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Abstract
Recent energy efficient illumination advancements have capitalized on using light-emitting diodes (LEDs) in combination with freeform lenses. However, the available freeform lens design methods are application specific. In addition, manufacturing of this class of optics is challenging. In this work, considering manufacturing constraints, we apply a customized algorithm to design freeform lenses for (1) transforming LED radiation into a uniform rectangular illumination pattern and (2) shaping collimated LED light beams into complex image target irradiance distributions. The algorithm is based on a numerical solution of the elliptic Monge–Ampère equation. Then, we proposed manufacturing these lenses using a modified ink-jet 3D printing technology called Printoptical technology. The demonstrated optical performance of printed lenses, exhibiting surface roughness of $\textrm {RMS} = 10\pm 2$ nm, is in good agreement with the simulation. We also explored an industrial application of the 3D-printed lens matrix for low-cost illumination of a paper milling machine.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
An estimate of around 20$\%$ of global electricity production is consumed by the lighting industry [1]. Thus, energy-efficient solid-state lighting (SSL), especially using light-emitting diodes (LEDs), has become a trend due to its high efficiency compared to conventional lamps. However, the emitted radiation has to be directed into a specific area in order to save energy and reduce light pollution. Such re-distribution of light radiation can be achieved using freeform optics.
Freeform-optics with no-axis of rotational symmetry provides improved optical performance with small packaging size for different applications [2,3]. Though used for transforming an extended light source into prescribed target irradiance, the methods available for designing freeform optics are application specific. Most commonly one treats the primary light source as a point source [4–6]. For instance, Ries et al. have proposed one of the most effective and efficient point-source-based freeform optics designing method that is available as commercial software [6]. Various research groups have proposed algorithms to relax the point source assumption: an extended light source is introduced by using optical phase space concept [7] and an iterative adaptation technique on the target illumination to design a freeform reflector for a small $(1\times 1~\textrm {mm}^2)$ LED source [8]. Oliker has also proposed to design freeform multifocal lens using supporting quadric mathematical method for collimated beam source [9]. The Simultaneous Multiple Surface (SMS) method works relatively well for an extended source by introducing more than one freeform surface [10]. Nevertheless, freeform optics design methods are limited by the size of LED sources when compact packaging is considered, and the selection of the design method has to be based on the application demands. Commercial software tools like LightTools and ffOptik include freeform lens design algorithms. However, the supporting side surface to the tailored freeform surface has steep slopes greater than 90 degrees, for instance, when a flat entrance surface is assumed [11]. Moreover, when a spherical entrance surface is considered for highly efficient designs, the freeform lens geometry usually looks like a dome. These conditions lead to further post-processing of the CAD designs if 3D printing is considered as the fabrication process [12]. Thus, the design has to be customized for the optic 3D-printer that is capable of printing less than 90-degree-slope side surfaces in a single printing process. Once these design issues have been adequately resolved, challenges remain in the manufacturing process.
The recent development of diamond-turning machines with c-axis or slow-servo motors open a door for the realization of complex freeform lenses. However, the overall process for prototyping and small series production with injection molding is expensive and time consuming [13]. As a result, alternative fabrication processes such as 3D printing deserve attention as they promise a low-cost and rapid prototyping and small-series production of free-form optics.
Additive manufacturing, or 3D printing, of optics has been demonstrated using two-photon direct laser writing to realize sub-millimeter-size freeform optics [14]. PolyJet and Stereolithography (SLA) based 3D printing technology have been used to demonstrate lightguide and freeform optical sensors [15,16]. However, optics printed by these technologies suffer from poor surface quality: strip lines due to the UV curing process can be seen on the optical surface by naked eye [16]. Thus, to achieve the optical functionality, post-processing like surface polishing has been applied on the optical elements. Furthermore, 3D-printed transparent freeform optics has been claimed using pulsed infrared (IR) laser and thermally curable optical silicones in layer-by-layer printing methods. Hong et al. presented optics with surface roughness of $\textrm {RMS} = 15$ nm and surface profile deviation of $\pm 20~\mu$m at the center of a printed lens [17]. However, as true freeform optics is in general rotationally non-symmetric, controlling the print droplets precisely is necessary for resolving the details on the freeform surface. Luxexcel has demonstrated ophthalmic lenses with ISO 8980-1:2004 focal power using Printoptical technology [18]. Recently, we have applied this technology and demonstrated 3D-printed freeform lenses as a proof of concept [12]. Here we further investigate this technology and demonstrate the complete process chain from designing freeform lenses using a custom-made algorithm up to fabricating and characterization of the optical surface for an industrial application. We have also investigated the optical performance of the design freeform lens for randomly generated surface profile deviation considering printing error.
