## 2 Principle and Design of Wave Field Synthesis 3D Imaging System  
### Optical System Principle  
In a real environment, the surface of an object can be assumed to be composed of a huge number of dipoles. Each of them is excited when illuminated and becomes the source of dipole radiation. Because the dipoles are unordered, the scattered light can be approximated as a set of mutually incoherent spherical waves, as shown in Figure 1. This simple gedankenexperiment shows that spherical waves can be very useful as a polynomial basis to imitate and synthesize wave fields generated by natural scenes. Figure 1: The light source irradiates an object. The surface dipoles start to oscillate and emit mutually incoherent spherical waves as dipole radiation that can be detected by the observer. We will employ this principle by using the display scheme depicted by Figure 2. It consists of a liquid crystal display (LCD) and an lens array (LA). The lateral pixel position is given by \(x_{n}\) while the lateral position of the virtual object point is defined as \(x_{0}\). The wavefront behind the lens propagates in the direction determined by \(\alpha\). In the system, light originating from each LCD pixel will form a spherical wave. The curvature of the spherical waves, which also defines the distance to its origin, can be controlled by either changing the optical path between the LCD and the LA (the distance \(d\)) or, more conveniently, by changing the focal length of the lenses using a dynamic LA. The setup resembles an LF display. However, the basic principle is completely different. While an LF display seeks to generate rays (i.e., plane waves), we use a set of spherical waves to synthesize a wave field. Compared with rays, spherical waves exhibit the additional parameter of curvature. Therefore, the curvature has to be dynamically varied during the display process using one of the above-mentioned techniques. In our work, we change the curvature by mechanically varying the optical path between LCD and LA. Additionally, since the aim is to generate wide-field spherical waves rather than rays (narrow plane waves), the individual lenses of the LA exhibit a much larger aperture. If one of the spherical waves hits the eye of the observer, the eye lens adjusts properly to image the origin of the spherical wave, thus creating the sensation of a point floating in space. Multiple points can then be used to create surfaces of objects or even complex scenes. Different depth layers can be addressed by changing the curvature, while lateral coordinates can be selected by the pixel position on the LCD panel. For the rendering, we use a ray trace approach and follow the line that intersects the center of a lens and the center of a corresponding LCD pixel. Whether the pixel is switched on or not depends on the virtual distance \(d_{\varphi}\) to the center of the spherical wave. If it is in close vicinity to an object's surface and nothing is blocking the path (object occlusion), the pixel is switched on. For the renderer, it is, therefore, necessary to calculate \(d_{\varphi}\) from the distance \(d\) between the LCD and the LA and the focal length \(f\) of the lenses. The separability of the problem allows us to treat it in 2 dimensions, \(x\), and \(z\). To further simplify the calculation, the parabolic approximation is used to describe the complex amplitude \(U_{S}\) of light originating from a specific LCD pixel in front of its associated lens [22], which is given by  
\[U_{S}(x,y)=\frac{A_{n}}{d}\text{exp}\left[\text{i}kd+\text{i}\frac{k}{2d}(x-x_ {n})^{2}\right]. \tag{1}\]  
Figure 2: Schematic 3D display system and geometrical quantities. The lens manipulates the wavefront of the light emitted from the red pixel in a way, that the generated spherical wave cutout seems to originate from a point in the virtual distance \(d_{\varphi}\) and propagates in the direction determined by \(\alpha\). Multiple points can be used to create the sensation of a virtual object or even a scene.