Iterative freeform lens design for prescribed irradiance on curved target Download: 528次
1 Introduction
Manipulating the irradiance distributions of artificial light sources are very crucial for lighting and laser applications. For example, a Gaussian laser beam needs to be converted into a flattop one for improving laser material processing abilities. Compared with traditional spherical and aspherical optics, freeform optics has much more freedom, which can produce very complex irradiance distributions that are previously unimaginable. In fact, diffractive optical elements (DOEs), metasurfaces or graphene oxide lenses, could also realize the same goal while remaining flat1-3. However, freeform optics is still an energy efficient and cost-effective choice especially for macro dimensions. Freeform optics design for irradiance tailoring on a given target is a very difficult inverse problem. Komissarov, Boldyrev and later, Schruben showed that the design of a freeform reflector for a point source could be formulated as a second order nonlinear partial differential equation (PDE) of Monge-Ampère (MA) type4, 5. In Schruben's formulation, the MA equation is derived by mainly merging two types of equations. The first type is the energy conservation between the source intensity and the target irradiance. The second type is the ray tracing equations that describe the coordinate relationships from source to target. In addition, the reflector surface is constrained to have continuous second derivatives. Unfortunately, Schruben did not present the final expression of the MA equation and gave no hint on the numerical calculation. This is probably because that the derivation process is too complicated and the final MA equation is very difficult to solve. Wu et al. made a great effort to formulate a freeform refractive surface using the direct determination and employed Newton's method to solve the final MA equation6. Ries and Muschaweck created a different formulation process and solved a set of equivalent nonlinear PDEs, but they kept silent on the numerical techniques7.
Many other numerical methods have been developed for designing freeform reflectors and lenses. A common way is to approximate the freeform surface with sufficient quadric surfaces and to optimize their geometry8-12, but the computations may become slow for high-resolution irradiance tailoring. Ray mapping methods are also commonly used, and they simplify the design with two separate steps: i) ray map computation and ii) surface construction following the ray map13-20. The traditional ray mapping methods are only accurate for paraxial or small angle approximations. Larger surface errors could occur for off-axis and non-paraxial cases because the surface integration from the approximate ray maps are no longer integrable21, 22. Fournier et al. pioneered the work on computing an integrable ray map in freeform reflector design with the help of the method of supporting ellipsoids10. Bruneton et al. presented an efficient ray map optimization procedure, which allows the design of multiple freeform surfaces23. Bösel et al. directly solved the first order nonlinear PDE system related with the energy conservation and ray tracing equations24. Desnijder et al. acquired an integrable ray map by modifying an initial ray map using a symplectic transformation25. Doskolovich and Bykov et al. reduced the calculation of an integrable ray map to finding a solution to a linear assignment problem26, 27. Besides, least-squares ray mapping methods created by Prins et al. and modified by Wei et al. could also be employed to acquire an integrable ray map28-30. In our previous work, we introduced the iterative wavefront tailoring (IWT) method to obtain an integrable ray map through immediate construction of a series of outgoing wavefronts31.
Most of the above methods focus on producing prescribed irradiance distributions on planar targets. Very limited work has been done for curved targets. Aram and Wang analyzed the freeform reflector design for non-planar targets and suggested a weak solution based on approximating the required surface using piecewise ellipsoidal surfaces32. Bykov et al. showed that their linear assignment method is applicable for curved targets, although no supporting examples are provided and the method is intended for collimated incoming beams27. Wu et al. extended the direct determination of freeform lens design for irradiance tailoring on highly tilted target planes, which is still not applicable for curved targets33. Sun et al. employed a ray mapping method for producing a uniform irradiance distribution on a non-planar surface34. As mentioned before, such a ray mapping method may suffer from large surface errors when the design geometry deviated much from small angle approximations22.
To address the problems above, we develop a new IWT-based method applicable for a curved target. The new method can artfully dissolve the difficulties that arise from the fact that a curved target has varying z-values. In addition, the new method is developed under the stereographic coordinate system with an additional mesh transformation, which is applicable for light sources that emit light in semi space. The proposed method is described in details in the Theoretical model Section. To verify this method, two freeform-lens designs are demonstrated in Results and discussion Section for producing a rectangular flat-top and a circular non-uniform illumination patterns on a very undulating surface. A short conclusion is then provided in the final Section.