In section 2, we describe the custom ray-mapping-based freeform lens design method. We then design freeform lenses as an example for different case studies in section 3. The experimental demonstration is presented in section 4, followed by conclusions in section 5.
2. Designing freeform lenses
In the design of freeform optics we have chosen ray-mapping approach based on optimal mass transport (OMT) method [19–21]. The method consists of two separate steps: The determination of a ray mapping between the input and the output irradiances and the reconstruction of the freeform surface with the help of Snell’s law (Fig. 1). The ray-mapping can be determined by solving numerically the OMT problem, also known as Monge-Kantorovich problem, which can be defined as follows: two positive density distributions must satisfy the conservation condition
where $S(x_s,y_s)$ and $T(x_t,y_t)$ denote the source and target irradiance distributions. The two distributions are mapped to each other according to Jacobi equation where $\det (\nabla \Phi (x_s,y_s))$ is the determinant of the Jacobian matrix of a smooth bijective mapping $\Phi (x_s,y_s)$. There may be many mappings which satisfy Eq. (2) but we are interested in a mapping which minimizes the transport cost according to Kantorovich–Wasserstein distance This mapping is unique [22] and characterized by a vanishing curl [23]. Furthermore, the optimal mapping $\bar {\Phi }$ can be expresses as a gradient of a convex function, $\bar {\Phi } = \nabla \Psi$. Substituting this into Eq. (2) leads to the elliptic Monge–Ampère equation We solve this equation using the method developed by Sulman et. al [21], where the numerical solution of Eq. (4) is obtained by finding a steady-state solution of the logarithmic parabolic Monge–Ampère equationAfter the calculation of the ray-mapping, the freeform surface is reconstructed by a least-squares optimization procedure following the steps shown in Ref. [19]. The surface is approximated by a triangle mesh with vertices whose positions are given by vectors
where $\textbf {r}_0(i)$ is the origin of ray $i$, for example, the position of a point source, $\textbf {s}(i)$ is a unit vector, and $\lambda (i)$ is a scalar parameter which defines the surface point $i$. The normal vectors at the vertex points are obtained as weighted average of the normals of the neighbor faces. The ray positions at the target plane can now be calculated and the surface can be computed by minimizing the objective function where $x_T(i)$ and $y_T(i)$ are the actual local coordinates at the target plane for a given vector of parameters $\lambda$, and $x_t(i)$ and $y_t(i)$ are the coordinates computed by the mapping algorithm. In our case we used Levenberg–Marquardt algorithm to minimize Eq. (7). To ensure that the algorithm converges, the initial guess of the surface shape has to be sufficiently close to the optimum.3. Design examples
3.1 Case study 1: uniform rectangular illumination
We have designed a freeform lens for ideal LED based uniform rectangular illumination using the customized algorithm. The design was extended to an industrial illumination application for inspection of defects in paper mill machine camera system. The freeform-lens design parameters are listed in Table 1.
Table 1. Design specifications of free-form lenses for rectangular and paper web uniform illumination
The designed lenses for ideal and paper web machine illumination cases are illustrated in Fig. 2. The 3D-CAD design in Fig. 2 (a & c) depicts the rotationally non-symmetricity of the designs. In addition, the top-view contour of the design presented in Fig. 2 (b & d) show clearly the difference between the two designs geometries.
Fig. 2. Freeform lens designs. (a) 3D-CAD format and (b) top-view contour of the freeform lens for an ideal uniform rectangular illumination; (c) 3D-CAD format and (d) top-view contour of the freeform lens for paper web illumination.
The designed freeform lenses were analyzed with ZEMAX OpticStudio 16 software using $20 \times 10^6$ rays. The result for the uniform rectangular illumination is shown in Fig. 3(a). The designed freeform lens is seen to direct light accurately from the LED source into a rectangular form. The relative irradiance uniformity at the center of the target irradiance distribution is around $80-90\%$ as shown in Fig. 3(c).
Fig. 3. Simulation results: relative irradiance distribution for (a) the ideal rectangular and (b) the paper web illumination case studies at the target plane; cross sections in $x$ and $y$ direction for (c) the ideal rectangular and (d) the paper mill machine illumination cases.
