Surface topology and functionality of freeform microlens arrays

Nonsymmetric (also known as freeform) optical components have attracted a great deal of academic and industrial attention due to the substantial benefits they have demonstrated in imaging and nonimaging optical systems. Additionally, freeform microlens arrays (FMLAs) are very promising with regard to the growing demand for device miniaturization and cost reduction. As a flip side, FMLAs entail specific challenges in design, manufacturing, and characterization. Here we report on the latter and present an innovative characterization strategy that makes it possible to assess the quality of FMLAs quickly and accurately. The precisely measured surface topology of FMLAs was accurately represented using nonuniform rational basis-spline (NURBS) and its optical response was predicted by means of ray-tracing simulations. We show that for reliably measured surface topology, the results are in excellent agreement with the experimental measurements. We also show that, compared to previous studies, illuminance levels displayed in a logarithmic scale are more adequate for low light levels and represent a closer match to nonlinear human visual perception. We believe that the method presented here will contribute to speeding up the FMLA manufacturing process, one of the current downsides of this promising technology. © 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Freeform optical components are nowadays applied in several fields such as automotive and professional lighting, aerospace, virtual and augmented reality, and biomedical engineering. Designed and manufactured with no symmetry constrains, FMLAs are preferred to conventional, rationally symmetric components to, for example, reduce aberrations [1,2] and expand the field of view [3,4] of imaging systems as well as to realize lighting systems with nonsymmetric illumination patterns. The use of freeform optics often leads to optical systems with a better optical response, a lower number of components, a smaller size [5], and a higher efficiency [6]. There are different classes of freeform surfaces [7], and in this article we cover structured surfaces, in contrast to, for instance, single-surface freeforms. Origination technologies for freeform micro-optics include laser ablation, gray-scale laser lithography, two-photon polymerization, and ultraprecision mechanical micromachining [8][9][10][11][12][13][14].
Micro-optics foster system miniaturization and can be manufactured over large areas using cost-competitive (e.g., roll-based) processes. Freeform microlens arrays, FMLAs, combine both aspects, and their industrial interest is consequently growing rapidly. However, manufacturing them is often time-consuming and cost-intensive and therefore usually restricted to small-area masters, which are thereafter used for large-area replication.
In addition to assessing the quality of the final products (e.g., the large-area replicas), intermediate quality checks are required on the masters and the replication tools. Because of their small area, their handling and alignment in optical setups is prone to large experimental errors or simply unfeasible. Also, replication tools (e.g., nickel shims) are typically non-transparent and hence optically non-functional. For this reason, quality is commonly assessed only from perspective of surface topology, that is, the surface form of manufactured FMLAs (masters or replication tools) is accurately recorded and compared to the nominal design [11,[13][14][15][16][17][18].
Although this approach offers fundamental information on manufacturing deviations, it provides little or no insight into their impact on optical performance. As a result, resources may be unnecessarily expended on optimizing the manufacturing processes with a negligible performance enhancement in return. Our method fills in this gap and enables functionality predictions based on the experimentally measured FMLA surface.
Creating 3D CAD models from scanned surfaces, often referred as reverse engineering [19], is widely applied for rapid manufacturing in fields such as mechanical engineering, architecture, and systems biology [20,21]. Reverse engineering has been used to investigate the optical performance of a freeform Alvarez surface [22] using fourth-order polynomials. Sieber et al. used measured topology of a freeform optical component and converted the point cloud into a computational model for optical simulations on a single lens. However, reverse engineering FMLAs, which have hundreds to thousands of intricately shaped microstructures, requires innovative surface reconstruction and representation strategies. FMLAs usually contain thousands of complexly shaped (often different) microstructures and hence demand for more efficient surface representation methods.
Our characterization approach uses recorded surface topology to build 3D CAD models based on nonuniform rational basis-spline (NURBS) surfaces. These models' optical performance can be accurately predicted using ray-tracing simulations. NURBS is a mathematical model that uses B-splines to represent curves and surfaces. Their attractive mathematical properties to represent complex freeform surfaces make them the first choice in many areas such us e.g., computer graphics [23]. NURBS surface representation was chosen because of its multiple advantages. First, it is widely implemented in CAD and optical simulation software tools. In addition, NURBS has a superior ability (than, e.g., polygonal meshes) to represent complex nonflat surfaces with manageable low file sizes. Moreover, contrary to analytical expressions, NURBS makes it possible to represent local deformations (e.g., manufacturing deviations).
We have recently demonstrated the potential of this method [24], and here we demonstrate it further using more intricately shaped microstructures, different surface characterization tools, and advanced data-processing. We also report on its ability to correlate the predicted performance degradation with specific manufacturing deviations. Our method is a new approach for making accurate performance predictions based on the experimentally measured FMLA surface topology.
The article is organized as follows: First we present the setups of the experimental and optical simulations. Then we describe the method used to create ray-traceable 3D models, how we validated them by comparing the predicted and measured optical performance, and we report on the potential of the proposed method for tolerance analysis. Finally, we report how we correlated the observed performance degradation with specific manufacturing deviations.

