Incoherent beam shaping plays a very important role in the areas of non-imaging optics and is applied today in different fields: light illumination, solar concentrators, multimode laser marking system, etc. Different types of optical component are employed for reshaping incoherent light beam: freeform system ^[1-5], digital mirror devices ^[6-8] or calculated roughness structure ^[9]. For example, one single free-shape mirror can be designed with a freeform shape, but the calculations are sometimes huge with complex surfaces ^[1,3]. Digital micro-mirror devices in connection with a telescope system are also used to reshape the incoherent light ^[6-8], but arbitrary targets without symmetry have not been reported in a static approach. Weyrich et al. obtain the custom reflectance by series of algorithms, in order to transform the highlight shape to produce the height fields ^[9]. Whereas the performance seems adapted to specific fields, the previous solutions seem difficult to implement for custom illumination system. We can note also the use of Fresnel lenses or equivalent structure in the field of solar energy where the idea of beam shaping is currently employed and a notion of symmetry appeared ^[10]. Another close case is the beam shaping of LED with lightpipe, but here, the goal is to obtain uniform illumination ^[11]. The method of design is really placed in the general case where the optical component and the result have no symmetry and the source is totally incoherent. The choice of the irradiance distribution is here two focalized spots to demonstrate the concepts and see the limitations.

We develop an optimization method to reshape incoherent light using Zemax OpticStudio 16 (Zemax) with a non-sequential approach ^[12]. At each loop inside the optimization process, the sum of the irradiance data on the non-target area of the detector is calculated automatically with the internal ZPL macro language. The two dimensional tilt angles of facets are modified iteratively by minimizing the sum of this irradiance value, while maximizing the target illumination. When the sum of the irradiance value is smaller to a preset standard minimum, the desired irradiance distribution is achieved. The verification of the optimization result is also realized with LightTools Illumination System Design 18.2 from Synopsys.

Using beam quality criteria in diffractive optics ^[13-16], the quality of the irradiance distribution is assessed in this paper by four factors including efficiency, uniformity, the root mean square and the correlation coefficient. Results on irradiance distribution are compared between Zemax and LightTools. The comparison is done by transferring the optimized model from Zemax into LightTools and setting up the same optical system. Identical non-sequential has been performed in both software tools leading to a more convincing comparison of the performances. Thus it is more convincing for providing the optimized model and to compare the performance.

As we will see, several parameters play a very important role in the quality of the irradiance. Among these are the wavelength, the incidence angle, the divergent angle and the facet size. The influence of these parameters is evaluated through the quality factors to see the real performance of our new component. Discussions about the acceptable tolerances will also be made with some recommendations.

In Section 2, details are given about the optical model. In Section 3, the optimization method and the optimization results are described. In Section 4, the numerical measurement theory is stated. Section 5 evaluates the influence of the parameters based on the beam quality assessment both with Zemax and LightTools.

The optical configuration is composed of an incoherent light source (ILS), the faceted surface (FS) and the detector (D). To avoid the influence of the dispersion, the material of the faceted surface is defined to be a perfect mirror. The reflective sample is made of a number of square facets with an equal size. As the tilt angles of each facet are zero before the optimization, the reflective surface in the initial configuration is flat. As we aim at redistributing the incoherent light beam, the size of the facet needs to be large enough in respect to the wavelength in the case of the non-sequential optics. For the feasibility of the following optimization process, the distance between the detector and the faceted surface is chosen to be 10 times larger than the size of the complete faceted surface.

As shown in Figure 1, the width of the incidence beam L is defined by two parameters, the width in X direction L_x and the width in Y direction L_y .Thecenter of the coordinate system is placed at the center of the faceted structure (FS). The divergence angle A₂ is also determined by the angle in X direction A_2x and theangle in Y direction A_2y .In the initial configuration, A_2x and A_2y are defined to be zero. In our study, the incidence angle A₁ is fixed to be 45 degrees, but this angle could be changed. In our choice of design, the reflective surface is made of 6×6 facets, and the size of each facet is 2 mm: the total size is then 12 mm by 12 mm. For totally fitting the overall reflective surface, the width of the source in direction X and Y needs to comply with the mathematical relation as, L_x² =W²/2, L_y =W. The size of the incidence beam is thus 8.5 mm by 12 mm. The incidence beam illuminates the faceted surface uniformly. The number of rays for the non-sequential ray tracing is 1 million. The number of pixels on the detector is 200 by 200 and the size of detector is 100 mm by 100 mm. The total power of the source is 1 Watt. The data type is incoherent irradiance, whose unit is watts/cm². The choice of all these parameters is explained by a good convergence of the optimization algorithm explained below. Before optimization, the initial irradiance distribution on the detector is rectangular with the same size of the incidence beam. After iterative optimization of the tilt angles of the facets, different kinds of irradiance distribution could be achieved. The optimization process will be depicted in the following section.