In the paper-web machine illumination case illustrated in Figs. 3(b) and 3(d) the irradiance pattern is highly uniform in the $y$ direction but less satisfactory in the $x$ direction, where the required beam divergence is larger. This is mostly due to the failure of the point-source assumption as the LED is now at a distance of 1.5 mm from the flat entrance surface of the free-form lens and its assumed size is $3.45 \times 3.45~\textrm {mm}^2$. In order for the point-source assumption to hold, the distance between the source and lens surface has to be around five times the source size [24]. Indeed, this criterion is not fully satisfied in the uniform rectangular case either. Video 1 associated with Fig. 4 illustrates the effect of LED source size on the uniformity of the rectangular target illumination. The uniformity degrades especially after the source size exceeds $6\times 6~\textrm {mm}^2$. The target illumination becomes almost circular when the LED size increases towards $10 \times 10~\textrm {mm}^2$.
Fig. 4. LED size effect on the target illumination (See Visualization 1, MP4, 378 KB). The video show the simulated irradiance pattern as the source size increases from $1 \times 1~\textrm {mm}^2$ to $10 \times 10~\textrm {mm}^2$.
3.2 Case study 2: complex target irradiance distributions
The customized algorithm is also applied to design freeform lenses to display a complex Plato and Aristotle and Lenna images as the target illumination pattern. A collimated beam from a rectangular source of size $20\times 20~\textrm {mm}^2$, located at a distance of 10 mm from the flat entrance side of the lens, is assumed in both cases. The size of the lenses is $20 \times 20~\textrm {mm}^2$, the target is at a distance of 200 mm from the lenses, and the sizes of the target irradiance distribution are assumed to be $50 \times 50~\textrm {mm}^2$ for Plato and Aristotle and $40 \times 40~\textrm {mm}^2$ for Lenna cases. Figures 5(a) and 5(b) show the freeform design and its sub-microns feature surfaces, respectively, for the required Plato and Aristotle target irradiance distribution illustrated in Fig. 6(a). Similarly, Figs. 5(c) and 5(d) show the freeform design and its micrometer size feature surfaces, respectively, for the required Lenna target irradiance distribution illustrated in Fig. 6(d). The feature size of the freeform lens for Lenna target image is two times bigger than Plato and Aristotle target image, that makes it relatively less difficult to fabricate. The calculation of the initial ray-coordinate mapping, for example, for the $256 \times 256$ pixel Plato and Aristotle target image using 10,000 iterations took around 150 seconds in MATLAB R2016b on a 8 GB RAM laptop with an Intel Core i5-7200U at 2.5 GHz processing speed. However, depending on the initial parameters, the mapping could be iterated repeatedly until it converges, which usually takes a few minutes for each case. Similarly, the surface construction took a few minutes for each case considering a good initial guess surface.
Fig. 5. Freeform lens design for complex target irradiance distributions: (a) 3D-CAD format and (b) sub-microns features of freeform lens design for Plato and Aristotle target image; (c) 3D-CAD format and (d) sub-microns features of freeform lens design for Lenna target image.
Fig. 6. Numerical simulation of the freeform lens designs for complex target images : (a) & (d) ideal prescribed target irradiance distribution of Plato and Aristotle and Lenna images, respectively; (b) & (e) ray-traced target image irradiance distributions and (c) & (f) the absolute difference between the ideal and simulated target image irradiance distributions of the freeform lenses designed for Plato and Aristotle and Lenna images, respectively.
Figure 6(b) and 6(e) show the numerical simulations in ZEMAX OpticStudio 16, with $100\times 10^6$ rays. The simulation show the details of the target images convincingly, though some blurring is visible as one may expect due to the limitations of the optimal mass transport map algorithm [25]. Another reason might be an insufficiently precise design conversion process in ZEMAX software from the grid sag format of the design [26]. Figure 6(c) and 6(f) show the absolute difference between the required and simulated target irradiance distributions. The performance of the resulting output irradiance using relative root-mean square deviation (RRMSD) is calculated to be 0.0902. The simulated image quality is limited by the diffraction and geometrical resolution parameters [27].
