1. INTRODUCTION

One of the most important functions of freeform optics is the generation of a prescribed illumination pattern on the target plane from a given light source (such as LEDs), which plays a significant role in high-quality and energy-efficient solid-state lighting and machine vision illumination. Designing freeform illumination optics is a highly challenging task. In point source and geometrical optics approximations, together with the assumption that all the rays are bent in a monotonic way, the design problem of freeform illumination optics can be transferred into a second-order partial differential equation of Monge-Ampère type [1,2]. Various numerical methods have been proposed to address the point-source design problem, enabling the generation of complex irradiance distributions, such as letters and images (see, e.g., [3–13]). The point source approximation assumes a light source with zero size, but its validity diminishes for compact designs.

Compact freeform optics are increasingly demanded for saving materials, reducing costs, and enhancing system integration, but they present a significant challenge for existing design methods due to the necessity of accounting for the source’s dimensions. There have been a few freeform optics design methods for irradiance tailoring of extended sources such as parametric optimization methods [14–18], over compensation methods [19–21], deconvolution methods [22–26], edge ray mapping method [27], 3D simultaneous multiple surface (SMS) method [28] and its further development wavefront tailoring method [29], the expectation-maximization algorithm applied for pinhole image arrangement [30], and the local surface correction method [31]. Most of these methods, although requiring profound knowledge of optics and mathematics, focus on the modest goal of obtaining uniform or flat-top irradiance distributions. Few methods could generate complex irradiance distributions [22–24], but they are only suitable for light sources without large divergence angles. It is very necessary to develop a flexible, general, and efficient design approach for irradiance tailoring of extended sources.

The parametric optimization approach is a very promising candidate for the universal solution to the freeform illumination optics design problem for extended sources. However, most current parametric optimization methods are not efficient because of the repeatedly time-consuming ray tracing processes. Thus, the number of parameters for representing a freeform surface is limited to balance the design efficiency and optical performance [17]. Muschaweck applied a noise-free, non-Monte-Carlo method to evaluate the irradiance by integrating the luminosity of the virtual source image over its finite projected solid angle [18]. This integral equation method can considerably improve the optimization speed because it guarantees differentiability. However, this approach requires a complicated and lengthy formula derivation of parameter gradients and only currently achieves a single freeform surface design.

Differentiable ray tracing has played an increasing role in evaluating and designing optical systems [32–45]. Such a technique can fast calculate the gradients with respect to surface parameters through one backpropagation based on the computational graph, which could significantly speed up the optimization. However, many previous works mainly employed differentiable ray tracing for imaging purposes. In non-imaging optics, the design of freeform surfaces requires consideration of more degrees of freedom, even when only a single lens is used. Wang et al. [44] realized the optimization of freeform surfaces based on differentiable ray tracing to form complex irradiance distributions. This method only applies to point sources and requires an initial solution generated by an optimal transport algorithm.

Effective differentiable ray tracing methods for extended sources with large divergence angles are still rare and challenging. The light field of an extended light source is 4D, which complicates the sampling and tracing process. As the freeform surface size approaches that of the light source, rays are more likely to intersect the surface multiple times mathematically. Using the intersection points of rays with virtual planes as initial solutions in methods like dO [38] is not applicable here. More complex procedures for determining intersection points are required. Furthermore, irradiance tailoring for an extended source is not only about achieving an expected illumination pattern but also about optimizing for energy efficiency and compactness. Parallel work from Heemels et al. [45], published during the review process of this paper, addresses the extended source design problem using differentiable ray tracing of Mitsuba3 [37] and truncated hierarchical B-splines. However, their method, which fails to envelop the light source fully, compromises energy efficiency and overlooks constraints critical to achieving optimal surface compactness.

