## 1. Introduction

Nowadays, light-emitting diodes (LEDs) are playing an increasingly important role in the world market of lighting devices. To obtain the highest optical performance, LED–based light sources are used with secondary optics – refractive or reflective optical elements redirecting the emitted light flux into the illuminated area. Computation of the optical surfaces (refractive or reflective) producing a required light distribution is a challenging problem. Even in the case of point light source and single optical surface the problem reduces to solving a complex nonlinear differential equation in partial derivatives of Monge–Ampère type [1, 2]. Analytical solutions of this equation are known only for simple cases with rotational or cylindrical symmetry [3–5]. For generation of complex asymmetrical light distributions (e.g. in rectangular, hexagonal, elliptical regions), the optical elements with free-form surfaces are used. Many different optimization methods were developed for the computation of such surfaces [6–9]. Optical elements with a single working free-form surface [6–9] provide good optical performance in case of light distributions having medium or wide angular size (80° and more). But for the generation of narrow light beams such optical elements are inefficient since the deviation of lateral rays by single refractive surface is performed with a high Fresnel loss (e.g. for a ray incident at an angle of 41.4° on the PMMA-air interface the deviation angle is about 40° only and the Fresnel loss is more than 45%). To improve the lighting efficacy of the optical elements, two working free-form refractive surfaces (inner and outer) are usually used [10–16].

Paper [11] presents Cartesian ovals method for computing piecewise-smooth surface for the generation of required illuminance in an arbitrary set of points. The application of the presented method for generation of continuous illuminance distributions is problematic due to a high computational cost. Also, while this method takes into account the presence of several refractive surfaces in the optical system, it is intended for the design of a single surface and cannot be used for simultaneous design of multiple surfaces. In paper [12] Simultaneous Multiple Surface design method is presented. It allows computing two refractive or reflective surfaces performing required transformation of two incident wave fronts into two outgoing wave fronts. Reducing of arbitrary illumination problem with extended light source to the problem of wave fronts transformation is not formalized in [12], so its usage for the generation of a required illuminance pattern is quite challenging in most cases.

In papers [13, 14], the authors consider a general approach to the design of optical elements with two free-form surfaces for shaping the light beam from a point source. The solution consists of two parts: computation of the ray-correspondence function and reconstruction of the optical surfaces implementing the obtained ray-correspondence. Unfortunately, the description of the approach presented in [13, 14] is too brief that makes it impossible to repeat the presented results. In particular, the authors in [13, 14] state that the utilization of two free-form surfaces enables infinite set of solutions and the most efficient one (or the most compact) can be chosen from this set, but the algorithm of choosing the solution is not discussed. Thus, the problem of computing light-efficient optical elements with two working surfaces remains relevant.

In this study, a novel optimization method for designing double-surface free-form optical elements is presented. As an example, a solution for generation of uniformly illuminated rectangular area with the angular size of 60° by 40° is computed. Simulation data shows that the light efficacy of the optical element is about 88.5%, while the nonuniformity of the generated illuminance distribution is 8.4%.

## 2. Formulation of the problem

Let us consider the design of a refractive optical element with two free-form surfaces generating a prescribed illuminance distribution 𝐸0(𝑢,𝑣) in the exit plane𝑧=𝑓, where (𝑢,𝑣) are local Cartesian coordinates. The point light source with intensity function 𝐼(𝜑,𝜓) is located at the origin of coordinates. The inner surface of the optical element is defined by its radius-vector 𝐑(𝜑,𝜓). The outer surface is given by scalar function 𝑙(𝜑,𝜓) that defines the distance between the point 𝐑(𝜑,𝜓) of the inner surface and the point 𝐌(𝜑,𝜓) of the outer surface along the direction of the ray 𝐚1(𝜑,𝜓) refracted on the first surface (Fig. 1). Thus, the outer surface is represented by the following vector equation:

𝐌(𝜑,𝜓)=𝐑(𝜑,𝜓)+𝑙(𝜑,𝜓)𝐚1(𝜑,𝜓).

The above definition of the outer surface reduces the design problem to the computation of two scalar functions 𝑅(𝜑,𝜓)=|𝐑(𝜑,𝜓)| and 𝑙(𝜑,𝜓).Fig. 1

Arrangement of light source, optical element and exit plane.

## 3. Design of free-form surfaces

The proposed design method consists of two steps: the analytical computation of the initial surfaces and the subsequent optimization of their parameters using obtained analytical expression for the estimation of generated illuminance distribution.

