## Abstract

Energy accumulating optimization based on dynamic programming is proposed to design non-rotational 3D high-compactness freeform optical surface for extended source. Each small piece which constructs the freeform optical surface is treated as a stage, and the normal vectors of the small pieces are treated as the decision variables. Then each small piece with a normal-vector-selection-range is calculated stage-by-stage, which is different to the common used loop-iterations-optimization-strategy. The state of the accumulated light distribution on the target plane is varied with the evolvement of the calculations. The optimal decisions are ascertained in a retrospective way only after all the calculations are finished, which are ensured by the principle of optimality. Moreover, several treatments are proposed to confine the normal vector selection range, and the feedback adjustment is developed as well. The effectiveness of this method is demonstrated by designing the 20 mm height freeform optical surfaces for 10 mm diameter Lambertian sources to achieve uniform illuminance distributions with dual-axial symmetry and single-axial symmetry, respectively. The energy utilization ratios are above 82% with Fresnel loss, while the relative standard deviation of the illuminance distribution can be less than 0.2.

© 2016 Optical Society of America

## 1. Introduction

Freeform optical surfaces (FOSs) are usually applied in solid state lighting which uses light emitting diodes (LEDs) as the source. Assuming *h* is the height of the FOS and *D* is the diameter of the source, different *h/D* ratio could indicate the source scale type and the compactness of the optical system. If$h/D\ge 10$, the light source can be treated as a point source, and the optical freeform surface designing methods in this case are successfully developed in decades [1–10]. If *h/D* ratio is less than five, the size of the light source is necessary to be considered, and the light source can be called as extended source [11]. Generally, it is inappropriate to use the methods based on point sources to design for Lambertian extended sources directly, otherwise it would lead to severe degradations of the light distribution [12]. The more compact of the optical system (e.g. smaller *h/D*), the harder to achieve the exact prescribed light distribution with high efficiency.

At the present time, some FOS designing methods for extended sources have been proposed [11,13–28]. Simultaneous multiple surface (SMS) method [13,14] can design two FOSs simultaneously and precisely control two wavefronts, which is effective to design collimators and concentrators. A numerical direct optimization [15] is proposed to design double FOSs, and it analyzed that single FOS is apt to be more difficult to achieve high efficiency than double FOSs system. In addition, a direct optimization for rotation-symmetry compact double FOSs [16] is recently reported, which achieves very high efficiency and performance. Multiple freeform surfaces can increase the designing degrees of freedom to achieve good results, on the other hand, they increase the interfaces that the rays encounter. Many single FOS designing methods are proposed as well. Tailored edge-ray design method [18,19] is utilized to design the optical system with translational symmetry or rotational symmetry. The feedback compensation method [20–22], which modifies the prescribed light distribution according to the simulated result and constructs the freeform surface based on one-to-one mapping idea, is usually applied to design the optical systems with the *h/D* ratio between 3 and 6. Numeric optimization in [23] can design non-rotational single FOS. Besides, the freeform surface overlapping optimization method [24] is simple to operate to design high-compactness rotational optical system, but not for the non-rotational 3D case, due to its try-and-error strategy.

Moreover, light spots superposing strategy [11,25,28] is an interesting idea, where the light distribution on the target area is composed by superposing many light spots or extended source images. For example, there are methods refer this idea to optimize the rotational-symmetry lens’s profile parameters (e.g. the polynomial coefficients [11] or the slope angles of control points [25]), and achieve good results.

The number of optimization parameters for a 3D non-rotational freeform optical system is usually far larger than that in a rotational case, and single FOS with high compactness (e.g. *h/D =* 2) would make the design case even more difficult to achieve satisfactory results. In this paper, based on the light spots superposing idea, the optimization strategy—dynamic programming [29,30] is utilized to optimize large parameters (e.g. hundreds or even thousands) and design 3D non-rotational high-compactness single freeform surface. Different to the loop-iteration-optimization strategy (i.e. each loop includes updating a set of surface parameters, construction of the entire FOS, and simulation with Monte Carlo method for a judgement, then, with many loop iterations, the optimal result would be finally obtained when the judgement is approval.), in this proposed method, the normal vectors of the facets on the FOS are ascertained through numerical discrete sequential decisions in a bottom-up way without Monte Carlo ray tracing. Under the condition of *h/D* = 2, the designed 3D non-rotational single FOS via this method can achieve high energy efficiency and satisfactory illumination uniformity simultaneously.