We propose an effective and efficient optimization framework of freeform lenses for extended-source irradiance tailoring based on differentiable ray tracing, which combines a representation of freeform surfaces using multi-level spherical radial basis functions (SRBFs) and a multi-scale optimization with key feasibility constraints on lens shape and size. This framework, which sequentially introduces finer details on top of coarse bases, can provide necessary global and local surface features required to form a complex irradiance distribution, and make the optimization insensitive to the initial solution. Here, we focus on a challenging case where both the inner and outer surfaces of the lens are freeform surfaces. Double-freeform surfaces could be more powerful than a single one for controlling the light rays from an extended source with a large divergence angle. Note that our method is also applicable in cases where either the inner or outer surface is a predefined spherical, aspherical, or freeform surface. Using our method, we have achieved an irradiance distribution with complex boundaries and analyzed the relationship between compactness and control effectiveness, which were challenging to realize with previous methods.

The details of our method are introduced in Section 2. Two design examples are provided in Section 3 to demonstrate that our optimization framework can efficiently design compact lenses, generating not only simple illumination patterns but also complex illumination patterns from extended light sources.

2. METHOD

We consider an irradiance tailoring system comprising a double-freeform lens, a target plane, and an extended light source with semi-spherical light emission, as illustrated in Fig. 1. The light source $s$, whose center is placed at the original point, has a radiance distribution $R({\boldsymbol x},\theta ,\varphi)$, where ${\boldsymbol x}$ denotes the position of an arbitrary point on the emitting surface, and $\theta$ and $\varphi$ denote the zenith and azimuthal angles, respectively. For a typical LED source with a Lambertian radiating surface, $R({\boldsymbol x},\theta ,\varphi)$ is a constant. The light source is supposed to emit light rays to the space of $z \gt 0$. The light rays passing through the inner and outer freeform surfaces are directed to the target plane at ${z_{\rm t}}$, forming a desired irradiance distribution ${{\boldsymbol E}_{\rm t}}$. The inner and outer freeform surfaces are described as radial distance functions $\rho (\theta ,\varphi)$ and ${\rho ^\prime}(\theta ,\varphi)$, respectively. Based on parametric representations, $\rho (\theta ,\varphi)$ and ${\rho ^\prime}(\theta ,\varphi)$ can be defined by parameters ${\boldsymbol \nu}$ and ${{\boldsymbol \nu}^\prime}$, respectively.

 

Fig. 1. Schematic of an irradiance tailoring system comprising a double-freeform lens, a planar target, and an extended source with semi-spherical light emission.

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The forward problem of the light transport from the extended source to the target plane can be formulated as ${\boldsymbol E} = {{\boldsymbol H}_s}({\boldsymbol \nu},{{\boldsymbol \nu}^\prime})$, where ${\boldsymbol E}$ is the simulated irradiance distribution on the target plane, and ${{\boldsymbol H}_s}$ denotes the operator of the light transport from the source $s$ to the target. The deviation between the current irradiance and the target one is evaluated using the loss function ${\cal L}$. The inverse problem for generating a desired irradiance ${{\boldsymbol E}_{\rm t}}$ from the source $s$ can be simply formulated as

We will describe these three aspects in the following, noting that they are not independent but interconnected with each other.

A. Representing the Freeform Surfaces Using Multi-level SRBFs

RBFs have been applied successfully for describing freeform imaging surfaces [46,47]. RBFs also appear to be good candidates for representing and optimizing freeform illumination optical surfaces. Two issues must be considered for RBF surfaces. The first issue is the choice of the shape parameters. Severe ill-conditioning may occur with a small range parameter when attempting to make the RBF description global [47]. The second issue is the number of RBFs and their locations. Having more RBFs does not mean better performance, especially for those surfaces dominated by low spatial frequencies. For a given surface fitting, the parameters of RBFs can be determined with the help of the surface properties. However, in the parametric optimization, the data of the freeform surface is not available beforehand.

We therefore propose a multi-level SRBF representation in the parametric optimization of illumination optics. This method draws inspiration from the concept of the woofer-tweeter deformable mirror (DM) system, aiming to approximate a calculated freeform reflective surface for shaping the focal-plane beam, where a high bandwidth tweeter DM is used to compensate for the residual error when fitting the required surface using the low-bandwidth woofer DM [50]. It is worth noting that similar multi-level RBF approximations have been successfully applied in scene rendering [51], 3D data interpolation [52], fitting of given vector fields [53], solving partial differential equations [54,55], etc.