This design limitation has been analyzed and improved by controlling both the phase and the irradiance part of the light source. For example, Feng et al. used an iterative Fourier-transform algorithm for further processing the solution of Monge–Ampère type equation on the formation of a smooth freeform lens [25]. Bösel et al. used two freeform surfaces for directing both the phase and irradiance independently beyond paraxial domain [28]. However, the fabrication error is a bottleneck for realizing the performance improvement in the design. For example in Fig. 7, we have shown the optical performance of the freeform lens for Lenna target image with randomly generated surface profile deviation of $\pm$100 nm, $\pm$ 250 nm and $\pm$500 nm that assigned to the printing error. The simulation result demonstrate that the optical performance of the lens is very sensitive to the printing error within sub-$\mu$m scale, so that the lens has to be printed with precision nano-meter scale in order to observe a good target image qualitatively. However, the effect of the manufacturing error on the optical performance of the design could be minimized by modifying the design for shorter target image distance since the distance is almost inversely proportional to the sag value of the freeform surface. Moreover, the surface form deviation introduces on the freeform surface from the 3D-printing errors are random, in the first instance, due to the variation of the droplets sizes. However, the printing errors become increasingly deterministic in the second and consecutive printing trials, which opens an opportunity for correcting the 3D-printing error by introducing the measured surface deviation on the design surface.
Fig. 7. Numerical simulation of the freeform lens design with randomly generated surface profile deviation for Lenna target images case study: optical performance of the design for additional surface profile error of (a) $\pm$ 100 nm, (b) $\pm$ 250 nm and (c) $\pm$ 500 nm.
4. Experimental demonstration of 3D-printed freeform lenses
The fabrication of the freeform lens is accomplished using a custom-made 3D-printer (see Fig. 8) based on Printoptical technology that uses Opticlear, a PMMA like polymer material.
The Opticlear$^\textrm {TM}$ material optical transmission without including fresnel reflection is 99.4$\%$ for 1 mm thick flat plate with refractive index of $n = 1.53$ at 587 nm wavelength [12,29]. We employed a drop-on-demand printing technique with a series of modified ink-jet printer heads that have a nozzle droplet volume of $2.6~$pl and print layer thickness of $4.1~\mu$m. The printing takes place layer by layer and, in order to solidify the droplets, we have used a UV-curing lamp. The feature sizes on the designed freeform lens surface for the Plato and Aristotle are in nanometer. Relatively for Lenna target image case, the feature sizes of the freeform lens surface are bigger but still around micrometer. Thus, it is difficult to realize these designs since it is below our current, Printoptical technology, 3D-printer nozzle droplet size, i.e. ($\thickapprox 17~\mu$m diameter). As a result, we chose to demonstrate the freeform lens designed for the paper web machine illumination case study, due to its simplicity and application value. Figure 9(a) shows a 3D-printed freeform lens with a thickness of 8.15 mm, a width of 12.6 mm, and a length of 13.8 mm, which is printed in less than three hours. No post-processing, such as polishing, or coating, is used to improve the surface quality. The surface roughness, measured using white light interferometry, was found to be around $\textrm {RMS} = 10\pm 2$ nm. This is around three orders of magnitude better than the roughness attainable with other 3D-printing techniques [16].
Figures 9(b) and 9(c) demonstrate the surface profile deviation of the printed lens from the design in two and three dimensions, respectively. The surface profile measurements were performed with a Keyence VR-3000 series wide-area 3D measurement system. The measurements show that the surface profile variation is around $\pm 40~\mu$m at the center and $\pm 100~\mu$m at the edge of the lens. The higher deviation at the edge of the lens might be due to the limitation of printing 84-degree or higher side slope for thicknesses above $200~\mu$m, c.f. Figures 9(d) and 9(e).
Fig. 9. Surface measurement of 3D-printed freeform lens. (a) 3D-printed freeform lens top view; the surface profile deviation between the design and printed lens (b) top-view and (c) 3D-view; cross-section surface profile thickness at the center of printed lens in (d) horizontal-direction and (e) vertical-direction relative to the design.
Array of freeform lenses can be straightforwardly printed, as shown in Fig. 10(a). Such arrays can be useful in improving the illumination in, e.g., the camera system of the paper web case study by using LED arrays. Polycarbonate sheet of thickness 1.5 mm was used as a support for the 3D-printed lens matrices. Figures 10(b) and 10(c) depict the layout and actual illumination in the paper-web industrial camera system with an LED-optic module, which is used to detect web breaks, defects over web width, and edge defects like wrinkles and cracks.