## 2. Energy accumulating optimization (EAO) method

#### 2.1 Brief introduction to EAO method

Generally, an FOS can be divided into numerous small pieces which approximate planar facets. In this paper, each facet on the FOS is briefly called an F facet. The target plane is parallel to the source surface. As shown in Fig. 1(a), owing to the non-negligible source’s scale, the light energy received by an F facet will be redistributed onto the target plane as a light spot. When two adjacent F facets are considered as illustrated in Fig. 1(b), there are two corresponding light spots overlapping on the target plane. Similarly, Fig. 1(c) illustrates the three F facets case. The more F facets are considered, the more light spots are overlapping on the target plane. From Fig. 1(a) to Fig. 1(d), with the number of F facets increase, the light distribution on the target plane is gradually extending, and the light energy is accumulating on the target plane. Finally, the whole FOS is constructed by all of the F facets, and the ideal case is that the ultimate accumulated light distribution within the target area approximates the prescribed one as shown in Fig. 1(e). The nature of this method is obtaining all the normal vectors of the F facets which construct the 3D FOS by optimizing the light spots overlapping process.

Assuming *L* is the number of the FOS parameters for optimization (e.g. the number of F facets), and each of the parameter has a selection range for searching (e.g. each F facet has a normal vector selection range). For the case that *L* number is large (e.g. hundreds even thousands), it is supposed to be more favorable to calculate each parameter only once instead of calculating repeatedly as that in the loop-iterations-strategy. Dynamic programming is an algorithm which can divide the *L*-parameters-optimization problem into *L* single-parameter-optimization problems, which are performed by one-by-one calculations with discrete sequential decisions.

In the proposed EAO method, the ingredients in dynamic programming for the case of designing FOS are defined and characterized as following:

- 1) A
*stage*is defined as the calculation for an F facet, and the calculation is based on that of its previous one F facet. The*stages*are calculated one-by-one with recursion through a specific order (from*1*to*L*). - 2) A
*state*in the proposed method is defined as one of the accumulated light distributions on the target plane during the calculation process. A set of*state*variables are utilized to characterize a physical system at any*stage*[29]. In Fig. 1(a) to 1(e), the light distributions on the target plane illustrate different*states*. The initial*state*of the optimization corresponds to the situation that no*stage*is calculated and the target area is blank without any light illumination. - 3) A number of decisions are within each
*stage*[29], where the decision variable is the normal vector. The effect of a decision is a transformation of the*N**state*variables [29], which can be concretely explained as following: supposing$k\in [2,L]$,${S}_{k-1}$is one of the*states*for the (*k-1*)-th F facet, ifis a decision for the*N**k*-th F facet, the light spot of the*k*-th F facet withis superimposed onto the light distribution of${S}_{k-1}$, then a new accumulated light distribution would be obtained, which is the*N**state*variable for the*k*-th F facet with.*N* - 4) A subproblem is defined as the orderly calculations from a
*stage*to the final*stage*(*L*) to achieve optimal result. The number of the subproblems in the optimization is also*L*, and each of two subproblems are partly overlapping. - 5) During the dynamic programming optimization, the past history of the system cannot influence the future decisions directly, but via the current
*state*. This also means the past history is of no importance in determining future actions [29]. To explain this concretely, assuming that${S}_{k}$is the current*state*which corresponds to the*k-*th F facet with, and the$(k+1)$*N**-*th F facet is going to be calculated. Due to${S}_{k}$is the outcome of a past energy accumulation history from the*1*-th F facets to the*k-*th F facet, if we take${S}_{k}$as the initial*state*of the subproblem which starts from the$(k+1)$*-*th F facet, the energy-accumulation information required to start the subsequent subproblem calculation would be adequate, and there is no necessity to consider the past history. This characteristic also ensures the discrete sequential decisions can be implemented. - 6) To pick up one allowable
for each F facet indexed from*N**1*to*L*, respectively, the series of the decisions is called a*policy*. The optimal choices offor each F facets constitute the*N**optimal policy*. - 7) The principle of optimality [29,30] ensures that within an
*optimal policy*, whatever the past*states*and decisions are (i.e. the past*states*and decisions correspond to the*stages*indexed from*1*to*k*,$k\in [1,L]$, no matter which specific number of*k*is), the remaining decisions (within the*stages*indexed from*k*to*L*) must constitute an*optimal policy*with regard to the*state*resulting from the past decisions. This suggests that one can determine the optimal policy in a piecemeal manner [30]: firstly, determines the*optimal policy*for the “tail subproblem” only including the last*stage*, then extends the optimal policy to the “tail subproblem” involving the last two*stages*, and continues in this manner until the*optimal policy*for the entire problem is determined [30]. This means that the*optimal policy*can be ascertained via a bottom-up strategy in a retrospective way only after all the*stage*calculations are finished.

