Freeform Optics is a change in the lighting industry's ability to redirect light to a target area. Non-uniform rational B-splines, commonly referred to as NURBS, are widely used to represent free-form surfaces and surfaces.

Free-form surface optics is a change in the lighting industry's ability to redirect light to a target area. Non-uniform rational B-splines, commonly referred to as NURBS, are widely used to represent free-form surfaces and surfaces. There are some optical systems that require local modification of the surface during the design or optimization phase. In this case, NURBS cannot provide this conversion. But a new mathematical expression called T-splines makes this possible. Although its potential has been well described, it has not been implemented in any optimization procedures so far. Annie Shalom Isaac, Jiayi Long and Cornelius Neumann from the Karlsruhe Institute of Technology demonstrated the advantages of local refinement capabilities by performing T-splines in the optimization program and evaluated the results. The results show that the T-spline provides a more uniform and uniform light distribution and a faster convergence rate than NURBS. This makes optical design or optimization using T-splines an intuitive method for future free-form design tasks.

3x3 OFFD grids are closed and optical surfaces before (left) and (right) deformation

The design of free-form optics relies heavily on one of the following methods: based on point source assumptions [3], SMS design [4] and cropping of source target maps based on equal flux grids [5] to create an initial Optical surface. Because these mathematical methods do not guarantee accurate results for extended LED sources and do not provide a versatile solution, optical designers still rely on any optimization tool to improve results. The speed improvement in the ray tracing algorithm and the complex intelligent optimization algorithm make the application of the optimization method more extensive. However, the shortcomings of free-form surface optimization are mainly due to its complex mathematical expression and the existence of many parameters.

Wendel et. Al proposed a method called optimization, using free deformation (OFFD) to overcome this difficulty, placing the optical surface in the mesh and deforming the closed mesh instead of acting directly on the mesh [1]. This method uses NURBS to represent optical surfaces, and the results show that with less optimization variables, they can achieve global deformation well, which makes manufacturing easier. However, in some cases there is a significant tilt in the light distribution, or the path of the light must change significantly. In this case, a slight local deformation will bring about a significant improvement. But for the current OFFD, this is not possible because the surface is represented below. Another surface representation called T-spline can overcome this shortcoming [2]. Bailey et al. Al has demonstrated the potential of T-splines and its application to optical surfaces [6]. But so far, this method has not been applied to any optimization procedures, nor has its optical performance been analyzed and compared with NURBS.

Therefore, this work takes this issue into consideration and provides an alternative to solving this problem. Part 2 introduces the OFFD technology. The mathematical surface representation of the optical surface is described in Section 3. The results of the T-spline and the results of the comparison are shown in Section 4, followed by a conclusion in Section 5.

Optimize with OFFD

The OFFD method uses the free-form deformation (FFD) technique proposed by Sederberg [7] and incorporates an optimization program. The relationship between the mesh and the optical surface is well established using the FFD algorithm [7]. Figure 1 shows a grid with optical surfaces before and after deformation. For the sake of brevity, only an overview of the OFFD method will be introduced.

A 3x3 OFFD mesh envelop and optical surface (left) and back (right) deformation

The algorithm first selects an input surface whose optical performance must be improved, which is usually far from the target. In this way, the optical surface is enclosed within a grid of 27 grid control points, and the user can select any combination from them. This is provided as a variable to the optimization algorithm. The optimization algorithm has a wide search space for selected grid point combinations and provides movement along a three-dimensional closed grid. As the closed mesh changes, it also changes its internal optical surface. The deformed surface is then photometrically evaluated, and the optimization algorithm determines the variation of its optimized variables based on this result. The algorithm is repeated over and over until the target lighting requirements are met.

Free deformation optimized workflow (OFFD)

The most important step in this program is the definition of the quality factor of the deformed optical surface, since the entire optimization is based on this single value, called Q. In this article, we use two different evaluation functions.

Deviation evaluation function Qdev, which corresponds to the deviation of the current simulated distribution from the expected distribution and is expressed as

G is the area of ??interest, Eideal(x) is the desired illuminance distribution target, and E(x) is the current light distribution.

The flux-preferred function Q flux corresponds to maximization, which is quantized to the flux in the target region required for the ratio of the Φ flux, and the φC available flux is collected by the optics.

Mathematical representation of the optical surface

NURBS

NURBS technology is very mature and can be used in computer-aided graphics systems as well as ray tracers. Because of its flexibility, you can easily manipulate or modify surfaces by changing control points or their weight during the optimizer. The NURBS surface is a parameter tensor product surface and is defined as follows:

Where P ij is a rectangular array of control points, where P ij is a (n + 1) × (m + 1) matrix, wi, j is a weight, N ip (u) and N jq (v) are bases of u and p The function, the v direction, is associated with the knot vector, respectively.

Where r = p + n + 1 and s = m + q + 1 holds. When a control point must be added in NURBS, the junction insertion method is used to complete the control point. Adding a single node requires adding an entire column or a row of control points. It is also impossible to remove the knot without using NURBS without changing the geometry. This local refinement is mainly limited to NURBS because its tensor product structure is shown in EQ3. As shown in Figure 3, the NURBS surface is repeated line by column horizontally and vertically. To satisfy this balance, if you add a new control point, add an entire column or a row of control points.

