## 3 Lenslet Array Homogenizer
### Lenslet Array Homogenization  
Although Kohler illumination can effectively provide the homogeneous illumination pattern, it cannot deform the light from any light source into a rectangular shape to match the dimension of the modulator without an aperture. Due to this reason, a basic Kohler illumination structure results in either a waste of light (with an aperture or a more massive illumination range than the modulator surface without an aperture) or partial use of the modulator's resolution (a smaller illumination range than the modulator surface). Therefore, another method for simultaneously shaping and homogenizing light is needed.  
For this purpose, LAs, which are based on Kohler illumination principle, can be applied in such headlamp systems. An LA homogenizer is constituted of a LA with a conventional lens [20]. The LA has multiple lenslets with identical dimensions, so that it firstly separates the incident light into beamlets and focus each beamlet on a focal plane. After that, the conventional lens located at the distance of its focal length with respect to the focal plane acts as the relay lens, as in Kohler illumination systems to collimate each beamlet and converge all parallel beam bundles in the homogenization plane. This homogenization plane is namely the display surface of the light modulator, and it is situated at  
Figure 2: Typical Köhler illumination principle in microscopy systems,it provides a homogeneous illumination pattern for the specimen by imaging the filament.  
the focal plane of the relay lens on the opposite side to the LA. Thus, each lenslet with the relay lens is working as an independent Kohler illumination system. A one-dimensional illustration of an LA homogenizer with the interplay of the geometry relationships is shown in Figure 3.  
On the left side of the focal plane in Figure 3, D represents the width of the incident beam in this dimension, d represents the dimension of each lenslet, and \(f_{1}\) is the focal length of each lenslet. The interplay of these parameters is:  
\[\tan\alpha=\frac{d}{2\cdot f_{1}} \tag{1}\]  
On the right side of the focal plane, the focal length can be determined once the dimension of the display surface of a modulator \(H\) is known. It can be calculated by:  
\[f_{2}=\frac{H}{2\cdot\tan\alpha} \tag{2}\]  
For different lighting modulators, there are different requirements on the incident angles \(\beta\) on the modulator surface. An appropriate incident angle helps the modulator to have a good efficiency and modulating performance [21; 22]. Once this incident angle is pre-defined, assuming \(n\) is the minimum number of lenslets to cover the incident beam, the focal length on the modulator side can be calculated, as below:  
\[f_{3}=\frac{(n-1)\times d}{2\cdot\tan\beta} \tag{3}\]  
For a rectangular lighting modulator that is two-dimensional, \(f_{1}\)s, \(f_{2}\)s and \(f_{3}\)s in both dimensions of the homogenizer are identical, and other geometric quantities of both dimensions can be calculated using Formulas (1)-(3). Therefore, it can be seen that each lenslet on the LA has the same aspect ratio as the modulator's display surface.  
In conclusion, the homogenizer divides the incident beam into multiple sub-beamlets with a specific aspect ratio in the first place. It superposes all of the beamlets on to the modulator with a series of defined geometric quantities. By doing so, both particular beam shaping and homogenization of the incident light can be realized. According to geometric optics, the more lenslets that are covered by the incident light bundle, the more homogeneous the superposition pattern will be.