#### 3.1. Computation of initial surfaces

The convergence of any optimization procedure significantly depends on the chosen starting point. To design the initial surface, we propose to use an analytical solution of a similar axisymmetrical problem [4] to compute several ″basic″ profiles of the free-form optical element. Furthermore, the initial inner and outer surfaces are obtained by fitting scalar bicubic spline functions [17] 𝑅(𝜑,𝜓;𝐜𝑅) and 𝑙(𝜑,𝜓;𝐜𝑙) on the computed 2D profiles, where 𝐜𝑅 and 𝐜𝑙 are the vectors containing the spline parameters. Thus, in this case the specified initial surfaces of the optical element are defined by two scalar functions 𝑅(𝜑,𝜓;𝐜𝑅) and 𝑙(𝜑,𝜓;𝐜𝑙).

As an example, let us consider the computation of the initial approximation for the optical element generating a uniformly illuminated rectangular area with the size of 1200 mm by 750 mm at the distance of 1000 mm from the light source (such linear sizes correspond to the angular size of 60° by 40°). At first, using the analytical method of [4] we calculate two different profiles of the axisymmetrical surfaces generating uniformly illuminated circles with diameters of 1200 mm and 750 mm (Figs. 2(a) and 2(b)). The initial points of the calculated profiles coincide on the axis 𝑧 (at 𝑧=0.8 mm and 𝑧=8 mm) that allows placing the profiles in two mutually perpendicular planes as shown in Fig. 2(c). Figure 2(d) demonstrates fitted spline surfaces containing the computed profiles. The latitude and longitude lines show borders between the patches of the fitted bicubic spline. Thus, the outer spline surface contains 8 patches in 𝜑 direction and 7 patches in 𝜓 direction. Each patch has 16 parameters including the values of the function (𝑅(𝜑,𝜓) or 𝑙(𝜑,𝜓) for inner or outer surface, correspondingly), its first and mixed derivatives in four corner nodes [6]. Due to common borders of neighbor patches and symmetry properties, the overall number of independent parameters is 56 for the considered case.

Fig. 2

(a) and (b) profiles of axisymmetrical surfaces generating uniformly illuminated circular areas with diameters of 1200 mm and 750 mm, respectively; (c) combined profiles; (d) fitted spline surface.

Illuminance distribution generated by the optical element in Fig. 2(d) with a point source is shown in Fig. 3. The obtained spot has elliptical shape; its sizes are close to the overall dimensions of the required rectangular spot. The generated illuminance distribution is not uniform and requires further optimization.

Fig. 3

Illuminance distribution generated by the optical element in Fig. 2(d) for point light source: (a) grayscale distribution; (b) profiles of illuminance distribution (solid line – v = 0, dashed line – u = 0).

#### 3.2. Optimization of the optical surface

As a rule, the illuminance distribution produced by the initial surfaces differs from the required one, and the parameters of the surfaces need to be optimized. We propose to use sequential quadratic programming method to minimize the relative-root-mean-square deviation (RRMSD) of the generated light distribution from the required one. This merit function is represented by the following expression:

𝜀(𝐜)=‖𝐸(𝑢,𝑣;𝐜)−𝐸0(𝑢,𝑣)‖=1‖𝑆‖∬𝑆(𝐸0(𝑢,𝑣)−𝐸(𝑢,𝑣;𝐜))2𝑑𝑢𝑑𝑣−−−−−−−−−−−−−−−−−−−−−−−−−−−−⎷→min𝐜,

where ‖𝑆‖ is the area of the illuminated region, 𝐸(𝑢,𝑣;𝐜) is the illuminance distribution generated by the optical element and 𝐜 is the vector of parameters describing the spline surfaces and containing the parameters 𝐜𝑅 and 𝐜𝑙 of inner and outer surfaces, respectively.The solution of the problem (2) requires an effective way to compute the illuminance distribution 𝐸(𝑢,𝑣;𝐜) generated by the optical element. Analytical expression for the illuminance distribution 𝐸(𝑢,𝑣;𝐜) was derived previously in paper [6] and has the following form:

𝐸(𝑢,𝑣;𝐜)=∬𝜑,𝜓𝐼(𝜑,𝜓)sin(𝜓) 𝛿𝜎(𝑢−𝑢̃(𝜑,𝜓;𝐜),𝑣−𝑣̃(𝜑,𝜓;𝐜))𝑑𝜑𝑑𝜓,

where 𝛿𝜎(𝑢−𝑢̃,𝑣−𝑣̃)=1𝜋𝜎2exp(−(𝑢−𝑢̃)2+(𝑣−𝑣̃)2𝜎2). The functions 𝑢̃(𝜑,𝜓;𝐜) and 𝑣̃(𝜑,𝜓;𝐜) in Eq. (3) represent the coordinates of intersection between the ray emitted in the direction (𝜑,𝜓) and the exit plane 𝑧=𝑓. The proposed definition of the outer surface through the scalar functions 𝑅(𝜑,𝜓;𝐜) and 𝑙(𝜑,𝜓;𝐜) makes possible the derivation of analytical expressions for the coordinates 𝑢̃(𝜑,𝜓;𝐜) and 𝑣̃(𝜑,𝜓;𝐜).Unit vector of the ray emitted in the direction (𝜑,𝜓) is 𝐚0(𝜑,𝜓)=𝐑(𝜑,𝜓;𝐜𝑅)/𝑅(𝜑,𝜓;𝐜𝑅). According to the Snell’s law, the vector of the ray refracted on the inner surface can be written as