Our multi-level SRBF representation for a freeform illumination surface is described as

The center directions ${\boldsymbol c}$ can be sampled with the Fibonacci grids [56], realizing uniform sampling on the unit sphere. Please refer to Supplement 1 for more details. The relationship between the number of SRBFs and the order of layers can be set as ${m_i} = {(2^{b}} + {1)^2},b \in \mathbb{N}$, which could balance the number of parameters and the richness of the surface details. Figure 2 illustrates the Fibonacci sampling grids for the first four layers of SRBFs. The shape parameter of SRBFs at the $i$-th layer can be set as

 

Fig. 2. Illustration of the Fibonacci sampling grids for the first four layers of SRBFs.

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B. Differentiable Ray Tracing with the Multi-level SRBF Surfaces and Extended Source

We implement differentiable quasi-Monte-Carlo (QMC) ray tracing to evaluate the irradiance performance of the double-freeform lens with an extended light source to improve the optimization efficiency. Such a differentiable simulation can achieve fast calculation of partial derivatives with respect to all the surface parameters through just one backpropagation after one forward simulation based on the computational graph, which can effectively reduce the forward simulation times.

Previous differentiable ray tracing methods for illumination mainly focus on designing a single freeform surface, which is represented by B-spline or its variants [38,40,44,45]. Our method is a customized simulation engine for extended sources and double-freeform surfaces described by the multi-level SRBFs. We will first introduce the forward simulation process, and then describe the backpropagation in the following.

We have proposed a Monte-Carlo-based parallel ray tracing method to achieve quick irradiance evaluation for freeform lenses with NURBS surfaces [57], and many of its strategies are adopted here for QMC ray tracing with multi-level SRBFs. The forward ray tracing starts from a uniform ray sampling of the extended light source, detailed in Supplement 1. A ray is represented by ${\boldsymbol \xi} = ({{\boldsymbol x}_{{\rm ray}}},{{\boldsymbol r}_{{\rm ray}}},{w_{{\rm ray}}})$, where ${{\boldsymbol x}_{{\rm ray}}} \in {\mathbb{R}^3}$ represents the starting point coordinates, ${{\boldsymbol r}_{{\rm ray}}} \in {\mathbb{R}^3}$ represents the unit direction vector, and ${w_{{\rm ray}}}$ represents the energy weight.

The main operation of ray tracing is the acquisition of the intersection points between rays and surfaces. To determine if a point of a ray lies on a SRBF surface, we can rely on its distance ${\rho _{{\rm ray}}}$ from the original point. ${\rho _{{\rm ray}}}$ can be calculated by

With fixed surface parameters ${\boldsymbol \omega}$ and $h$, we can solve this intersection equation for $t$ using Newton’s method, which requires a certain degree of accuracy of the given initial solution to ensure convergence. Since there can be multiple intersection points between the ray and the freeform surface, we employ linear seeking with an optical path interval of $\Delta t$ to determine the range of the first intersection. Then we employ the dichotomy method to obtain an initial solution of the intersection. Finally, Newton’s iteration is implemented to find the exact solution:

After tracing all the rays to the target plane, the discrete intersection coordinates ${\boldsymbol p} \in {\mathbb{R}^{2 \times N}}$ and the corresponding energy weights ${{\boldsymbol w}_{\rm t}} \in {\mathbb{R}^N}$ are converted into a two-dimensional irradiance distribution ${\boldsymbol E}$, where $N$ is the total number of rays, as detailed in Supplement 1. Our simulation results closely align with those of Zemax and Mitsuba3 [34,37], validating the high accuracy of our method. A detailed comparison is available in Supplement 1.