Methods
For this study we selected three FMLAs (see Fig. 1) with feature sizes covering a large range (10 to 300 µm) in the lateral and vertical directions. Also, form-wise, they contained sharply edged micropyramids as well as smoothly curved microlenses. Sample 1 was a commercially available (microstructured) deglaring solution (Polyscale GmbH and Co. KG). Sample 2 was a replica of sample 1 and was fabricated with two-photon polymerization. Sample 3 was an engineered diffuser again fabricated by two-photon polymerization. In all cases, the microstructures were fabricated on the (top) surface of an otherwise flat substrate.
The experimental setup used to measure the optical performance of the selected FMLAs is displayed in Fig. 2. The optical bench was composed of a 532 nm wavelength laser source coupled to a beam expander (Thorlabs BE06R/M), a Lambertian diffuser (referred to here as the observation plane), and a CCD Luminance camera (Konica Minolta CA-S25w), which was focused on it. A cage system was implemented to prevent unintentional sample tilt and to facilitate the FMLA-beam alignment. The CCD camera (980 × 980 pixel array) placed behind the observation plane records the luminance distribution over a 19.5 × 19.5 cm 2 area of the observation plane. The distance between the FMLAs and the observation plane was adjusted to achieve a 19.5 × 19.5 cm 2 irradiance distribution on the observation plane and hence a maximum resolution (19 cm for samples 1 and 2, and 12 cm for sample 3). All samples were placed with their microstructured surface facing the incoming light beam. The surface topology was measured with a confocal (λ = 404 nm) laser scanning microscope (Keyence VK-X1100). A 2 × 2 mm 2 area was measured in the stitching mode using a 50x magnification objective. The measured surface form (provided as an XYZ point cloud) was converted into a NURBS-based ray-traceable 3D solid model using Rhinoceros surface-modelling software (version 6) and python scripts. In order to increase the surface accuracy in the fitting process, the measured surface was divided in small pieces of 0.2 × 0.2 mm 2 , and each one was fitted independently. The size of the divided surface parts was a compromise between increasing the accuracy by minimizing the size and decreasing the computational effort so that the commercial software could handle the task. Finally, the fitted NURBS surfaces were merged into a single NURBS-based 3D model. This process, sketched in Fig. 3, produced 3D models with standard deviations below 0.2 µm (from the experimentally measured data; computed using the Rhinoceros deviation analysis tool, which analyzes the point deviation of the measurement point cloud from the created surface model). Finally, the 3D model was imported into LightTools ray-tracing software (version 9.0) and ray-traced using over one million rays under nominally the same conditions used in the optical characterization bench (Fig. 4). The refractive index values used in the simulations were set according to the information given by the material and sample suppliers for λ = 532 nm.

Freeform surface representation based on measured surface topology
Previous studies have shown that surface shapes recorded with accuracy levels in the order of tens of micrometres may be insufficient for studying the optical performance of freeform optics [22]. Indeed, small topological deviations may translate into unacceptable performance drops. Accurate surface topology is therefore essential to making accurate predictions. Our approach of creating a computational model using NURBS allowed higher fitting accuracy compared to polynomial fitting.
Analytical expressions such as Zernike and other higher order polynomials are widely used to represent freeform optical components with surfaces that deviate slightly from the base sphere, as is often the case in imaging applications. From a design perspective, analytical representation is advantageous because optical performance can be maximized using parameter-optimization algorithms. However, this is of little advantage with regard to characterization, since correlating manufacturing deviations and polynomial coefficients may be far from evident or even impossible, whereas NURBS can naturally describe local deformations and deviations. Finally, complex microstructured surfaces (like those used here) can hardly be expressed analytically. For these reasons NURBS representation is more appropriate for the characterization of FMLAs.

Comparison between predicted and measured optical performance
Currently, quality-control processes focus solely on comparing the measured and nominal surface topology. As highlighted above, this approach may lead to erroneous conclusions. For example, considerable resources may be devoted to reducing manufacturing deviations that have negligible (or at least acceptable) impact on optical performance. In this section, we demonstrate that this can be avoided using ray-traceable 3D models.