2 Theoretical model
Fig. 2. The rectangular (u , v ) grid (a) is transformed into a circular (u' , v' ) grid (b) which is used as the stereographic coordinates (c).
(uʹ, vʹ) then become the stereographic coordinates and their relationships with (X, Y, Z) can be expressed as:
We will describe the establishment of the outgoing wavefront equation in the following. Assume that the intensity of the light source is denoted as I(uʹ, vʹ), and its stereographically projected irradiance (SPI) on the (uʹ, vʹ) plane can be acquired as19:
The SPI on the (u, v) plane can be calculated according to energy conservation between the (u, v) and (uʹ, vʹ) planes:
From the differential form of Eq. (4), we can obtain the SPI on the (u, v) plane as:
Energy conservation between source and target can then be expressed as:
where dσ denotes the differential area element of the target surface. Generally, Eq. (6) can be written in the differential form as:
Next, we will link Eq. (7) to the properties of the outgoing wavefront
Since s and t are both functions of (u, v), according to the chain rule, we have:
Combining Eqs. (8) and (9), we can describe (x, y) as:
According to the previous IWT method31 for a planar target where z is constant, Eq. (10) could explicitly determine x and y as functions of u, v, s, t, w and the first derivatives of s, t and w. We can eliminate the two variables x and y by inserting Eq. (10) into Eq. (7) and thus obtain the final MA equation of w(u, v). However, since we concern a curved target here, z is no longer a constant and becomes a function of x and y: z=z(x, y). Even for a simple case, e.g., z=x2+y2, it is very difficult to express x and y explicitly, not to mention the derivation of the final MA equation. Such a fact could also explain why a direct determination of the freeform surface for a curved target is no easy work. However, we can avoid this difficulty artfully by involving z in the following iterative procedure. We consider z as a function of (u, v) and retain it in the right side of Eq. (10). We then insert Eq. (10) into Eq. (7) to obtain a MA equation of w(u, v):
where the coefficients Ai(i=1, 2, 3 and 4) are functions depending on u, v, s, t, w, z, ∂w/∂u, ∂w/∂v, ∂z/∂u, ∂z/∂v and the first and second derivatives of s and t. A nonlinear boundary condition can be specified by applying Eq. (10) for the boundary points.
It is noted that we cannot solve Eq. (11) with the nonlinear boundary condition unless we know s, t and z in advance. That is why we must employ an iterative procedure as shown in
Based on the above iterative procedure, the design complexities could be greatly reduced. Although a sequence of MA equations of the wavefront need to be solved, a multi-scale strategy, which is successfully used in the previous IWT algorithm for planar targets31, could be employed to speed up the computation. It is noted that the proposed method is also applicable for more complicated lens geometries, such as plano-freeform, aspherical-freeform or even double freeform lenses.
3 Results and discussion
To verify the proposed method, we first design a freeform lens for producing a flat-top illumination on an undulating surface shown in
Fig. 4. The desired curved target. The side length of the target is 240 mm and the z values range from 73.80 mm to 132.42 mm.
where (x, y) is confined within the domain Σ={(x, y)| -120≤x≤120, -120≤y≤120} (mm). The target irradiance distribution has the form of a super Gaussian function:
Suppose that the light source located at the origin has a Lambertian intensity distribution, and E(u, v) can be determined using Eqs. (3) and (5). The inner surface of the lens is set as a 12 mm-radius semi-sphere. The refractive index of the lens is set as 1.4932.
