## 5 Concepts for Inhomogeneous Beam Shaping
### Modification on Lenslet Dimension  
Another method to create the required illumination hotspot besides lengthening the focal length is through the modification on the lenslet dimension. The focal lengths of individual lenslets stay the same. When the dimension of a lenslet decreases, it results in a centralized distribution area, as illustrated in Figure 11.  
In this figure, \(d^{\prime}\) is the modified lenslet aperture dimension, which leads to a smaller image of itself with the size of \(H^{\prime}\) on the display surface. The inhomogeneous distribution pattern is therefore generated. This smaller lenslet dimension also leads to a smaller incident angle \(\alpha^{\prime}\) on the virtual focal  
Figure 10: Simulation model and its result for the hotspot generation based on the modification of focal length. (**a**) Simulation model; (**b**) Detector view located at the modulator plane [14].  
Figure 9: Principle of a focal length changed LA inhomogenizer [14].  
plane. Thus, the geometry relationships of the modified and the resulted parameters can be confirmed while using Formulas (8) and (9).  
\[\tan\alpha^{\prime}=\frac{d^{\prime}}{2\cdot f_{1}} \tag{8}\]  
\[f_{2}=\frac{H^{\prime}}{2\cdot\tan\alpha^{\prime}} \tag{9}\]  
A simulation model modified from that in Section 4 is also used for the validation of this method. The inner lenslets are half the size of the outer lenslets. The simulated system and its result are shown in Figure 12. The optical efficiency, according to the simulation, is 75.0% with PMMA components.  
Figure 13 shows acomparison of the simulation results with the high beam requirement. The dotted curve in the figure shows the requirement in terms of the minimum illuminance in a headlamp's FoV (field of view). The illuminance distribution of a headlamp must not be below this curve in the FoV. Both of the resulted distributions in Sections 5.1 and 5.2 are further shaped by a projection lens to achieve an FoV of \(\pm 15^{\circ}\) in the horizontal direction and \(\pm 5^{\circ}\) in the vertical direction. The simulation results are quantified to the same level of the regulation, and the comparison result is as illustrated. According to the comparison, the two hotspot generation methods can fulfill the requirements in the regulation for headlamp systems.  
Both of the methods described in Sections 5.1 and 5.2 are on the base of the fundamental principle of the LA homogenizer. By modifying this basic principle, centralized illumination hotspots can be customized. Depending on the specific design requirement of a high-resolution headlamp, optional methods for homogeneous or inhomogeneous lighting distribution can be correspondingly applied.  
Figure 11: Principle of a aperture size changed LA inhomogenizer.  
Figure 12: Simulation model and its result for the hotspot generation based on the modification of lenslet dimension. (**a**) Simulation model; (**b**) Detector view located at the modulator plane.  
A twisted nematic LCoS is selected as an example because of its potentiality in headlamp applications. First, an LCoS has high aperture ratios and reflectances, which are beneficial for the optical efficiency and the thermal management in high power systems, i.e., headlamp systems in the context [30]. Next, the pursuit of efficiency and compaction of a headlamp suggests the appliance of laser diodes as the light source [4]. A laser diode emits linearly polarized light, which fits the working mechanism of the LCoS well. The collaborative work of laser diodes and LCoS modulators can improve energy efficiency and contrast performance in a headlamp. Nevertheless, except for the LCoS, the two proposed methods for the central hot spot generation can be applied for other area-based modulators, e.g., DMDs and LCDs, in high-resolution headlamp systems.