𝐚1=𝑛𝐚0+(1−[𝑛𝐚0,𝐧1]−−−−−−−−−−−√2−𝑛(𝐚0,𝐧1))𝐧1,

where 𝐧1(𝜑,𝜓)=[∂𝐑∂𝜑,∂𝐑∂𝜓]/|[∂𝐑∂𝜑,∂𝐑∂𝜓]| is the unit normal vector to the inner surface at the point 𝐑(𝜑,𝜓) and 𝑛 is the refractive index of the optical element. As the point 𝐑(𝜑,𝜓;𝐜) is the start of the refracted ray, the intersection point of the ray and the outer surface is given by Eq. (1). To derive the vector of the ray refracted on the outer surface, let us again use the Snell’s law:𝐚2=𝑛𝐚1+(1−[𝑛𝐚1,𝐧2]−−−−−−−−−−−√2−𝑛(𝐚1,𝐧2))𝐧2

,

where 𝐧2(𝜑,𝜓)=[∂𝐌∂𝜑,∂𝐌∂𝜓]/|[∂𝐌∂𝜑,∂𝐌∂𝜓]| is a unit normal vector to the outer surface at the point 𝐌(𝜑,𝜓). After the computation of the ray vector 𝐚2(𝜑,𝜓) the intersection point of the ray and the exit plane can be written as:𝑢̃(𝜑,𝜓;𝐜)=𝑅(𝜑,𝜓;𝐜𝑅)𝑎0𝑥+𝑙(𝜑,𝜓;𝐜𝑙)𝑎1𝑥+𝑑(𝜑,𝜓;𝐜)𝑎2𝑥,𝑣̃(𝜑,𝜓;𝐜)=𝑅(𝜑,𝜓;𝐜𝑅)𝑎0𝑦+𝑙(𝜑,𝜓;𝐜𝑙)𝑎1𝑦+𝑑(𝜑,𝜓;𝐜)𝑎2𝑦,𝑑(𝜑,𝜓;𝐜)=(𝑓−𝑅(𝜑,𝜓;𝐜𝑅)𝑎0𝑧−𝑙(𝜑,𝜓;𝐜𝑙)𝑎1𝑧)/𝑎2𝑧,

where 𝑑(𝜑,𝜓;𝐜) is the distance from the point of the outer surface 𝐌(𝜑,𝜓) to the intersection point in the exit plane.Expressions (3)-(6) enable analytical computation of the illuminance distribution generated by the optical element with two working free-form surfaces. The computation procedure is reduced to the numerical integration of Eq. (3) and can be performed in a split of a second.

## 4. Example of designing optical element

The proposed method of optical design was implemented in Matlab®; for optimization we used the standard function fmincon. Below we continue the consideration of the computation example started in the paragraph 2.1 (generation of uniformly illuminated rectangular spot with the size of 1200 mm by 750 mm at the distance of 1000 mm).

The outer surface of the optical element shown in Fig. 2(d) was optimized using expressions (2)-(6). Totally, 56 parameters were changed during 93 iterations, and the merit function was computed 5618 times. As the overall optimization time was 205 seconds (Intel® Core i7-3770K CPU @ 3,50 GHz), the average time of evaluation of the merit function (3) did not exceed 36 milliseconds, that became possible due to the usage of analytical expressions (3)-(6).

The optimized optical element is depicted in Fig. 4. Figure 5 shows the resulted illuminance distribution simulated for Lambert point source. Lighting efficacy (the part of the emitted light flux reached the exit plane) of the obtained solution is 89.1% and RRMSD is 7.5%.

Fig. 4

Optimized optical element.

Fig. 5

Illuminance distribution generated by the optical element in Fig. 4 for a point light source: (a) grayscale distribution; (b) profiles of illuminance distribution (solid line – v = 0, dashed line – u = 0).

Expressions for the computation of generated illuminance distribution (4)-(6) were derived under the assumption of the point light source. Nevertheless, the designed optical element has good optical performance even in the case of the extended light source. Figure 6 demonstrates the simulated illuminance distribution for the light source with the size of 1x1 mm. In this case the values of lighting efficacy and RRMSD are 88.5% and 8.4%, respectively.

