1. Introduction

Freeform optics has nowadays gained worldwide recognition in the optical design community [1,2]. It is widely accepted that freeform geometries permit unprecedented levels of performance and compactness, both in the fields of Imaging and Nonimaging Optics [3–6]. In the last 20 years, fabrication and metrology techniques for freeform optics have progressed enormously and are hot topics in today’s research [7–11].

One strong advantage offered by freeform optical components is the capability of coping with peculiar shape and mechanical constraints, without compromising performances. Keeping the size of the optics to a minimum, or being able to fit them inside complex geometries, hence leaving space for non-optical components, is often a strong advantage in industrial applications. In this paper, we present the design, tolerancing and fabrication of a miniature freeform optical lightguide for sensing applications, whose shape and size are strongly limited by the geometrical constraints of the available volume. The main optical feature of the lightguide is a 45° acceptance half-angle at its entrance port, which faces the incoming radiation. The design uses a method originally developed in Nonimaging Optics, namely the concept of flow lines applied to optical concentrators [12–17]. In Section 2, we describe the design process in detail, building the “theoretical” lightguide as a 3D combination of two flow-line concentrators. Subsequently, in Section 3 we analyze its performance and study the impact of parameters related to fabrication (draft angles, fillet radii, surface finish). The theoretical optic is then modified according to this tolerance analysis and the mold is prepared for prototyping the optic. The mold design phase deserves special attention, given the small size of the component (indicatively: 1.3 × 2 × 20 mm³) and the requisite of keeping the fillet radius smaller than 0.030 mm. In Section 4, we describe a special approach for the realization of the mold, which is obtained as a sort of 3D puzzle where each component (or “insert”) carries one external surface of the lightguide. This external surface is machined using ultraprecision diamond milling, which allows to attain the desired optical quality on all the lightguide’s surfaces and a fillet radius which is practically zero, thanks to high-accuracy milling and assembly of the various mold inserts. At present, the first prototypes have been fabricated and the coating process is being optimized.

2. Optical design

The lightguide optics presented here is the “receiver” component of a “sender and receiver” pair forming an optical sensor. The sender part is designed to send light from an LED to the region to be analyzed and provide a certain light distribution. Since it does not present significant interest from the design standpoint, we will not discuss it further.

The peculiar geometrical shape of the receiver lightguide is determined by the performance and the mechanical requirements that it should meet. These requisites can be better understood looking at Fig. 1. In detail:

 

Fig. 1. (a) Lateral and (b) 3D views showing the volume available for the optical system, the position of the optic’s entrance port and of the detector (in contact with the exit port) and the orientation of the x-y-z coordinate system used throughout the presentation.

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The choice of a lightguide-type optic permits to keep the number of components to a minimum and to enhance the robustness of the component. Since the lightguide will be injection molded in Poly(Methyl MethAcrylate) polymer (PMMA), in the following the index of refraction will be n = 1.484 (PMMA at 850 nm, which is the wavelength of interest).

The requirement of an acceptance angle suggests a design approach based on the theory of optical concentrators from the field of Nonimaging Optics [18]. In effect, a simple 3D concentrator with square ports, obtained by joining two 2D Compound Parabolic Concentrators (CPCs), one aligned along x and one along y, would immediately provide the desired acceptance. However, this simple solution presents some drawbacks:

To fulfill these additional requirements, we employed an approach based on the use of the flow-line design method of Nonimaging Optics. Rigorous mathematical treatments on flow lines (or lines of flux) can be found in [12,19]. Here, we introduce flow lines exploiting one of their crucial properties: at each point of a 2D optical system, flow lines bisect the angle formed therein by the edge rays of the system.

Consider first the requirement of a flat lateral surface. Figure 2(a) shows a generic 2D Compound Parabolic Concentrator (CPC) with acceptance half-angle θ. This CPC accepts the bundle of rays specified, for instance, by the points along its entrance aperture and the directions comprised between +θ and -θ from the vertical. In Fig. 2(a), a special subset of this ray bundle is shown: the “edge ray” subset comprising parallel rays impinging on the entrance port with angle of incidence -θ. Another edge ray subset exists and is formed by parallel rays hitting the entrance aperture with incident angle +θ (not shown). Figure 2(b) shows some flow lines inside the same CPC, obtained by considering different internal points of the optic and tracing the segment that, locally, bisects the angle formed by the edge rays belonging to the two subsets. The flow lines in this case are continuous unions of

The system of flow lines is perfectly symmetric with respect to the central straight flow line.

 

Fig. 2. (a) A standard 2D CPC and the sub-bundle of edge rays relative to incident angle -θ. (b) Structure of the flow lines in the interior of the CPC. (c) A half-CPC concentrator with flat lateral side, where the right side is the central vertical flow line of the full CPC. The acceptance is the same of the original CPC.

