## 3 Simulation  
An AD prototype developed by our lab [6] is employed to verify our theory. This prototype is based on a light emitting diode (LED) display. Every pixel of the LED display is composed of a red,a green, and a blue LED chips in a vertical line, with the size of 0.18-0.23 mm. The size of every pixel is 2 mm. Other specifications of the prototype are shown in Table 1.  
Note that our prototype employs cylindrical lenslets as the light splitting elements; thus, the surface function of the lens _ms_ is:  
\[\frac{1}{\lambda}Q(ms)=exp\bigg{\{}-\frac{i\pi}{\lambda f}(\varepsilon-ms)^{2} \bigg{\}}. \tag{8}\]  
According to the parameters of the prototype and the Panum theory [7], the maximum SD that can be performed by this prototype is 1420 mm and the corresponding maximum screen parallax of this prototype is 18.5 mm. However, as there is imaging aberration by lenslets, viewers in margin areas may not experience the SD of 1420 mm. We can find the maximum viewing angle by using the model in Section 2. First, we calculate the diameter of light spots in the viewing plane. We study the light spots formed by two pixels in the display panel as a stereo pair with the maximum screen parallax 18.5 mm. Each pixel emits a spherical wave to the corresponding lenlets, and two light spots are formed in the viewing plane, with the distance of 65 mm. After being multiplied by the transmittance function of the lenlets (\(R=1\)), a diffraction integral is done to obtain the light intensity at a single point in the viewing plane according to Equation (6). In the simulation, only the light with the wavelength of 550 nm is considered. We calculated 512 \(\times\) 512 points in the area of 265 mm \(\times\) 265 mm in the viewing plane for each stereo pixel. The full width at half maximum (FWHM) value of the spot intensity is considered to be the width of the spot. We calculate the spots with the viewing angles from 0\({}^{\circ}\)-20\({}^{\circ}\), with an interval of 2\({}^{\circ}\). Some of the spots' shapes are shown in Figure 2, and to show the width difference of the spots at different viewing angles, we present their horizontal cross section of the spots' intensity in Figure 2. As we use cylindrical lenslets in the AD prototype, \"spots\" are actually lines, as shown in Figure 2. However, for convenience, we still use the name \"spot\" in the following parts. We further show the relationship between spot sizes and viewing angles in Figure 3. The similar factors of the perceived stereo pair at each viewing angle are calculated according to Equation (7). The relationship between the similar factors and the viewing angles are shown in Figure 4. As seen in Figure 4, the similar factor value remains larger than 0.95 when viewing angles increase from 0\({}^{\circ}\) and decreases sharply at the viewing angle of 16\({}^{\circ}\), which shows the distinct SD reduction effect of imperfect lenlets. This breaking point shows the viewing angle limitation caused only by lens aberration. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Viewing Distance (\(D\)) & 5000 mm \\ Pixel Pitch (\(t\)) & 2.5 mm \\ Lens Aperture (\(s\)) & 2 mm \\ Focal Length (\(f\)) & 38.46 mm \\ Number of views & 4 \\
3D Resolution & 288 \(\times\) 300 \\ Size & 2400 mm \(\times\) 1440 mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specifications of the autostereoscopic display prototype. According to the simulation result, with the sudden decrease of the similar factor when the viewing angle exceeds \(16^{\circ}\), viewers are expected to feel a sudden decrease of SD. A similar result can be obtained if the positions of the stereo pair pixels are fixed and the viewer moves along the \(x_{0}\) axis on the viewing plane. According to the geometrical relationship, if the viewer stands at point O\({}_{3}\), he/she will experience a full SD (the maximum stereo depth of the AD system) within the full width of the display panel, and the SD decreases sharply from the outside area of the screen.