1. Introduction

In 2008 Moreno and Sun [1] proposed an analytical model to describe the far field radiation pattern of a LED. It was shown that most radiation patterns could be modeled with the sum of two types of functions, the Gaussian type and the cosine-power type. All presented cases were modeled with the sum of maximum 3 functions, which is consistent with the authors’ previous statement that there are three main factors or components of the LED that influence the radiation pattern. These are the LED chip, micro-reflector and the primary lens (encapsulating silicon dome). This article will describe an attempt to apply the mathematical model of [1] to a LED with attached secondary optics as presented in Fig. 1. The application of the model would allow us to further pursuit the research of developing a fast and reliable computation method to aid in the design process of a light engine that consist of an array of LEDs and different secondary optics. This would then allow to fully customize and optimize the luminaire to meet the constraints given by the area of installation and the designer. As we will be dealing with the general illumination of real objects (objects and surfaces in rooms, streets, pedestrian ways, outdoor park areas, billboards, etc.) we can safely assume that all of these targets are at a distance that would correspond to the far-field radiometric model (distance to the target is larger than 20 mm). This simplifies the radiation pattern modeling as the normalized pattern in the far-field does not change with the source to target distance. Consequently, the radiation pattern in far-field is described as the luminous flux per solid angle in a given direction. The source can be then characterized as a point source. This corresponds to the method in which the LED optics manufacturers (LEDIL, Carclo, LEDlink, etc.)[2, 3, 4] describe the radiation pattern of the LEDs with attached secondary lenses.

 

Fig. 1 Schematic representation of a LED with mounted secondary optical element.

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The optics manufacturers usually provide the measured data of the far-field region of different lens-LED combinations. The data is provided in a standard photometric format. This will be our input and reference data for the analytic model. We will describe the standard digital photometric formats in detail in Section 2. We give our modification made to the analytical model proposed by Moreno and Sun in Section 3 and then describe the use of the model with different fitting algorithms in Section 4. In Section 5 we will showcase eleven real lenses from LEDIL, compare the analytical models with the input data and finally give our conclusions and future work expectations in Section 6.

2. Standard photometric data formats

Currently there are two main digital photometric data formats. The IESNA (.ies) [5] format used mainly in the USA, and the European ELUMDAM (.ldt) [6] file format. Both formats transfer data in almost the same way. The only difference is in the coding of the luminaire general information data. The IESNA format uses special code words in the beginning rows of the file to incorporate the information about the manufacturer name, date, product name/number, laboratory test information etc. There is no limit to the number of information rows. After the last code word the data in rows bellow is specified in the standard way and is not allowed to be written differently. In the contrast the ELUMDAT file format has a fixed and specified content in every row from the beginning till the end. Also it is not allowed to skip any rows because the file format depends on the row number count. In both standards, the radiation pattern data is the same. Basically the radiation pattern is described with spatial vectors in spherical coordinates which all have a common source point that in fact is the fixed space point of the measurement. Every vector has three components, the azimuthal angle, the polar angle and the luminous intensity at that angle combination (presented as the radial distance in the graphical plot). The number of vectors depends on the complexity of the pattern and the demanded accuracy. In general, a minimum of 3240 (45 polar on 72 azimuthal) vectors are provided. The radial distance at every vector is determined as the luminous intensity at that angle set. The luminous intensity is measured absolutely but in most cases normalized with the luminous flux. This step is done to enable the comparison between radiation patterns with different absolute values (important for visual chart plot comparison). In this article we will use the standard data format (.ies) provided by the optics manufacturer LEDIL, to obtain the input data for the model curve fit. First the algorithm will denormalize the data and extract the list of vectors that will be used in the fitting process. The fitting will be accomplished with different algorithms which will utilize different heuristics and meta-heuristics. We shall describe those briefly in Section 4.