We can employ the normalized mean-square error (MSE) as the loss function to characterize the deviation of the simulated irradiance distribution from the target one:

We now turn to describe the backpropagation based on the technique of computational graphs, which is the basis of the autograd function of deep learning platforms such as MindSpore and PyTorch. A computational graph is a directed graph of a computational flow that employs elementary operators, with their derivative expressions, to decompose complex operations. The backpropagation computational flow can be automatically constructed when the forward QMC simulation is defined using computational graph techniques. However, a direct auto-construction for the backpropagation of the intersection-finding process will result in low efficiency because it involves linear seeking, dichotomy iteration, and Newton’s iteration. Several methods have been proposed to solve this problem. Volatier et al. derived the gradients formula from the implicit equation that the extremum of optical path length equals zero based on Fermat’s principle and demonstrated that the derivatives only depend on the intersection coordinates, irrelevant to the intermediate variables of the intersection-seeking process [33]. Wang et al. performed a pre-operation to obtain the accurate intersection results without automatic differentiation, which is then used to build the computation graph by an extra Newton iteration [38]. Such strategies could reduce the memory occupation required for acquiring the initial solution and implementing differentiable Newton iterations.

Inspired by [33,38], we directly derive the backpropagation formula using the intersection equation. Here, the input of the intersection operator is the ray parameters $({{\boldsymbol x}_{{\rm ray}}},{{\boldsymbol r}_{{\rm ray}}})$ and the freeform surface parameters ${\boldsymbol \omega}$ and $h$. The output is the intersection parameter $t$. According to the intersection equation as shown in Eq. (5), we get the partial derivatives through the concept of implicit function differentiation:

The intersections of rays on the target plane form two-dimensional point clouds. The ray-energy statistical operator should determine the pixel location on the target plane where an intersection point is. However, such a round operation renders the process non-differentiable. Li et al. described the energy distribution of a light ray using a Gaussian function, enabling differentiability in ray tracing [36]. In this method, the energy of each ray is distributed to all the pixels on the target plane, resulting in a large amount of memory occupation for the intermediate results. Therefore, this method is only suitable for cases where the number of rays is small. Wang et al. utilized a linear interpolation scheme that involves rounding the fractional intersection point to the closest set of four neighboring pixels and applying linear weights to their respective pixel values [38]. This may be able to reduce memory usage significantly. However, they do not consider Fresnel losses of light energy.

Based on [38], we regard the energy distribution of a ray as a 2D rectangle function and allocate the energy to the four pixels closest to its intersection according to the area weights. As shown in Fig. 3, we find the nearest four pixels: A, B, C, and D of the intersection point $({x_{\rm t}},{y_{\rm t}})$. We then linearly distribute the energy weight ${w_{\rm t}}$ of the ray to the irradiance values of these four pixels according to the area weights ${S_{\rm A}}$, ${S_{\rm B}}$, ${S_{\rm C}}$, and ${S_{\rm D}}$, establishing a differentiable relationship between the irradiance and the ray coordinates (see Supplement 1). Although such a differentiable statistical strategy may deviate slightly from the standard QMC simulation model, it can significantly reduce the memory requirement for backpropagation. In addition, such a strategy can reduce the influence of the QMC simulation noises.

 

Fig. 3. Schematic diagram of the differentiable ray-energy statistical operator. The energy of a ray is distributed to the four pixels closest to its intersection coordinates.

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The computational graph of our differentiable QMC simulation is shown in Fig. 4, which contains the main flow and the details of how to trace rays through a freeform lens. In the forward simulation (marked by the blue arrows), the ray parameters ${{\boldsymbol \xi}_0} = ({{{\boldsymbol x}_0},{{\boldsymbol r}_0},{{\boldsymbol w}_0}})$ sampled from the extended source are input to the program. When the rays are traced through a freeform surface, the intersection points, corresponding surface normal vectors, refraction directions, and energy weights after Fresnel losses are calculated sequentially. The ray parameters are changed to ${{\boldsymbol \xi}_1} = ({{{\boldsymbol x}_1},{{\boldsymbol r}_1},{{\boldsymbol w}_1}})$ and ${{\boldsymbol \xi}_2} = ({{{\boldsymbol x}_2},{{\boldsymbol r}_2},{{\boldsymbol w}_2}})$ after passing through the two surfaces, respectively. Here, ${{\boldsymbol \xi}_0}$, ${{\boldsymbol \xi}_1}$, and ${{\boldsymbol \xi}_2}$ belong to ${\mathbb{R}^{7 \times N}}$. After that, the coordinates and weights ${\boldsymbol \kappa} = ({{\boldsymbol p},{{\boldsymbol w}_{\rm t}}})$ of discrete intersections at the target plane are counted as irradiance distribution ${\boldsymbol E}$. Finally, the loss function is calculated according to the target irradiance ${{\boldsymbol E}_{\rm t}}$. In the backpropagation (marked by the red arrows) based on the chain rule, we can get the gradients of the loss function with respect to the surface parameters, leading to efficient freeform lens design for extended light sources.