The framework of using dynamic programming to design an FOS is illustrated in Fig. 2. The construction of the FOS are directly via numerical sequential calculations without Monte Carlo ray tracing. The details of the optimization process would be described in sections 2.2 and 2.3, and the proposed techniques to facilitate EAO method would be described in section 2.4.

#### 2.2 The ribbons construction and the order of indexing the F facets

The F facets are calculated in a specific order. The determining of the indexes for the F facets is according to a ribbons construction as described in the following.

To design a non-rotational 3D optical system, the source angle can be divided with the *u-v* angle system [3, 20] as illustrated in Fig. 3(a), and the coordinate origin locates at the center of the extended source. As shown in Fig. 3(b), each lattice within the *u-v* angle grid corresponds to the solid angle of a specific F facet, and it demarcates the boundary of that solid angle. Moreover, the angles which correspond to the centers of the F facets can be approximately obtained by further bisecting the *u-v* angle grids shown in Fig. 3(b).

After giving the dimensional parameters of the optical system, along the *u* direction while *v =* 0, an initial curve which is a curve on the FOS calculated by the method based on point source can be obtained by [3], where this above mentioned FOS aims to achieve the same prescribed light distribution. Then as illustrated in Fig. 3(c), the F facets calculations start from the points on the initial curve and along the *v* angle direction. A ribbon is named as an assemblage of the F facets with equal *u* angles and monotonically varying *v* angles, which is illustrated as one of the red arrows in Fig. 3(c).

Figure 4(a) shows the construction for a ribbon, where the position of an F facet is related to its normal, and also be related to the position and normal of its previous F facet. For example, the center position *p _{2}* of an F facet is geometrically determined by the normal

*N*and the edge position

_{2}*pp*(this geometrical construction has been described in [3] and [20] with details), where

_{1}*pp*can be geometrically determined by the position

_{1}*p*and the normal

_{1}*N*of its previous F facet in a similarly way. Therefore, within a ribbon, any two adjacent F facets have a common edge point, which guarantees the connections of the F facets. Moreover, the starts of all the ribbons come from one smooth initial curve. Thus this ribbon-construction way can preliminarily avoid constructing the FOS with severe discontinuities.

_{1}According to the ribbons construction, the F facets’ calculating order in the optimization can be determined. Figure 4(b) shows the way to index the F facets for a dual-axial-symmetry optical system, where each bin corresponds to a solid angle of an F facet, and it merely needs to design a quarter of the whole FOS by treating each calculated light spot with symmetrical operation along *x* and *y* axes. Figure 4(c) gives a simple example to illustrate the way of indexing the F facets for a single-axial-symmetry optical system which is symmetrical along x-axis direction and asymmetrical along y-axis direction, and it needs to design half of the whole FOS. The one-by-one calculating order of the F facets should be fixed so as to conveniently utilize dynamic programming optimization.

#### 2.3 Dynamic programming optimization process in EAO method

If the target area is rectangular, the bins of the target area can be obtained by equally dividing the length and width of the area as shown in Fig. 1(d). The way of obtaining the illuminance distribution of a light spot for a certain F facet with a specific normal vector *N* and position is described in Appendix. When more and more F facets (*stages*) are calculated, the light energy illuminated on the target plane is monotonously increased. Each accumulated light distribution (*state*) is obtained by counting the accumulated light energy within each bin on the target plane.

Assuming *L* is the number of the F facets for calculation, and *M* is the number of the normal vector decisions for each F facet, where *M>*1. In Fig. 5, each small box represents that a certain F facet with a specific normal vector decision, and each box would correspond to a current-light-distribution on the target plane. Figure 5 illustrates the process of how to find the *optimal policy*, i.e. how to choose the serial of optimal boxes for the F facets.