Display an example of the initial 5x5 NURBS patch (left), how to add control points along rows and columns when NURBS (middle) and T-spline (right) are completed after modification.

T-spline

The disadvantage imposed by NURBS can be solved by an alternative mathematical representation of a freeform surface called a T-spline. The T-spline summarizes the B-spline and assigns specific row and column parameters to a particular control point by adding a T-junction to the associated B-spline in Figure 3. This makes the T-spline a more advanced technique in local deformation, an unwanted control point. T-splines are based on points rather than grid-based tensor product B-splines. The control grid is called the T grid, and the definition of the T-spline surface is given by

Where P i is the control point. N i(u,v) is a basic function and is given by

The basic functions N ui(u) and N vi(v) are associated with the knot vector, respectively

When a person inserts a new control point or a knot, the spacing between the other control points must be improved without changing its shape. This is done by satisfying EQ6 to refine the two univariate basis functions N ui(u) and N vi(v) respectively. This local refinement does not increase the number of control points, nor does it change the geometry so that the T-spline naturally becomes ideal for implementation in the OFFD and for local deformation.

The initial optical surface is divided into 6 sections (left side) and an OFFD grid (numbered) with numbered control vertices to prepare for local deformation.

Application of T-shaped spline in OFFD

The final section introduces the theoretical background of the T-spline and the advantages of using it in local deformation. This section describes the use of T-shaped splines in the free deformation system optimization program briefly introduced in Section 2. In our case, we used the same optical design task as the design streetlight lens used in [1]. A Cree XPG2 LED with 100 lumens was used as the light source, and the initial surface before optimization was as shown in FIG. In order to evaluate the photometric properties, the evaluation functions expressed in Equations 1 and 2 were applied.

To start the T-spline implementation in OFFD, you must add more control points to the desired area and the shape of the surface remains the same. More local deformations are achieved in areas where more and more dense control points are concentrated. A study was then conducted to find out which part of the lens the deformation process had more influence on, and if the sensitive local deformation on that part resulted in better results.

The entire optical surface is divided into six sections, as shown in Figure 4. This step is based on intuitive assumptions. The symmetry in the y direction is due to the fact that the target street and lights stay in the middle of the y direction (Fig. 5). For direct comparison, select the grid point [1, 3, 13, 15]. More control points are added to each of the segments from 1 to 6. Finally, six new optical surfaces were generated. The only difference between these generated new T-spline surfaces and the initial surface is the difference in the number of control points. The shape of the optics remains the same without any expected changes. These optical surfaces act one after the other as the initial surface of the OFFD and are evaluated using the merit function described. Preliminary results indicate that more impact on the light distribution can be seen when more control points are added to segment 5. Thus, an optical surface with more control points at segment 5 and fewer control points at the remaining segments is considered an initial system for optimization and compared to NURBS. This result is then compared to the NURBS-based OFFD and the results are discussed in the next section.

A schematic diagram of a street lighting installation with a pole distance of 10 m, an extremely high height of 6 m and a road distance of 1 m. The yellow rectangle shows the area to be illuminated.

Initial system

The optical properties of the initial surface of the performance are shown in Figure 6, with only 15% of the total internal flux of the target, and the distribution shape is away from the desired rectangular distribution marked white.

Illumination distribution of the target street area (white rectangle) of the initial surface displayed on the left side

T-spline using bias evaluation function (b) flux-based evaluation function (c) using bias-based NURBS and (d) flux-based evaluation function with more control on the fifth part using T-spline Illumination distribution of point street lamp lens.

Comparison of NURBS and T-spline

Two important photometric methods for comparing NURBS and T-splines for analyzing street lighting shots are the total luminous flux in the target, and uniform illumination is distributed over the target. NURBS and T-splines performed the total luminous flux in the target at the same level and found it to be 55%, as shown in Figures 7b and 7d, yielding a 40% improvement ([Delta]n). However, the performance of the T-spline in shaping the target distribution as needed is better than the simulation results in Figure 7a. As shown in Figure 7b, the illuminance distribution using the T-spline is more uniform than the distribution with NURBS. The deformed optical surface using NURBS and T-splines is shown in Figure 8. Slightly different is the T-spline near the edge, labeled as segment 5 in Figure 4.

Deformed optical surface (right) represented by a false color based on the OFFD NURBS (left), based on the T-spline (middle) and the shape change between the NURBS and the T-spline.

If NURBS is used to achieve the same result, the control points in the grid are more sensitive to the user's choice. These results are available when selected accurately, but at the cost of optimized run time - almost twice the time required to use a T-spline. The biggest advantage of using a T-spline is that when the user knows the local deformation of the optical surface, he can have minimal trouble with the choice of control points in the grid.

In conclusion

This work highlights the use of T-shaped splines by first achieving local deformation of the optical surface in the optimization program. The results show that a more uniform light distribution than NURBS can be obtained using T-splines. When using a T-spline, the sensitivity of the FFD grid points is reduced. This is very important for intelligent optimization systems. Since the T-spline curve is a more advanced surface representation, it is not yet mature. CAD technology and ray tracer have not yet reached the level of import and use the T-splines file format directly. So at the time of this article, people still need to rely on T-spline to convert back and forth to NURBS to perform ray tracing. Another limiting factor, but not very large, because the control points must be added more precisely where needed. (Source: led-professional)