 

Fig. 4. Computational graph of the proposed differentiable QMC simulation.

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C. Optimization with Constraints on Freeform Lens Structure

We now describe the multi-scale optimization technique corresponding to the multi-level SRBFs. We employ the stochastic GD optimizer of MindSpore to update the parameters. When the parameters are updated once, we call it a GD step.

We start by optimizing the first layer of SRBFs, which has a global influence on the surface profile. The parameters of the higher-level SRBFs are then progressively added to the optimizer, which can involve more surface details in optimization. Such a process is repeated until realizing a satisfied performance. Aside from the natural link with the multi-level SRBF representation of the freeform surfaces, the major advantage of this coarse-to-fine optimization strategy is that the optimization could be insensitive to the initial solution and escape from premature iteration stagnation.

During optimization, we add three constraints on the structure of the freeform lens. The first constraint is aimed to ensure that the inner surface has enough dimensions to wrap around the light source. The maximum characteristic size of the square light source is ${r_{\rm s}} = \sqrt 2 d$, where $d$ is the side length of the square source. The coordinates $({x,y,z})$ of points in the Cartesian coordinate system on the inner freeform surface are required to satisfy that

The second constraint imposes a minimum distance between the inner and outer surfaces of the lens:

The third constraint limits the radial size of the outer surface of the freeform lens:

We employ the SUMT with Lagrange multipliers to realize the three constraints [49]. The total loss function with the Lagrange regularization term is

3. DESIGN EXAMPLES

In this section, we present two design examples to demonstrate the effectiveness of our approach. Both examples involve a square light source with a total power of 1 W. The material of the lenses is PMMA, which has a refractive index of 1.4936 at the wavelength of 564 nm [58]. During the optimization process, we sample ${2^{19}}({\approx 5.2 \times {{10}^5}})$ rays and configure the target plane with $64 \times 64$ pixels.

In each example, the inner and outer surfaces of the initial lens are both hemispherical surfaces, meaning that the vectors ${\boldsymbol \omega}$ and ${{\boldsymbol \omega}^\prime}$ are both set to zeros at the start of optimization. Throughout the multi-scale optimization process, the number of the optimized parameters gradually increases, and the learning rate $\tau$ of the GD optimizer decreases accordingly. At each GD step, the offset radii $h$ and ${h^\prime}$ are also optimized. During optimization, ${{\cal L}_{{\rm MSE}}}$ and ${{\cal L}_{{\rm unif}}}$ are employed in different stages.

All designs are implemented on the Intel i9-12900K CPU, Nvidia RTX3090 GPU, and the Ubuntu 22.04 system. Detailed optimization settings and time consumptions are provided in Supplement 1.

After optimization, the design results are evaluated by simulating the irradiance distribution using ${2^{24}}({\approx 1.6 \times {{10}^7}})$ rays and a receiver with $128 \times 128$ pixels. The simulation results are then filtered using a $3 \times 3$ Gaussian filter with a variance of one. Note that the ray-energy statistical operator used for the final evaluation does not employ the differentiable strategy.

A. Freeform Lens for Generating a Flat-Top Irradiance Distribution

As the first design example, we utilize the proposed design framework to design a double-freeform lens generating a square flat-top irradiance distribution. The side length $d$ of the light source is 1 mm. The target plane is positioned at ${z_{\rm t}} = 1000\;{\rm mm} $, with the target irradiance distribution represented by a 2D super-Gaussian function, as shown in Fig. 5(a). We aim at maximizing the irradiance uniformity within the area of $1155 \times 1155\;{{\rm mm}^2}$. The parameters of this target irradiance distribution and the uniform region are detailed in Supplement 1.