In Fig. 5(a), the current-light-distribution for each box in F-facet-*1* is its corresponding calculated light spot distribution. As for the first box in F-facet-*2*, superimpose its corresponding light spot onto each current-light-distribution of the boxes in F-facet-*1* respectively, so as to obtain *M* new accumulated light distributions (*states*). Then select the “best” one from these *M states* to be the current-light-distribution for the first box in F-facet-*2* (illustrated by the arrow with solid line in Fig. 5(a)), and record the correlating box index of F-facet-*1*. This “best” selection is based on choosing the minimum value of an evaluation function *f*, which would be described with details in the following paragraphs. Moreover, the *f* value would be increasing within a *policy* as the F facet index increased. In Fig. 5(b), the subsequent box in F-facet-*2* is calculated in a similar process. Figure 5(c) shows that all the boxes in F-facet-*2* have been sequentially calculated, and F-facet-*3* starts to be calculated similarly. Figure 5(d) give the illustration that the sequential calculations of all the F facets (indexed from *1* to *L*) are completed, and the cumulative superimposing of all the light spots within the target area are finished. Then among the current-light-distributions of the boxes in F-facet-*L,* the light distribution which has the “best” (i.e. minimal) value of the function *f* would be found as the final optimal one, as marked by “*optimal*” in Fig. 5(d), and this final optimal light distribution records its correlated box index in F-facet-$(L-1)$. Furthermore, this certain box in F-facet-$(L-1)$ was also recorded its correlated box index in F-facet-$(L-2)$, similarly as the red arrows illustrated in Fig. 5(d), the *optimal policy* can be obtained in this retrospective way. After the optimal normal vector for each F facet is identified, the optimal FOS can be constructed as introduced in Section 2.2.

To achieve a uniform illuminance distribution in the target area$A{}_{\text{target}}$, the ideal case is that the total flux emitted from the source${\Phi}_{source}$can be redistributed uniformly within the target area, where the ideal illuminance${E}_{0}$is a constant, which is calculated by Eq. (1). Equation (2) is the evaluation function *f*. To some degree, a smaller value of Eq. (2) suggests that the evaluated accumulated light distribution${E}_{evaluate}$is closer to the prescribed ideal case, due to the meaning of the first item in Eq. (2) is the deviations of the illuminance distribution from${E}_{0}$. Besides, Eq. (3) shows the way to obtain the current-light-distribution for each box in Fig. 5, and$opt(f\left({C}_{{j}_{k}}\right))$is also the second item in Eq. (2), which makes the evaluation function *f* have a recursive relationship.

In Eq. (2),${j}_{k}$and${j}_{k+1}$represent the box indexes in the F-facet-*k* and F-facet-$(k+1)$, respectively.${C}_{{j}_{k}}$represents the current-light-distribution for the box${j}_{k}$. ${S}_{{j}_{k+1}}$represents an accumulated light distribution (*state*) for the box${j}_{k+1}$, and${S}_{{j}_{k+1}}|{}_{{C}_{{j}_{k}}}$means this *state* comes from${C}_{{j}_{k}}$with the decision${j}_{k+1}$. ${E}_{evaluate}({t}_{x},{t}_{y})|{}_{{S}_{{j}_{k+1}},{C}_{{j}_{{}_{k}}}}$is the illuminance of the evaluated light distribution (the *state*${S}_{{j}_{k+1}}|{}_{{C}_{{j}_{k}}}$) in the target bin$({t}_{x},{t}_{y})$. The number of the bins in the target area is *num*. The value *f* for the *state*${S}_{{j}_{k+1}}|{}_{{C}_{{j}_{k}}}$is calculated by Eq. (2).

Then, the current-light-distribution${C}_{{j}_{k+1}}$for the box${j}_{k+1}$can be obtained by comparing the elements of$\{f\left({S}_{{j}_{k+1}}|{}_{{C}_{{j}_{{}_{k}}}}\right),{j}_{{}_{k}}=1,2,\mathrm{...},M\}$and choosing the minimum one, as similarly shown in Eq. (3). Besides, at the start of the calculations, the${C}_{{j}_{1}}$for each box in F-facet-*1* is the corresponding light spot distribution, where ascertaining$opt(f\left({C}_{{j}_{1}}\right))$is not by comparison. It should be emphasized that each$opt(f\left({C}_{{j}_{k}}\right))$in Eq. (3) do not necessarily correspond to the exact optimal decision within the *optimal policy*.