3. Modification of the mathematical model (Moreno and Sun, 2008)

In order to simplify the implementation of the algorithms, we slightly modify the model of Moreno and Sun. A new normalizing parameter is introduced, and consequently, all other parameters will have values in fixed intervals known in advance. It should be noted that the modified model is equivalent to the original, only the number of parameters and their meaning differ. In particular, it may be interesting to note that although we apply the model to a different phenomena, we will see that it works perfectly on LED’s with symmetric light distribution. The proposed model is a phenomenological model based on the light propagation characteristics of the LED (single or multi chip). Phenomenological models are mathematical models that are developed and based on experience gained from observing a real physical entity. These models are of up-most importance in all branches of scientific practice, because they link the theory with the real phenomenon. When observing the light emission from the LED at the microscopic level Moreno and Sun [1] found out that the emitted light is incoherent and its superposition is a linear combination of intensity. From photometric point of view, the source is a Lambertian type light source. However, the LED consist of different optical elements which play a role in shaping the final radiation pattern, as does the surface quality of that elements. In general there are three main elements. The source chip, collimating reflector and the silicone lens. The surface of the chip element is relatively rough which means that the emitting pattern will be likely Lambertian (cosine power). This pattern is then influenced by the collimating micro-reflector and the primary lens which finalizes the shape of the output. Taking all elements into account it can be determined that this combination generates a cosine power or Gaussian like pattern. Remembering that there are three main light guiding elements in the LED, Moreno and Sun deducted that the sum of three functions will probably be enough to model the pattern in most cases [1]. Hence i = 1, 2, 3 is sufficient in the model.

Here I (Θ) stands for the measured Luminous intensity at the polar angle of Θ, i is the number of function to sum and a_i, b_i, c_i are the function coefficients that we search for. We use a slight modification of the model (1), as explained below. The fitting test on two symmetric lenses from LEDIL showed that it should be possible to utilize the models on LED with secondary lens. The test showed that the sum of cosine power functions was able to describe the pattern more accurately than the Gaussian and was more intuitive to manual fit. This is the reason why we have decided to further investigate the possibilities of the cosine functions. We have determined that by adding a normalizing factor in the sum of functions we can limit the coefficient values inside specific intervals. Consequently, there will be three types of coefficient parameters, a_★ ∈ [0, 1], b_★ ∈ [−90, 90], and c_★ ∈ [0, 100]. Note that the coefficient parameters in the model are continous, so a feasible value can be any real number from the corresponding interval. However, in the optimization process we will consider a discretization of the interval. Based on the phenomena described above and the specific lens design for every LED we can again limit the sum to maximum of three functions. This can be supported by the fact that the lens manufactures optimize the inner surface of the lens in a way that this surface has a minimal impact on the efficacy and light bending of the LED output. This means that the rays passing out of the primary lens into the secondary loose minimal energy and a very small percentage of them is reflected. Accordingly it can be observed that both lenses act as a joint entity so they can be described as one and represented by one cosine power function.

4. Fitting algorithms

The mean square error defined above implies that we are solving a data fitting problem with error defined to be $\mathit{RMS}=\sqrt{M^{-1}}\sqrt{\mathit{LS}}$ where LS is the sum of the squares of the errors made. Hence minimization of RMS is equivalent to data fitting by the standard method of least squares. As there is no closed-form solution to a non-linear least squares problem, numerical algorithms have to be used to find the value of the parameters. The problem at hand is a continuous optimization problem and hence, compared to discrete optimization, there are even more possibilities to define a neighborhood for the local search based heuristics. Local search heuristics are basic heuristics both in discrete [7] and continuous optimization [8]. In fact, the neighborhoods used can be seen as variable neighborhoods, although they are all similar [9, 10]. We have started our experiments with two basic local search algorithms, steepest descent (SD) and iterative improvement (IF), where in both cases the neighborhoods were defined in the same way. We call this the fixed step-size neighborhood. The third local search algorithm is a variation of iterative improvement (IR) where we introduce random step-size, roughly speaking, given a step-size and direction as before, we randomly make a step in the direction that is at most as long as in the fixed size neighborhood search. Naturally, whenever local search is used, the multistart version is worth consideration. As preliminary testing of multistart version was not competitive with single longer runs, therefore we decided to use a more advanced heuristics that would on one hand take advantage of the seemingly successful local search and possibly accumulate information obtained by independent local searches. Our choice was to use a genetic algorithm (GA). Finally, we ran and compare results of a simple generation of random solutions (RAN). Table 1 shows the RMS error values calculated with different heuristics for eleven Ledil symmetric lenses, marking the best two values in bold. For more details, see [11]. LEDIL was chosen because it is one of the worlds biggest lens producers, with a large repository of different lenses for almost all LED types. That enabled us to choose several different lenses (11 are presented) to thoroughly test the algorithms.