Deglaring micro-optics
The recorded illuminance patterns produced by samples 1 and 2 are displayed in a linear scale in Fig. 5(a) and 5(d). As expected, both samples produced similar patterns: hexagonally arranged bright spots. However, when displayed in a logarithm scale (Fig. 5(b) and 5(e)) additional features became evident (see the connecting lines between the bright spots). Illuminance levels displayed in a nonlinear scale reflect human perception more accurately. Human perception of sensory stimulus (including light) is proportional to the logarithm of the stimulus (Weber-Fechner law [25]). Due to the different sensitivities of photoreceptors in the eye, the same difference can be perceived better at lower light levels than at brighter light levels. The nonlinear visual perception of light levels is well known and applied in postprocessing in imaging cameras [26] as well as in the field of biology and behavior [27][28][29][30]. For lighting applications, this aspect is often overlooked. It is common practice for most lighting research to display illuminance in a linear scale [31][32][33][34][35][36][37].
The experimentally measured and predicted illuminance distributions (displayed in Fig. 5) are in excellent mutual agreement, including the faint light lines. This is a clear indication of the reliability of the proposed method, which ultimately reflects the accuracy in the surface profile topology measurements as well as the high fidelity of the NURBS fitting process.
Since the method has been validated against experimental measurements, it can be used in a real application. As already mentioned, the optical microstructures of samples 1 and 2 were designed for deglaring Lambertian lighting panels, that is, to reduce the luminous intensity angles over 60 degrees with respect to the normal.
The distributions of simulated angular luminous intensity, which were recorded by a far-field angular detector, are presented in Fig. 6 together with the Lambertian distribution of the lighting panel. As expected from previous simulations and as can be seen in Fig. 6, both samples had similar deglaring capabilities.  6. Normalized angular radiant intensity of freeform structures 1 and 2. The intensity slice of the C-type detector in plane C0-180 is displayed. The freeform samples represent deglaring foils manufactured with two different fabrication technologies. The optical simulations showed a redirection of light rays above 60 degrees in a forward direction. The light emission in backward direction is due to the reflected light intersecting the first surface.

Engineered diffuser
The experimental and simulated illuminance distributions produced by sample 3 are shown in Fig. 7. This FMLA was designed to produce flat-top laser beams (uniform square-shaped light distributions).
When comparing higher magnitudes of illuminance seen in the linear-scale figures, one could observe a deviation in the exterior areas. This seemed to be the result of unreliable surface-form measurements. A close inspection of the areas of high angles showed that the confocal laser microscope was not capable of measuring the surface topology with sufficient accuracy. Figure 7(f) shows the 3D data used for modelling, and artifacts are visible. The surface topology measured with a better objective and higher accuracy (Fig. 7(e)) for the same sample showed a smoother surface and indicated that the form deviations were artifacts due to limitations of the 50x magnification objective. Unfortunately, the high magnification of the objective translated into a smaller field of view and ultimately to much longer measurement times for a given area and highlights the need for faster and more reliable surface characterization tools.

Tolerance analysis
The presented method can be applied to investigate tolerances in optical performance linked, for example, to rotation, tilt, and displacement. Such an analysis could also be done with a nominal design. However, the manufacturing deviations may translate into different tolerances for the manufactured samples. Our approach thus allows not only analyzing for tolerances with the optimal nominal designs but also analyzing fabricated prototypes without involving complex experimental setups prone to errors.

Optical relevance of manufacturing deviations
The proposed method can be used to compare origination and replication. For example, sample 2 was fabricated as a replica of sample 1. As a result, they produced slightly different illuminance patterns, as clearly seen in Fig. 5. Specifically, sample 2 produced broader lines.
Backwards ray-tracing simulations (Fig. 8) indicated that the edges of the micropyramids were responsible for light redirection to these areas. A close-up look at the profiles (Fig. 1) revealed that the edges in sample 2 were less sharp and were hence expected to produce less sharp light patterns. This is a valuable information for improving manufacturing if the observed broadening has to be minimized. Any effort to obtain sharper edges would therefore not pay off performance-wise.

Conclusion
We presented an innovative method for characterizing manufactured FMLAs based on experimentally measured surface topology and ray-tracing optical simulations. Reverse engineering is a well established process in various fields, but fairly new in micro-optics. To our knowledge, surface fitting FMLAs with submicrometer accuracy has never been reported. The described method can accurately simulate and successfully predict the optical performance of manufactured FMLAs without confronting the limitations of experimental optical setups.
The method exploits recent improvements in surface topology characterization tools, continuously increasing computational power, and the excellent mathematical properties of NURBS to represent complex surfaces.
This method demonstrated accurate predictions and is therefore capable of discriminating between relevant and irrelevant manufacturing deviations. When predicted performance is conveniently displayed (e.g., in a logarithmic scale), nonevident features can be investigated and corrected if necessary. The method therefore has a strong potential to speed up FMLA manufacturing and quality control and to reduce the overall manufacturing costs.
We have demonstrated an improved method for investigating microscale form deviations by displaying illuminance over several orders of magnitude. Logarithmic illuminance representation provides closer match to visual perception and can be beneficial to take into account for lighting applications, especially regarding low light levels. It can help to design lighting solutions with improved performance that are linked to human perception such as wall washers, road and interior lighting solutions.
Disclosures. The authors declare no conflicts of interest.