Fig. 6

Illuminance distribution generated by the optical element in Fig. 4 for an extended light source 1x1 mm: (a) grayscale distribution; (b) profiles of illuminance distribution (solid line – v = 0, dashed line – u = 0).

For comparison, let us analyze the optical performance of the element with single working free-form surface for the same illumination problem. The inner surface of such optical element is spherical; the center of the sphere coincides with the center of the light source. Figures 7, 8show the optical element designed with the usage of method [6] and its illuminance distribution. While RRMSD does not exceed the value of 7.1%, lighting efficacy of this solution is 75.6%. The shaping efficacy (the part of the emitted light flux that reaches the prescribed rectangular area) in this case is equal to 63.3% that is 19.7% less than in the case of double free-form surface. The rest part of the light flux that reaches the exit plane (12.3%) forms a halo around the rectangular region and plays negative role.

Fig. 7

Optical element with a single working free-form surface.

Fig. 8

Illuminance distribution generated by the optical element in Fig. 7 for an extended light source 1x1 mm: (a) grayscale distribution; (b) profiles of illuminance distribution (solid line – v = 0, dashed line – u = 0).

Figure 9 demonstrates dependences of the lighting efficacy on the source’s size for both designs. Even in the case when the size of the extended light source is 4 mm by 4 mm (that is just half of the height of the optical element), the lighting efficacy of the element with two working surfaces remains very high (close to 78%). On the contrary, lighting efficacy of the optical element with single free-form surface in this case decreases to 53%. Such ″stability″ of the optical element in Fig. 4 with respect to the size of the light source can be explained by reduced incident angles of rays. Indeed, when the redirection of rays is performed with two surfaces, the deviation angles and incident angles on each interface become smaller that eliminates high Fresnel loss and the undesirable effect of the total internal reflection.

Fig. 9

Dependences of the lighting efficacy on the source’s size for the designs with two free-form surfaces (square markers) and single free-form surface (triangular markers).

Let us consider one more example: an optical element generating a uniformly illuminated hexagonal region. This problem is of great practical importance in designing direct backlight systems [18]. Indeed, such optical elements placed in the nodes of a rhombic grid can provide uniform illumination of large areas with complex shape [18]. In Fig. 10, the computed optical element generating uniform illuminance distribution in hexagonal region with side of 580 mm at the distance of 1000 mm is presented. Totally, 28 parameters were optimized in 113 seconds, and the merit function (2) was evaluated 1642 times during 48 iterations. Figures 11 and 12 demonstrate the resulting illuminance distributions for point and extended (1 mm by 1 mm) sources, respectively. The lighting efficiency in both cases is more than 90% while RRMSD is less than 5.3%.

Fig. 10

Optical element with two free-form surfaces generating uniformly illuminated hexagon spot.

Fig. 11

Illuminance distribution generated by the optical element in Fig. 10 for a point light source: (a) grayscale distribution; (b) profiles of illuminance distribution (solid line – v = 0, dashed line – u = 0).

Fig. 12

Illuminance distribution generated by the optical element in Fig. 10 for an extended light source 1x1 mm: (a) grayscale distribution; (b) profiles of illuminance distribution (solid line – v = 0, dashed line – u = 0).

Generally, the convergence of any optimization procedure depends on many factors such the choice of the initial point, the used optimization algorithm, the merit function, etc. The proposed design method has several distinctive features providing quick and stable convergence. First, the proposed choice of the initial surfaces essentially reduces the chance of getting to a local minimum. Second, the utilization of the analytical expressions (3)-(6) for the computation of the generated illuminance distribution reduces the computational cost associated with the evaluation of the merit function (2) and allows avoiding a statistical noise that affects the convergence (which is typical e.g. for Monte-Carlo raytracing procedures). Third, the definition of the surfaces by scalar bicubic spline functions reduces the complexity of the optimization problem due to the locality of the spline parameters which affect only a part of a surface in contrary to the polynomial parameters.

## 5. Conclusion

In the present work we developed a novel design method for the computation of the optical elements with double free-form surface and high lighting efficacy. The computation procedure consists of two stages: the analytical design of initial surfaces and the subsequent optimization of their parameters. The chosen representation of the free-form surfaces enables fast computation of the generated illuminance distribution which, in its turn, makes possible performing the whole design procedure in few minutes.

Using the developed method, we compute and simulate the optical element producing uniformly illuminated rectangular spot with angular size of 60° by 40°. Despite the fact that the design procedure does not take into account the size of the light source, it is shown that the optical performance of the obtained solution remains high also for the extended light sources. In particular, in the case of 1x1 mm source relative root-mean square deviation is about 8.4%, while the lighting efficacy is more than 88%.

## Acknowledgments

This work was funded by Russian Science Foundation grant 14-19-00969.

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