Through evaluating all the current-light-distributions of the boxes in *stage L* as shown in Eq. (4), the optimal final light distribution and its box index can be found. Then the *optimal policy* is traced retrospectively.

The principle of optimality [29, 30] has been discussed in section 2.1. Based on the above descriptions, the reason why the *optimal policy* ascertained by the strategy illustrated in Fig. 5(d) has the minimum *f* value among the other *policies*, would be explained concretely. As illustrated in Fig. 5(d), in the “tail subproblem” which only involve the final *stage L*, the box corresponds to the minimal *f*-value (“*optimal*”) can be found among the boxes of F-facet-*L*. Then extends to the “tail subproblem” which involves the last two *stages*$(L-1)$and *L*. Firstly, according to the way that the solid arrow is determined, it is the first box in *stage*$(L-1)$that make the *j*-th decision in stage *L* to achieve “*optimal*”, and the other boxes in *stage*$(L-1)$cannot make the *j*-th decision in stage *L* to achieve any smaller *f* value. Secondly, it is impossible for any of the boxes in *stage L* exclusive of “*optimal*” to connect to any boxes in *stage*$(L-1)$can have a smaller *f* value. Thus, the *optimal policy* must include the first box in *stage*$(L-1)$and the “*optimal*” box in stage *L*. The discussion of further extending the “tail subproblem” is similar. The principle of optimality can guarantee that the *optimal policy* determined in this retrospective way has the minimum *f* value.

With this optimization strategy of dynamic programming, the search volume is only$(L-1)\cdot {M}^{2}+M$. In contrast, if all the parameters are optimized at the same time by the exhaustive method, the search volume would be *M ^{L}*. In this method, it is inconvenient to reduce the number of

*L*when constructing an elaborate FOS, thus decreasing the search volume is mainly based on reducing the number of

*M*.

#### 2.4 Skillful techniques for EAO method.

### 2.4.1 The treatments of reducing normal-vector-searching-number (TRNs)

Three treatments of TRNs are proposed to cut down the selection range of the normal vectors, so as to confine *M* within a small number (e.g. only dozens).

Firstly, a preliminary treatment would be introduced. As for one F facet illustrated in Fig. 6, ${I}_{c}$is the normalized vector of the incident ray which emits from the source center; and ${O}_{loca}$ is the corresponding normalized emergent ray which emits from the F facet and aims to a ray’s location on the target plane. Different emergent ray’s locations *loca1* and *loca2* determine the different normal vectors${N}_{1}$and${N}_{2}$of the F facet, according to $scalar\cdot N={n}_{2}\cdot {O}_{loca}-{n}_{1}\cdot {I}_{c}$ and ${I}_{c}\cdot N>0$for a refractive optical system, where *scalar* is a quantity. If we restrict the locations *loca* within the grids of target area, at least a part of each light spot would be included in the target area, then, the preliminarily restricted discrete normal vectors for selection are obtained. Additionally, the deflection angle of the vector${I}_{c}$and${O}_{loca}$is better to be restrained smaller than a certain angle which is dependent on the plastic refractive index [1], so as to avoid insufficient refraction.

Secondly, further controlling the FOS continuity can help to reduce *M*. The ribbons construction can help to avoid severe discontinuities, but some discontinuities might exist between the posterior parts of adjacent ribbons. This treatment is proposed to control the discontinuities between adjacent ribbons. A certain F facet’s position is related to its normal vector, and the arrows illustrated in Fig. 5(d) can be used to trace the corresponding *states* in previous F facets, thus the positions of an F facet and its adjacent corresponding F facet (in the previous ribbon) can be identified and compared, so as to judge the continuity and determine if the optional normal vector is selectable. Defining polar radius *R* as the distance between the origin point (0, 0, 0) and the position of an F facet, Eq. (5) presents the criterion of judging the continuity between two adjacent F facets in different ribbons, where *R*_{1} and *R _{2}* are the polar radius of the two F facets, respectively, and

*tol*is a very small tolerance value.