 

Fig. 5. Freeform lens design for generating a square flat-top irradiance distribution. (a) Target square-uniform distribution and uniform mask. (b) 3D view of the designed freeform lens and source, and the cross-sectional profiles for $\varphi = {0^ \circ}$ and 45°, respectively. (c) Illumination pattern generated by the lens and its sectional profiles along ${\rm y} = {0}$, ${-}{250}$, and ${-}{500}\;({\rm mm})$. (d) Evolution curve of ${{\cal L}_{{\rm MSE}}}$ during the optimization process. (e) Lenses (only the outer surfaces are illustrated) and corresponding illumination pattens of different multi-scale optimization stages (the color of the outer surface is according to $\partial \rho /\partial \theta$).

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The total number of SRBFs in the designed lens amounts to 790. Both the inner and outer surfaces of the lens comprise 395 SRBFs, which are distributed across three layers with the number of SRBFs in each layer being ${m_1} = 25$, ${m_2} = 81$, and ${m_3} = 289$. To initiate the optimization process, we establish the initial lens parameters as $h = 0.80\;{\rm mm} $ and ${h^\prime} = 2.45\;{\rm mm} $. We impose constraints on the lens dimensions as ${r_{\rm a}} = 0.20\;{\rm mm} $, ${r_{\rm b}} = 0.85\;{\rm mm} $, ${r_{\rm c}} = 2.45\;{\rm mm} $, and $\Delta r = 0.40\;{\rm mm} $, which is aimed to realize a high level of compactness.

Throughout the optimization procedure, the side length of the target plane is set to 1650 mm. The designed freeform lens exhibits a compact structure, as depicted in Fig. 5(b). The maximum height of the freeform lens is 2.17 mm, and the maximum radius, measured from the origin, is 2.42 mm. The inner surface is almost flat in the center and concave at the edge. This causes the source to appear bigger in the center and smaller near the edge when viewed from a pinhole on the outer surface. To achieve a sharp edge, the pinhole images near the edge need to be small, which is exactly what this inner surface achieves. Figure 5(c) displays the evaluated irradiance distribution, including the normalized line energy distributions around $y = 0\;{\rm mm} $, $y = 250\;{\rm mm} $, and $y = 500\;{\rm mm} $. These distributions closely match the target distribution. The irradiance uniformity that is defined as the ratio of the minimum irradiance value to the average one is approximately 0.93 within the area of $1155 \times 1155\;{{\rm mm}^2}$. Taking into account the Fresnel losses, the energy efficiency, which is used here to denote the ratio of the light power collected by a receiver with the size of $2400 \times 2400\;{{\rm mm}^2}$ to the total power of the source, amounts to 90.3%. It closely approaches the theoretical limit of 92.3% when all rays are perpendicularly incident on both the inner and outer surfaces. The loss function curve for the first three optimization stages is presented in Fig. 5(d). The curve initially exhibits a rapid decline, and then gradually flattens out as the iteration increases. When new parameters are added for further optimization, the loss function starts another rapid decline. This phenomenon endures until satisfactory results are attained. Figure 5(e) illustrates the intermediate and final results of the design process. It is evident that as the design degrees of freedom increase, the generated irradiance distribution progressively approaches the target one. Generally, the large-scale profile features of the freeform surfaces effectively converge the light energy towards the target region, while the small-scale features regulate the finer details of the irradiance distribution.

The performances of the SMS method [29], traditional optimization method [16], and our method in generating a flat-top irradiance distribution are compared in Table 1. Our design achieves results comparable to the state of the art in terms of uniformity, energy efficiency, and compactness.

Table 1. Performance Comparison: SMS, Traditional Optimization, and Proposed Method for Generating Flat-Top Irradiance Distribution

View Table

 

Fig. 6. Freeform lenses design for generating a $\pi$-shaped irradiance distribution. (a) Target irradiance distribution and the uniform mask. (b) Variation of ${{\cal L}_{{\rm MSE}}}$ with different lens size constraints ${r_{\rm c}}$. (c) Three of the designed freeform lenses and corresponding irradiance distributions, where the energy efficiency is calculated within the area of $300 \times 300\;{{\rm mm}^2}$. (d) Evolution curve of ${{\cal L}_{{\rm MSE}}}$ during the optimization process when ${r_{\rm c}} = 14\;{\rm mm} $. (e) Lenses (only the outer surfaces are illustrated) and corresponding illumination pattens of different multi-scale optimization stages when ${r_{\rm c}} = 14\;{\rm mm} $.