Fig. 10. 3D-printed freeform lens matrix for paper web illumination. (a) 3D-printed freeform lens array, (b) layout of paper web illumination, and (c) actual illumination of the paper web.
The printing area of the 3D-printer is currently 6 $\times$ 7 cm$^2$. But it can be scaled up to a larger printing area by aligning parallelly more than three print-heads, allowing many lenses to be printed at once. An industrial application of the printed lenses is illustrated in Fig. 11 where six times printed $5\times 3$ lens matrices are on top of the $15\times 6$ LED matrix. The experimental result demonstrates that the light distribution resembles the desired $45^\circ \times 35^\circ$ target distribution.
Fig. 11. LED based 3D printed freeform lens array for paper web illumination. (a) Illumination device using 3D printed lens arrays and (b) the experimentally demonstrated target distribution.
Depending on the application, a high UV-dosage curing process may be used in the printing process, which will lead to higher yellowness index of the printed lenses. The yellowness index can be decreased after the printing process using blue-LED cabinets or a tinting process where the lens is immersed in a container of a solution of organic dyes for an appropriate time. Even if the dyes fade too fast, the process can be repeated easily at low cost. In addition, the 3D-printed lens can be used to make highly transparent lens by using silicone molding and isophorone diamine (5-Amino-1,3,3-trimethylcyclohexanemethylamine) based epoxy casting [30,31], as demonstrated in Fig. 12. This method might reduce the precision of the lens. However, if performed in a carefully controlled environment, it could work for replicating smooth freeform lenses without abrupt surface features or slope changes. Such additional post-processing steps after the actual 3D-printing may add a few days to the overall fabrication process. Nevertheless, the overall duration of small-volume production is days instead of weeks, as is the case in injection molding processes.
Fig. 12. Freeform lens matrices. (a) 3D-printed freeform lenses with high UV-curing dosage and (b) silicon molded and epoxy resin casted freeform lenses.
5. Conclusion
We have designed freeform lenses using a custom made ray-mapping algorithm. The method was applied to design lenses for forming simple rectangular and more complex target-image illumination patterns from either extended or collimated light sources. The ray-tracing simulations show that the designed elements perform well for LED light sources if the distance between the light source and the entrance surface of the freeform surface is chosen large enough compared to source size.
We have also demonstrated a 3D-printed freeform lens for a paper-web illumination case study using Printoptical technology. The surface roughness of the 3D-printed freeform lens is within well-acceptable range without any post-processing like polishing. The experimental demonstration shows the feasibility of swift small-series production of freeform lenses using 3D-printing. This is useful for industrial illumination applications, where time, energy, and cost are constraints. Moreover, the efficiency of the illumination system can be improved by using spherical or oval entrance surfaces instead of plane surface while designing the freeform lens. However, the printing process will then grow to three stages: printing (1) the top part of the freeform lens, (2) the holder or the mold for the printed lens part, and (3) the spherical or oval entrance surface over the flipped freeform lens part.
The 3D-printed freeform lens can also be used as a prototype for silicon molding and vacuum casting in order to change the material properties of the printed lens, so that the lens can be more transparent and possess higher refractive index or different dispersive properties. However, the surface profile deviation of the printed freeform lens is still in a scale of tens of micrometers, which is acceptable for some industrial illumination purpose. The reliability of the printed lens under high temperature stress is another factor that we will consider in the future. Further research will also be carried out to control the printing process parameters so that the surface error will be reduced to an acceptable level for imaging-quality lenses. In conclusion, the lighting sector can benefit from using freeform optics and 3D-printing. 3D-printing could complement the existing fabrication technology by allowing a rapid prototyping of unique custom freeform optics designs at low cost.
Funding
Tekes (247126-4524); European Association of National Metrology Institutes (EURAMET) (29093); Academy of Finland (285880).
Acknowledgments
The authors wish to thank Henri Partanen, Olli Ovaskainen, Anni Eronen, Tommi Itkonen, Pertti Silfsten, and Pertti Pääkkönen for discussions on 3D printed optics. Our special regards go to Mikko Ruuska from Mill Optics Ltd. for bringing different optical designing challenges to our attention.
Disclosures
The manuscript is a modified and extended version of an article presented in SPIE Photonics West 2018, SPIE conference proceedings [32]. The internal parts of the optic 3D-printer cannot be shown in detail due to a non-disclosure agreement with Luxexcel company.
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