Table 1. Calculated RMS error for eleven lenses presented in % according to (3) with four million iterations which took approximately 30 minutes CPU time for each lens on a Core I3 4130 @ 3,4 Ghz (a standard home PC). Best two results in each row are emphasized.

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5. Results

Table 1 shows the quality comparison of different types of algorithm on a particular lens. We can observe that all of the lenses are modeled to have the RMS value lower than the limit of 5%. This means that all lenses are appropriately modeled according to the RMS value. Also most of the algorithms found good solutions, but the best one was the IF - iterative improvement algorithm that found the best solution in six instances and was in ten out of eleven instances one of the best two algorithms. An interesting instance is the CP12632 on which the best solutions was found with the RAN - random search algorithm and the second best was the GA - Genetic algorithm. This might be explained with the nature of the search of the two winning algorithms. It appears that the search space of CP12362 contains a lot of ”sink-holes” that do not provide a good local optima. In the case of algorithms with steepest descend or iterative improvement they cannot escape the hole once they are in it, but the Random search algorithm has no such restriction nor does the Genetic algorithm. More on the algorithms can be found in [11]. The low RMS values ensure that overall average approximation is good. However, what may be of more interest and surely worth observing is the maximal fitting error. The graphical presentation of the results on several (randomly selected) examples is simple and intuitive enough to be evaluated visually. Figures 2 and 3 show five lenses from the Table 1 rendered in 2D and 3D polar views. The 2D view presents the comparison among the best modeled function curve (in blue), worst modeled function curve (in green) and the source points (in red) from the photometric data.

 

Fig. 2 Above are presented the radiation patterns for three lenses. Viewed form the top to the bottom are C13353(a), CA11268(b), CA12392(c). On the left side is the comparison of the best modeled curve of each lens in blue, worst modeled curve in green and the measured data points from the photometry in red (relative intensity values). On the right side is the 3D representation of the best modeled solution and the function coefficients for that curve.

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Fig. 3 Above are presented the radiation patterns for two lenses. Viewed form the top to the bottom are CA11934(a), FP13030(b). On the left side is the comparison of the best modeled curve of each lens in blue, worst modeled curve in green and the measured data points from the photometry in red (relative intensity values). On the right side is the 3D representation of the best modeled solution and the function coefficients for that curve.

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6. Conclusion

In this paper it was demonstrated that the mathematical model (Moreno and Sun, 2008) can be applied to LEDs with attached secondary optics to model an appropriate spatial radiation distribution using functions. In addition, a minor modification to the model was applied with intention to limit the search space size. The change had a positive effect when used in combination with different fitting algorithms. Our experiments have shown that all four fitting algorithms that were tested provided acceptable solutions in short time. They all produced appropriate function parameters that in fact produced appropriate radiation patterns. The encouraging results presented here open a challenging avenue of further research. The natural next step is to extend the model in a way that it will be possible to fully describe free-form asymmetric light distributions, which is expected to be a very challenging task. The result may then be used in the study which has the ultimate goal of designing an optimization tool. The tool would take an available set of lenses and a user defined radiation pattern as an input and then return a combination of lenses that would achieve the desired pattern.