Thirdly, an approximate but effective treatment would be introduced to confine M number into only dozens. Assuming *M _{1}* is the amount of the potential normal vectors for each F facet (

*M*is equal to the

_{1}*M*in the above TRNs), and

*M*is the amount of the actual boxes for each F facet in Fig. 5, the evaluation function Eq. (2) can also be used to pick up

*M*normal vectors (dozens number) out of

*M*normal vectors (hundreds number) at the start calculation for each F facet. Supposing F-facet-

_{1}*k*starts to be calculated, the current-light-distribution${C}_{\mathrm{min}(k-1)}$which has the minimum

*f*value among the boxes of F-facet-$(k-1)$needs to be found firstly, then the light spots corresponding to the

*M*potential normal vectors for F-facet-

_{1}*k*would respectively add onto${C}_{\mathrm{min}(k-1)}$to obtain

*M*light distribution and

_{1}*f*values. Arranging

*M*values into a descending order for convenience, the last

_{1}f*M*values and the corresponding normal vectors can be picked up to be the actual decision variables for F-facet-

*k*, due to these normal vectors could approximately achieve lower

*f*value for the

*optimal policy*.

### 2.4.2 Discussion of the evaluation function and applying the idea of feedback compensation

When the FOS is constructed, the simulated result obtained by Monte Carlo ray tracing is usually assessed by two indicator separately: the energy utilization ratio (also called efficiency$\eta $, defined as the ratio of the energy within the target area to the energy emitted from the LED source); and the relative standard deviation (RSD) defined by Eq. (6) to characterize the uniformity, where${E}_{i}$is the illuminance of the simulated light distribution in each target bin, $\overline{E}$is the average illuminance within the target area.

On the other hand, in the optimization, the evaluation function Eq. (2) is used to evaluate and control the efficiency and the uniformity simultaneously, and it is hard to clearly distinguish the two factors. Accordingly, the one-merit-function evaluation in the optimization process would be somewhat different to the two-indicator evaluation for the Monte Carlo simulated results. Besides, when applying TRNs, it is possible that some useful normal vector options are missed, and it might lead to degradation of the optimal results. Therefore, the RSD value of the simulated result might need further improving. Then the idea of the feedback compensation is introduced and developed, which could become an accessorial treatment for EAO method to adjust the evaluation function *f*. It is emphasized that the feedback EAO method do not use the one-to-one mapping idea to construct the freeform surface as in [20–22].

The formula of the evaluation function with feedback is shown in Eq. (7). Compared with Eq. (2), the difference is that${E}_{0}$in Eq. (2) and Eq. (3) is substituted by${E}_{feedback\_iter}({t}_{x},{t}_{y})$, which is the illuminance in the bin$({t}_{x},{t}_{y})$of the feedback light distribution. The calculations for ${E}_{feedback\_iter}({t}_{x},{t}_{y})$are given by Eq. (8), where${E}_{simu}$denotes the simulated illuminance distribution within the target area for a specific feedback order. Moreover, the lowest illuminance bound of${E}_{feedback\_iter}({t}_{x},{t}_{y})$is compulsively restricted to be $c\cdot {E}_{0}$, where *c* is a specific constant choosing from$[0,1)$. For a certain *h/D* ratio, a trade-off is usually within the efficiency and uniformity performance. In this method, though the feedback can improve RSD, it would lower the efficiency. The iterations is suggested to be less than two.

## 3. Examples of the optical systems with two kinds of symmetries

As shown in Fig. 7, two high-compactness refractive optical systems with different kinds of symmetries are designed by EAO method to achieve uniform illuminance distribution in the target area. Figure 7(a) illustrates the dual-axial-symmetry optical system to design, where the center of the target area is right over the center of the FOS and the center of the extend source. Figure 7(b) illustrates the single-axial-symmetry optical system which has asymmetry along y-axis direction. In the two designing examples, the diameter *D* of the round Lambertian source is 10 mm, while the total outgoing flux of the source is 1000 lm in$2\pi $solid angle; the height *h* of the designed FOS is 20 mm, i.e. *h/D* = 2; the source is immersed in the freeform optical lens; the refractive indexes *n _{1}* is 1.49 for PMMA and

*n*is 1 for air; the distance

_{2}*H*between the parallel target plane and the source surface is 1000 mm. After constructing the designed FOSs, the simulated light distributions are obtained by tracing one million rays by Monte Carlo ray tracing method, and the Fresnel loss is considered. When the feedback adjustment is necessary to be used,

*c*= 0.5.