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Our differentiable method effectively utilizes the partial derivatives information, leading to fast convergence during the optimization process. Our method only requires 300 forward simulation evaluations to achieve this flat-top irradiance distribution. In [16], 19,296 forward evaluations were implemented in the design of a double-freeform lens with 98 parameters for generating a similar flat-top irradiance distribution based on the quasi-Newton optimization method. In [17], the design of a double-freeform lens involved the implementation of 2000 forward evaluations through the simulated annealing method. The design was accomplished using only 42 parameters, producing a uniform rectangular irradiance distribution.

The approach proposed in this paper minimizes the need for approximations in the simulation procedure, ensuring high accuracy. The total optimization time for this example was 2.43 h. When employing a total of 790 SRBFs, the average duration of a single GD step amounts to 39.27 s, with most of the time allocated to solving intersection points during the forward simulation. As the number of optimization parameters increases, the time per GD step also lengthens. Low-level code implementations or the fast ray tracing methods (for example, proposed in [16]) can be employed to speed up these computations. Additionally, thanks to our multi-level surface expressions and multi-scale optimization, it seems that our approach does not need a well-designed initial freeform lens.

 

Fig. 7. Experimental verification: (a) extended source; (b) machined lens (grid spacing: 0.25 mm); (c), (d) experimental setup; (e) normalized simulated irradiance distribution; (f) normalized experimental irradiance distribution; (g) comparison of simulated and experimental profiles across $y = 0$.

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B. Freeform Lens for Generating a Complex Irradiance Distribution

As for the second example, we design freeform lenses for generating a $\pi$-shaped irradiance distribution at ${z_{\rm t}} = 150\;{\rm mm} $, as depicted in Fig. 6(a). The target distribution contains a zero background. Such a high contrast and complex irradiance distribution is difficult to achieve by previous methods for an extended source with a large divergence angle.

During optimization, the size of the target plane is set as $300 \times 300\;{{\rm mm}^2}$. The side length $d$ of the light source is 2 mm. Both the inner and outer surfaces contain 404 SRBFs, distributed across four layers with the following number of SRBFs in each: ${m_1} = 9$, ${m_2} = 25$, ${m_3} = 81$, and ${m_4} = 289$. The optimization settings are described in detail in Supplement 1.

In this example, we conduct six designs with different size constraints. The minimum design time for these lenses amounts to 1.84 h, and the maximum time is 2.20 h (see Supplement 1). All the designed lenses achieve energy efficiency higher than 85% (the receiver size: $300 \times 300\;{{\rm mm}^2}$).

The variation of ${{\cal L}_{{\rm MSE}}}$ with the maximum radius constraint ${r_{\rm c}}$ on the lens is illustrated in Fig. 6(b). Additionally, three designed lenses are provided in Fig. 6(c). It is evident that as the constraint on the maximum lens size is gradually relaxed, the optimized irradiance distribution becomes closer to the target. This could be attributed to the fact that compacter structures only provide shorter optical path differences, thereby reducing the regulatory capabilities of freeform lenses. Consequently, in practical applications, it is essential to strike a balance between the irradiance performance and lens size in accordance with design requirements. For the case of ${r_{\rm c}} = 14.00\;{\rm mm} $, the loss function curve for the first four optimization iterations is displayed in Fig. 6(d). The intermediate and final results of the design process can be observed in Fig. 6(e).