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At this point, it is worth mentioning a second important general property: étendue is conserved between flow lines. A consequence of the two facts above is that placing a mirror along a flow line does not modify the radiation field (for a proof, see for instance [12]). Thus, we can freely choose specific flow lines inside the CPC and obtain different concentrators, all with the same acceptance angle of the “parent” CPC. In particular, we select the central vertical flow line, obtaining a half-CPC optic, which has a flat lateral side (Fig. 2(c)). This result will be useful for obtaining the horizontal acceptance (along x).

Focus now on the requisite of a tilted exit port for the concentrator part of the lightguide. We consider the case of a 2D concentrator for a wedge receiver ABC (see Fig. 3(a)); we particularize the example to our specific design case, so the concentrator is a solid PMMA one with 45° acceptance half-angle θ (which is refracted to 28.5° inside the dielectric). Some flow lines for this configuration are drawn: they are parabolas below the dashed lines r and s and straight segments above them. Again, by choosing different pairs of flow lines, different concentrators are built, with the same acceptance angle of the parent concentrator. In particular, we choose the flow lines in blue in Fig. 3(a), obtaining a concentrator with two partially parabolic surfaces which smoothly convey the light beam to an exit port tilted 61.5° with respect to the entrance port (shown in Fig. 3(b); at the end of this section we will comment further on this choice). This concentrator will provide the vertical acceptance (along y) and will be placed in the top right region of the available volume, with its entrance port coinciding with the entrance port of the lightguide. As a final step, we report that fixing the entrance port of the concentrator to 1.33 mm, as from design specifications, automatically sets the exit port to 0.63 mm. This would mean a minimum waist thickness of 0.63 mm in the lightguide. Such a tiny waist may lead to breakage of the optic during fabrication (when demolding) or manipulation, thus a larger minimum waist is preferred. To achieve this, we notice that the concentrators considered so far provide maximum concentration (i.e. they maximize the ratio of the entrance port to the exit port), but this property is not essential for our application. Thus, as exemplified in Fig. 3(c), we can take a smaller entrance port in the designed concentrator and cut it horizontally along a straight internal flow line until a certain point (dictated by the available volume in the neighboring region). The portion of the concentrator on top of this flow line is removed; the new edge is connected to the original parabolic flow line by a line whose inclination can be tuned based on the available space. To avoid spurious effects (like rays which should be rejected based on the acceptance value, but are instead reflected towards the exit port), the connecting surface should be absorption coated. Finally, the new profile is scaled up until the new entrance port has the same size of the original one (1.33 mm). With this procedure, we obtain an exit port which is 0.90-mm wide, a value which ensures a more robust minimum waist for the full lightguide in view of its fabrication.

 

Fig. 3. (a) A CPC for a wedge receiver, with some flow lines shown. (b) A concentrator for a tilted receiver, obtained by choosing the blue flow lines in the original concentrator. (c) The cut-and-rescale process which provides a wider exit port, increasing the robustness of the full optic. The concentrator has been rotated 90° clockwise with respect to the one in (b), since in the final configuration the entrance port is vertical.

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The construction of the full lightguide can be understood by looking at Fig. 4. It starts from the top, taking the final 2D concentrator of Fig. 3(c), placing it on the x = 0 plane and extruding it 1.33 mm along the x direction, obtaining the square entrance port. This first section, called Concentrator 1, provides the acceptance angle along y.

 

Fig. 4. Views of the designed lightguide. The origin of the coordinate system is located at the central point of the lightguide entrance port (coordinate axes are drawn only to show the lightguide’s orientation).

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Subsequently, a curved section is attached to Concentrator 1’s exit port; its lateral sides, parallel to the x = 0 plane, are flat. This section transports light downwards and aligns the beam along the desired angle (40° with respect to the vertical direction). Its curved surfaces are obtained by extruding two arcs with different radii (the radius of arc₁ is larger than the radius of arc₂) 1.33 mm along x. It is clear from the lateral view in Fig. 4 that the cross section of this segment of the lightguide increases going downwards. This way, the beam divergence on plane x = 0, which is maximum at the exit port of Concentrator 1, is reduced and light rays impinge on the curved front and back surfaces with a progressively larger incident angle. This will prevent rays from being trapped inside the optic by total internal reflection later on, when they will meet the exit port of the lightguide. At the end of the curved piece, the lightguide section measures 1.33 × 2 mm².

Here, we attach a second concentrator, obtained by extruding a 2D concentrator like the one in Fig. 2(c). The concentrator profile has been tuned for an acceptance half-angle of 45° in air and oriented to provide an acceptance angle in the x direction. Its flat rectangular side is aligned with one flat lateral side of the lightguide. Notice that the entrance port of this optic is larger than the lightguide section attached above: similar to what was shown in Fig. 3(c), we scaled up the concentrator to enlarge its exit port, which is now 0.95 × 2 mm². The surface in red should be absorption coated to avoid unwanted reflections.