For measuring the quality of the achieved irradiance distribution on the detector, numerical calculation is done with these four factors: the efficiency, the root mean square error, the uniformity and the correlation coefficient.

The efficiency η is given by Equation (3):

In Equation (3), I_n,m is the actual irradiance distribution on the detector. R is the target area. N and M are the one-dimensional number of pixels on detector. In our case, N = M = 200. The efficiency η is the ratio of optical power in the target area to the total incidence power on the detector: it is a measurement of the useful energy in the irradiance map. In our case, we consider in the calculation any other energy losses as the reflective coefficient or roughness.

The root mean square error ε_i is given in Equation (4) for measuring a mathematical distance between the target and the achieved irradiance distribution ^{[13, 14]}. A is the number of pixels in the target area R. O_n,m is the desired irradiance distribution. The term a_i is a constant that normalizes the calculated irradiance map in order to get consistent error values. The root mean square error in intensity is the most straightforward criterion to evaluate the quality result, and thus the convergence rate of the optimization process. The equation is given by:

where

The uniformity U is given by Equation (6), which is the standard deviation of the difference between the actual irradiance and the mean irradiance e_m^[14] :

with the value e_m is the average of the irradiance in the target area:

The illumination uniformity U is small if the difference between irradiance at each point and the mean value e_m is small. Bokor et al. provide an equation for measuring the fidelity of an achieved irradiance distribution ^[13-15]. They introduce a mathematical quantity which measures the correlation between the calculated irradiance map and the desired irradiance. This correlation coefficient is always in the interval [0,1] The expression of the correlation coefficient C is given in the reference ^[13].

The quality factors will be used in the following sections to measure the quality of the irradiance distribution on the detector and evaluate the influence of several important parameters on the reshaping result of the incoherent beam. These four factors contain the required information to analyze the irradiance map properties. It is important to note that the analysis in the next section should always be done by observing the 4 criteria together. Indeed, one can easily notice that the behaviour of efficiency and correlation coefficient are linked together, and so are also the uniformity and the error.

Of course, in some applications, factors are more relevant than others, for example uniformity in high quality illumination. The most important in studying the quality factor is to observe the evolution of these values in a large interval, knowing that only efficiency is possible to link to the experiment and easily measurable.

In conclusion, we demonstrate in this paper the design of a new type of optical component containing only 6 x 6 facets to do beam shaping of incoherent light source to obtain two focalized spots. An automatic optimization process using internal macro language is explained and result is presented with a simple but non-standard irradiance map. Numerical measurements are made to analyze the influence of different parameters on the optimized result. We show also that the numerical simulations under Zemax OpticStudio 16 are comparable to those obtained with LightTools 8.2. This study shows us the limitation in using our optical component. First, the incidence angle of the incoherent light beam needs to be 45° ± 1° range. The divergence angle of the light beam should be smaller than 0.5°. The acceptable deviation from the size of the facet must be more accurate than 0.5 mm. The choice of aluminium as material for the facet is optimal for visible spectrum, but the total reflectivity is smaller than 40 %. This last value can be probably improved by changing the optimization process employing as constraints the total reflectivity for a future goal and not the number of illuminated pixels. The tolerance of the angular position of the facet is about 0.1 degree. However, this optical component has advantages comparing diffractive optics or other type of solution for beam shaping where tolerances of fabrication are sometimes harder and difficult to obtain. The influence of the wavelength is low and only related to the Fresnel losses if the quality of the surface is good. The flexibility of the design and the optimization process prove that this component can replace other type of solution in this field of incoherent light technology or nonimaging optics.