We conduct experimental verification of our design method, as shown in Fig. 7. We select the CREE XLAMP XP-L HI LED, with an emitting surface of approximately $2 \times 2\;{{\rm mm}^2}$, as the light source [Fig. 7(a)]. Since the actual light source differs from the Lambertian source in both spatial and angular distribution, we load the rays data file of the light source product and import it into our framework for redesign, with the constraint set to ${r_{\rm c}} = 14.00\;{\rm mm} $. Additionally, we extend the lens’s base to create a cylindrical fixture with a one-inch diameter and a height of 2 mm for ease of clamping. Simulations indicate that this has a small impact on the design result. To enhance the manufacturability description, Supplement 1 analyzes sagittal deviations from best-fit spheres and details some specifics. The fabrication of the freeform lens [Fig. 7(b)] employs five-axis precision milling with a 1 mm milling cutter radius. Following accurate shaping of both surfaces, polishing is performed to attain a lustrous and smooth finish. The experimental setup, depicted in Figs. 7(c) and 7(d), includes a camera positioned behind the target plane (a light diffusing plastic panel) to record the irradiance distribution. We captured two sets of images with the LED in both on and off states to reduce the impact of background light through subtraction. Image processing involved locating the target plane’s corner coordinates and applying an affine transformation for calibration and scaling to correct for camera tilt. Figures 7(e) and 7(f) show the simulated and experimental irradiance distributions, respectively, with a structural similarity index measure (SSIM) of 0.81 indicating significant similarity. Figure 7(g) further confirms the match between actual and simulated distributions through a comparison of the central one-dimensional irradiance distributions. However, the experimental result exhibits relatively lower contrast, which could be attributed to precision errors in machining the freeform surface or mismatches between double surfaces. Additionally, deviations between the actual energy distribution of the light source and the standard product data may also lead to reduce the lens performance.

Supplement 1 also presents another example aimed at achieving an irradiance distribution like a plus sign. These examples demonstrate the effectiveness of our design approach for complex irradiance distributions. Through packaging software based on our design framework, we can enable designers to implement complex designs more easily. This feature is highly attractive in reducing the design cost of freeform lenses.

4. CONCLUSION

We have presented a differentiable design method of freeform lenses for extended sources. In this design method, we achieve an efficient combination of multi-level surface representation, differentiable simulation, and multi-scale optimization process with lens structure constraints. The proposed multi-level SRBFs can provide necessary global and local features to freeform surfaces for producing a high-quality beam pattern. In addition, the resulting freeform surfaces are very smooth, which can facilitate fabrication. The developed differentiable simulation, customized for extended sources and multi-level SRBF surfaces, can fast calculate the gradients of the loss function with respect to the surface parameters through just one backpropagation after one forward simulation. The multi-scale GD optimization corresponding to the multi-level SRBFs reduces sensitivity to initial states and improves design efficiency. The employed SUMT with Lagrange multipliers provides key feasibility constraints on the lens shape and size.

Using the differentiable design method, we designed a compact freeform lens generating a flat-top irradiance distribution. The energy efficiency amounts to 90% considering Fresnel losses and the uniformity (min/avg) within an effective region achieves 0.93. With three layers of SRBFs for either the inner or outer surface and a total number of 792 parameters, only 300 forward simulations were implemented to achieve this flat-top beam pattern. More importantly, our method is applicable to complex irradiance distributions, as demonstrated in the second example for generating a $\pi$-shaped beam pattern. Four layers of SRBFs for each surface and a total number of 810 parameters were adopted for this example. In addition, six designs with different compactness levels were provided, illustrating a trade-off between compactness and performance. Both simulation and experiment have demonstrated the effectiveness of our method.

The proposed method is also capable of optimizing lenses with fixed inner or outer surfaces, which can be spherical or aspherical surfaces. Such a design framework may facilitate the general application of optimization algorithms to the design of freeform illumination systems, forming a universal and standardized design process. It can enable designers to implement illumination designs for different requirements without too much mathematical and physical foundation, reducing design costs and promoting wide applications.

Some significant work remains to be done. Although our differentiable design method can significantly reduce the number of forward simulations required in the optimization process, it is still very necessary to reduce the time consumption based on improving the efficiency of surface calculation and intersection seeking process. Future work may also include the investigation of the effects of different optimizers and loss functions, the development of non-sequential ray tracing, and the sensitivity analysis. Furthermore, we also plan to take into account the diffraction effects in freeform lens design for coherent or partially coherent light sources, for instance, by employing a ray-based diffraction model [59].