as a single representative angle of \(\theta_{Center}\), which is given by Equation (2).  
\[\tan(\theta_{Center})=\frac{KP}{L} \tag{2}\]  
Equation (2) means that the incidence angle of the center ray to each convex mirror depends on the vertical location of the convex mirror (\(KP\)) and the distance between the point source and the CMA plane (\(L\)). Since the CMA consists of a number of same-sized convex mirrors with regular intervals along the \(x\) and \(y\)-directions, each bundle of rays incoming to each convex mirror has different incidence angles. According to Equation (2), as the number of convex mirrors of \(K\) increases, its corresponding incidence angle of \(\theta_{Center}\) can be increased up to \(90^{\circ}\). Now, when a point source is replaced with a beam projector, all rays coming from the projector carrying the DV-EIA can be approximated to be a set of bundles of parallel rays with corresponding DV-EIs, and the operational performance of the proposed system, which is simply composed of a beam projector and a CMA, can be analyzed based on the PRA method.