# Displays Based on Dynamic Phase-Only Holography
## 5 System Application
Due to the many-to-one correspondence between the pixels in the SLM and the points in the image, there is no longer a direct relationship between the pixels on the LCOS device and the pixels in the replay field. The size of the pixels in the replay field is determined by the physical process of optical diffraction. The aperture of the LCOS device is the limiting aperture so that the replay field pixels are ellipses with their long axis vertical, which corresponds to the short axis of the LCOS. When the illumination of the LCOS is uniform, these spots are sinc squared functions. This results in the most compact spot. Uniform illumination with no concomitant light loss is an \"ideal\" objective, which is difficult to achieve in practice. In order to assess the approaches to beam shaping, [PERSON] and [PERSON] introduced a dimensionless parameter \(\beta\)[26]. \[\beta=\frac{2\pi RD}{\lambda f}\]  
where \(R\) is the dimension of the input beam, \(D\) is the dimension of the output beam, \(\lambda\) is the wavelength, and f is the distance between input and output, which is commonly the focal length of the collimating lens. The \(1/\mathrm{e}^{2}\) radius of the Gaussian waist is a convenient measure for \(R\) and the half width of the LCOS aperture is convenient for \(D\). The small \(1/\mathrm{e}^{2}\) radius of a laser diode favors large radius output beams. The relatively small aperture of the LCOS leads to a small \(\beta\). The design of the beam shaping optics becomes more difficult as \(\beta\) decreases. The collimation optic design minimizes both the loss of intensity and the wave front error at the LCOS. The latter minimizes spot size increases due to pupil aberrations. A good compromise is to expand the Gaussian beam profile and apodise it using the LCOS aperture. The illumination beam profile is an elliptical Gaussian beam with a 1/e\({}^{2}\) waist radius, \(w_{1}\), at the corner of the LCOS. The horizontal and vertical waists, \(w_{1x}\) and \(w_{1y}\), are in the same ratio as the rectangular aperture of the LCOS (approx. 2:1) and the perimeter of the active area is 0.707 \(w_{1x}\) and 0.707 \(w_{1y}\). The spot size (3 dB diameter) will then be seen in the equation below [27]. \[\text{Spot size}_{x}=\frac{1.4\lambda\times f}{1.414w_{1x}}\]  
\[\text{Spot size}_{y}=\frac{1.4\lambda\times f}{1.414w_{1y}}\]  
The power captured by the LCOS aperture at this apodisation is 71% of the incident beam. One of the differentiating features of a phase-only holographic display is its ability to redirect light in the formation of the target image. For low information content images such as those used in head up displays, this causes a concentration of the light energy in the utilized pixels, which significantly increases the brightness of the illuminated areas of the display. This is termed \"Holographic Gain.\" Taking the reciprocal of the pixel utilization approximates the Holographic Gain. A typical figure for the Holographic Gain in a head up display application is 5. The size and resolution of a device such as an SLM can be expressed in a single parameter, which is a dimensionless product of the two and the Space Bandwidth Product (SBWP). The size of the LCOS device shown in Figure 7 is L\(\Delta\)uxM\(\Delta\)v. The resolution in the \(x\)-direction is \((\Delta\)u\()^{-1}\) and the resolution in the \(y\)-direction is \((\Delta\)v\()^{-1}\). Therefore, the SBWP is equal to the number of pixels in the device, LxM, when the resolution is not reduced by, for example, fringing fields. The spot separation in the replay field can be controlled by hologram repetition [28], which is also illustrated in Figure 7. If the size of the hologram is C \(\times\) B pixels within an LCOS aperture of M \(\times\) L pixels, then the repetition ratios are M/C and L/B in the \(x\)-axes and \(y\)-axes. We employ a repetition ratio of around 1.4 to give a pleasing replay field display.