Brief description of the optical principle of the hologram display

The optical principle of three dimensional representation of objects is based on a lenticular structure of vertical cylindrical lenses. At the focal line of the lenses a binary black/white pixel arrangement is locate on a color and brightness filter. A more detailed description of the arrangement is given in [1]. Behind them a background illumination is mounted. The lenticular structure has got a pitch of 500$\mu$m. Each black/white sub-pixel at focal layer has got a size of 5$\mu$m x 5$\mu$m. Hence 100 different views can be written horizontally side by side being emitted over the full angle 2$\beta$, cf. Fig. 1.

The focus of the lenses of the lenticular glass meets the plane, in which the digitally written black-white film or the liquid crystal layer is situated. Due to the high writing resolution of 5080 dpi, up to 100 different views of objects can be emitted over 100 neighbored ray angles. Geometrically, for each pixel a fan projection has to be calculated over a full fan angle of 2 $\beta$. This angle $\beta$ depends on the pitch $LS_{pitch}$ of the lenticular lenses, the focal depth point $Rl_{focus}$; and the light index of glass N (e.g. N=1.5). The horizontal pitch $FP_{pitch}$ on the focal plane is a little bit larger than that of the lenses itself.

$$FP_{pitch}=LS_{pitch}\left( 1+\frac{Rl_{focus}}{D_{obs}}\right)$$ (1)

where $D_{obs}$ denotes the optimal distance of the display screen from the observer's eyes, being equal to the distance between the z-position of lenticular plane and the z-position of the observer. The half fan angle $\beta$ is given by the following function:

$$\beta =\arctan \left( \frac{FP_{pitch}\cdot N}{2\cdot Rl_{focus}}\right)$$ (2)

Figure 1: Observer positions: viewing angles, full fan angle with respect to lenticular pitch.

For the system presented the following full fan angle is realized: $2\beta=19^{\circ}$. The number of views $N_{view}$ is distributed over the fan angle $2\beta$. Therefore the following view density at the observer distance $D_{obs}$ from the screen is achieved:

$$\Delta X_{obs}=\frac{2\cdot D_{obs}\cdot \tan \left( \beta \right) }{N_{view}}$$ (3)

For $N_{view}=100$ this density step is $VD=0.19^{\circ}$ per view. For each view point the image can be represented by a fan projection on the focal plane of the screen. Alltogether a number of $N_{view} \left(=100\right)$ fan projections have to be carried out. Fig.1 shows the relationship of fan angle, lense focus, pitches and optimal planned observer distance from the screen.

Figure 2: Fan projection onto the screen; sampling grid and 3D resolution

Fig.2 shows the fan projection for two special eye positions: the right view center and the left view center, being situated in the z-plane of the observer $z=Z_{obs}$. The screen is placed in the z-plane $z=Z_{screen}$. The pixels on the screen, on which the objects are projected, have the distance $LS_{pitch}$. This pitch is given by the width of the screen $SC_{width}$ divided by the number of horizontal pixels per view: $N_{pixh}$

$$LS_{pitch}=\frac{SC_{width}}{N_{pixh}}$$ (4)