#### 3.1 Design of optical systems with dual- axial-symmetry

As shown in Fig. 7(a), the length *X* of the target area is 4000 mm, and the width *Y* is 2000 mm. The target area is meshed within 40 × 20 equal area bins, *num* = 800. One quarter of the FOS, i.e.$u\in [0,0.5\pi ],v\in [0,0.5\pi ]$, is constructed with 20 × 20 F facets, i.e. *L* = 400. Three different FOSs are designed: with the first TRN only, with all the TRNs, and with all the TRNs and once feedback adjustment, respectively. The results are shown in Fig. 8.

The first case is only a contrast, where *M* is 150, and it would consume a relatively long time (about 29 hours) to finish the optimization with an Inter^{®} Core^{(TM)} i5-2400 CPU. The corresponding simulated illuminance distribution is shown in Fig. 8(a), and the white-dotted line represents the target area. Figure 8(b) shows the illuminance distribution along the *x-*axis and *y-*axis. The efficiency$\eta $of this optical system is 84.7% with Fresnel loss, while the RSD of the illuminance distribution is 0.193 as listed in Table 1. Figure 8(c) shows the shape of the designed FOS, where the parts marked by the red circle have some discernible discontinuities.

In the second case, *M* is reduced to 15 by TRNs (while *M _{1}* is 150),

*tol*is set as 0.005. The optimization time is only 28 minutes with the same CPU, which suggests that applying all the TRNs can decrease the optimization time effectively. The simulated light distribution are shown in Fig. 8(d) and Fig. 8(e), and the efficiency of the optical system considering Fresnel loss is 84.8%, while the RSD is 0.188. Figure 8(f) shows the designed FOS, and it is noticed that the continuities of the FOS are conspicuously improved.

In the third case, the simulated light distribution is given in Fig. 8(g) and Fig. 8(h). The once optimization time is the same as the second case. As listed in Table 1, the efficiency of this optical system with Fresnel loss is 82.1%, while its RSD value is decreased to 0.134 by the feedback adjustment. Comparing with the RSD of the first case and the second case, the RSD of the third case is improved by 30.6% and 28.7%, respectively. The corresponding designed FOS is shown in Fig. 8(i). We can see that the TRNs and feedback adjustment can help to design the FOS more effectively.

#### 3.2 Design of optical systems with single-axial-symmetry

It is usually more difficult to design compact optical system with asymmetry along one axis direction, due to the deflection angles of the rays are usually larger than that of the dual axial symmetry case, which is easier to cause total internal reflections. As shown in Fig. 7(b), the length *X* of the target area is 4000 mm, *Y _{1}* = 1000 mm,

*Y*= 2000 mm, i.e. the width is 3000 mm. The target area is meshed within 40 × 30 equal area bins,

_{2}*num*= 1200. In order to prove the effectiveness of the method to optimize for more parameters, half of the FOS, i.e.$u\in [0,0.5\pi ],v\in [-0.5\pi ,0.5\pi ]$, is constructed by 40 × 80 F facets,

*L*= 3200. All the TRNs are applied, while

*M*is 15 (

*M*is 150), and

_{1}*tol*is set as 0.005. The consuming time of the optimization is 3.5 hours with an Inter

^{®}Core

^{(TM)}i5-2400 CPU. Three different FOSs are designed: without feedback adjustment, with once feedback adjustment, and with twice feedback adjustments, respectively.

In the first case, the simulated illuminance distribution is given by Fig. 9(a), where the white-dotted line denotes the target area, and the dot-dashed lines mark the *x-*axis and *y-*axis of the system. Figure 9(b) shows the illuminance distribution along *x* and *y* axes. It indicates that the light distribution is not well uniformly distributed in the target area, and the energy in the target area of *y* < 0 needs to be enhanced. The RSD of the illuminance distribution is only 0.367, though the efficiency is as high as 86.8% with Fresnel loss, as listed in Table 2.

Whereas in the second case, the corresponding simulated illuminance distribution shown in Fig. 9(c) and Fig. 9(d) suggest that the light distribution uniformity is improved. As given in Table 2, its efficiency with Fresnel loss is 82.7%, while the RSD is improved to 0.201. Comparing with the first case, the RSD value of this case is improved by 45.2%, and the efficiency is decreased by only 4.1 percentage points.