The desired computation size is set as 256×256. We implement the computations using the multi-scale strategy run in MATLAB 2019b. The initial computation size is set as 32×32, and a uniform rectangular grid of the target is adopted as the initial ray map. After implementing the procedure shown in
Fig. 5. (a ) The (u ′, v ′) grid corresponding to a uniform (u , v ) grid on Ω ={(u , v )|-0.94≤u ≤0.94, -0.94≤v ≤0.94} and (b ) the final target grid for the first design (only showing 64×64 grid points for better visualization); (c ) The final 3D freeform lens model and (d ) its simulation results for a point like source (size: 10-3 mm×10-3 mm). (The unit of the irradiance: W/mm2)
Fig. 6. (a ) The (u ′, v ′) grid corresponding to a uniform (u , v ) grid on Ω ={(u , v )|-0.8≤u ≤0.8, -0.8≤v ≤0.8} and (b ) the final target grid for the second design (only showing 64×64 grid points for better visualization); (c ) The final 3D freeform lens model and (d ) its simulation results for a point like source (size: 10-3 mm×10-3 mm). (The unit of the irradiance: W/mm2)
The second design is more challenging, which aims for generating a letter 'π' and its approximate value '3.14159∙∙∙' in a circular region with the radius of 120 mm on the undulating surface shown in
In the following, we will illustrate the influence of the source size and the distance from the source-lens system to the target surface on the performances of the two designed freeform lenses.
Fig. 7. Simulated irradiance distributions for the first lens design when the source size is changed into (a) 1 mm× 1mm and (b) 2 mm× 2 mm respectively; simulated irradiance distributions for the second lens design when the source size is changed into (c) 1 mm × 1mm and (d) 2 mm × 2 mm respectively. (The unit of the irradiance: W/mm2; the unit of the length: mm)
Since we concern irradiance tailoring on a non-planar target, it is necessary to show the effects of the distance from the source-lens system to the target on the simulated irradiance distributions.
Fig. 8. Simulated irradiance distributions for the first lens design when the source-lens system is (a) 5 mm and (b) 10 mm closer to the target. Simulated irradiance distributions for the second lens design when the source-lens system is (c) 5 mm and (d) 10 mm closer to the target. (The unit of the irradiance: W/mm2; the unit of the length: mm)
4 Conclusions
A new IWT-based method is proposed for designing freeform lenses that can produce prescribed irradiance distributions on curved targets. The method gradually refines the ray map and its corresponding z-coordinates of the target based on solving a sequence of parameterized wavefront equations. The ray map computed at the i-th iteration is obtained by solving the parameterized wavefront equation which imbeds the scattered z-coordinates of the target at the (i-1)-th iteration, and then the z-coordinates of the target is immediately updated according to the i-th ray map. The high performance of the proposed design method is confirmed by providing two examples of generating a rectangular uniform illumination pattern and an image with circular boundary on an undulating surface from a Lambertian light source. The method was developed under stereographic projection coordinates system, which adopted a special coordinate transformation of the source domain for obtaining reasonable results. In fact, this method may also be applicable for the spherical coordinate system.
In many cases the target cannot be considered as a perfect plane e.g. road surfaces on mountain area, sand tables, surfaces of sculptures and cultural relics. Therefore, we believe that irradiance tailoring on curved targets, which can be generally regarded as 3D surface lighting, can extend the applications of freeform illumination optics.
5 Acknowledgements
We are grateful for financial supports from National Key Research and Development Program (Grant No. 2017YFA0701200) and National Science Foundation of China (No. 11704030). The author Z X Feng thanks the valuable discussions with Xu-Jia Wang and Rengmao Wu.
6 Competing interests
The authors declare no competing financial interests.
[6] R M Wu, L Xu, P Liu, Y Q Zhang, Z R Zheng, et al.. Freeform illumination design: a nonlinear boundary problem for the elliptic Monge-Ampére equation. Opt, 2013, 38: 229-231.
[7] H Ries, J Muschaweck. Tailored freeform optical surfaces. J Opt Soc Am A, 2002, 19: 590-595.
[32] A Karakhanyan, X J Wang. On the reflector shape design. J Differ Geom, 2010, 84: 561-610.
Article Outline
Zexin Feng, Dewen Cheng, Yongtian Wang. Iterative freeform lens design for prescribed irradiance on curved target[J]. Opto-Electronic Advances, 2020, 3(7): 07200010.