As for the third case, its simulated light distribution is shown in Fig. 9(e) and Fig. 9(f). Its efficiency with Fresnel loss is 80.4%, while the RSD value is further improved to 0.188. Comparing with the first case, the RSD of this case is improved by 48.8%, though the efficiency is decreased by 6.4 percentage points.

Although the third case has a smaller RSD value, its efficiency is lower than that of the second case by 2.3 percentage points. In our opinion, the second case is more favorable to be the optimal result for this asymmetrical example. Figure 10(a) gives the shape of the FOS for the second case in the planform view, and Fig. 10(b) shows the side view.

## 4. Summary

The EAO method provides an optimization way to design high compactness 3D FOS for a given large-size LED source. In this proposed method, a light spot distribution on the target plane corresponds to a facet on the FOS, and dynamic programming optimization are utilized to efficiently control the light spots superposing procedure via ascertaining the optimal normal vectors of the facets in a numeric sequential-decisions solution. Moreover, three TRNs are proposed to confine the selection range of the normal vector and control the continuity of the FOS. The feedback adjustment is developed as well, which can improve the uniformity of the simulated light distribution when it is necessary to be used.

In the given examples of high compactness single FOS with *h/D* = 2, the results achieve high energy utilization ratio and well uniform illumination distributions. In the dual-axial-symmetry example, the energy utilization ratios of the cases are all higher than 82% with Fresnel loss, which are high efficiency for non-rotational single FOS with such high compactness, while the best RSD value is 0.134. In the more complicated single-axial symmetry example, the case with one feedback adjustment has high energy utilization ratio of 82.7% with Fresnel loss, while the RSD value is 0.201.

To clarify, based on the present manufacture techniques, the fabrication of faceted freeform lens is more difficult than that of a smoothed one, and further improve the smoothness could be a direction to develop this method.

## Appendix *The way of calculating the light distribution of each light spot*

The extended source is fractionized into a large number of small-equal-area source facets (S facets). In this paper, the 10 mm diameter source is divided into 100 S facets. When calculating for an F facet as shown in Fig. 11, the narrow beam emitted from an S facet to the F facet is approximated as a single ray, whose incident direction is determined by the center positions of the F facet and its corresponding S facet. Table 3 lists the descriptions of the parameters.

With a specific F facet, the rays emitted from different S facets and received by the F facet are redirected into their certain bins in the target area respectively. Considering one S facet, the energy carried by the ray is filled in its corresponding bin approximately, and the flux carried by the incident ray${I}_{i}$can be calculated by Eq. (9) [31] for a Lambertian source case.

When$\beta \left(i\right)$is less than the angle of total reflection in the case *n _{1}*

_{>}

*n*,${O}_{i}$is derived from a vector form of Snell’s Law as given by Eq. (13) and Eq. (14). The position${p}_{t}(i)=\left({x}_{t}\left(i\right),{y}_{t}\left(i\right),{z}_{t}\left(i\right)\right)$where${O}_{i}$casted into the target plane can be located via Eq. (15).

_{2}When$\beta \left(i\right)$is equal to or larger than the total reflection angle, the flux carried by${I}_{i}$would be thought as a loss and excluded from the light spot. Moreover, if the position${p}_{t}(i)$ locates outside of the target area, the flux carried by ${I}_{i}$would be neglected. Then only the energy within the target area would be evaluated in the optimization.

In this way, after calculating the rays emitted from all the S facets of the extended source, and combining their carried fluxes in the bins of the target area, the light spot distribution formed by a certain F facet from the Lambetrian extended source can be obtained.

If the emission of the source is not Lambertian, the constant luminance *L _{S}* in Eq. (9) would be changed by${L}_{S}(i)$, where${L}_{S}(i)$would be varied with different S facet according to the specific source emission.

## Funding

National Basic Research Program of China (Grant No. 2015CB351900); Guangdong Province Science and Technology Program (Grant No. 2014B010121004); National Natural Science Foundation of China (NSFC) (Grant Nos. 61210014, 61321004, 61307024, 61574082, and 51561165012); High Technology Research and Development Program of China (Grant No.2015AA017101); Independent Research Program of Tsinghua University (Grant No. 2013023Z09N).

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