Finally, one final straight section connects the exit port of Concentrator 2 to the exit port of the lightguide. The x-cross section of this part increases linearly going downwards, to reduce the horizontal divergence of light rays and ensure that they will not be trapped inside the optic by total internal reflection at the exit surface. The size of the exit port is 2 × 2 mm²; this surface is attached to the detector by means of index-matching adhesive to minimize Fresnel reflections on the interfaces, which otherwise would lead to a decrease in efficiency. The optic has a fully flat lateral surface (the right side in the front view in Fig. 4), which will provide a parting line for injection molding fabrication (see Sections 3 and 4).

In the lateral view of Fig. 4 it is shown how the lightguide fits inside the available volume. Notice that Concentrator 1 tilts the light beam even more than the required 40°. Indeed, as already mentioned, the steering angle of Concentrator 1 is 61.5°, fixed solely by the acceptance angle and by the index of refraction. With this approach, we achieved beam steering already in the top part of the lightguide, which is the most “difficult” section given the local shape and available volume. In the central region of the lightguide we have much more space available for finely adjusting the beam steering angle towards the desired 40°, using the curved section which smoothly connects Concentrator 1 with Concentrator 2.

3. Performance analysis and fabrication tolerancing

The key parameters of the lightguide performance are the accuracy in providing the acceptance angle in the x and y directions and the efficiency in transferring light to the exit port. In this Section, we assess the impact of the fabrication parameters (draft angle, fillet radius, surface finish, potential geometric adjustments to ease mold design and demolding) and of coating choices on the performance.

Since light rays hit the concentrators’ internal surfaces with random orientation, a high-reflection coating needs to be applied on the lightguide lateral surfaces (the only uncoated surfaces will be the entrance and exit ports, plus the “unused” surfaces colored in red in Fig. 4: this point will be discussed later). To estimate the efficiency of the lightguide with different optical coatings, an optical simulation has been set up: a constant-radiance source emitting rays with ±45° maximum divergence along x and y is placed right in front of the entrance port of the optic. The size of this source is identical to the size of the entrance port. A detector placed immediately after the exit port collects the light exiting the optic. To set a benchmark, 100% efficiency corresponds to the energy reaching the detector when the optical coating is 100% reflective and only Fresnel losses occur at the entrance and exit ports. Simulations show that an aluminum (Al) coating, with 85% reflectance at 850 nm, gives an efficiency of 20%, which is definitely too low. With protected silver (Ag), which has a reflectance close to 98% at 850nm, the efficiency reaches 81%, which is satisfactory for our application. A dielectric protection layer is added on the Ag coating, to avoid tarnishing and improve the handling robustness of the coated lightguide. Notice that the best reflection performance would be attained with a gold (Au) coating. However, to ensure good adhesion of Au to a PMMA surface, a Cr layer is typically required in-between, which would significantly impede the optical performance of the lightguide coating and, as such, is not a viable option in our case.

Next, we check the impact of the fabrication parameters on the acceptance. To do so, a collimated beam, travelling in the -z direction and with a total power of 1 Watt, is sent through the entrance port of the optic: to check the horizontal (vertical) acceptance, the incident angle of the beam on the port is varied from -70° to +70° in the y = 0 (x = 0) plane, where 0° means normal incidence. For each incident angle, we collect the energy reaching the detector after the exit port and then build acceptance vs. (incident) angle diagrams. An acceptance value of 1 for a specific incident angle indicates that all the incident light is transferred to the detector. To “fill” the entire entrance port with incident light, while at the same time minimizing light losses caused by parts of the beam which miss the port, the beam has a square cross section with 1 × 1 mm² area.

First, we consider the draft angle. A draft angle is required to allow demolding the part after injection in the fabrication via injection molding. In our case, the parting line is the red-colored flat lateral side in the lightguide model in Fig. 5. We added several draft angle values to the surfaces in edge contact with the parting line. In particular, we considered the optic with Ag coating with no, 1° and 2° draft angles. The simulation results for the horizontal and vertical acceptance as functions of the incident angle of the collimated beam are displayed in Fig. 5. The “Ideal” reference case corresponds to the theoretical lightguide with 100% reflective coating and without losses (neither Fresnel nor absorbtion). We notice that with 2° draft angle the horizontal acceptance develops a high “tail” at angles less than -45° and becomes smaller for angles between 40° and 45°. The 2°-draft vertical acceptance is still very steep at the extreme angles, but it gets slightly lower in the central region. For these reasons, we target a draft angle of 1° or lower during fabrication. In the following, the optics used in simulations will have a 1° draft angle.

 

Fig. 5. Simulated impact of the draft angle on the acceptance. Draft angles are calculated referring to the flat surface in red (parting line) of the lightguide. For the sake of visualization, in the displayed lightguide the draft angle is 3°: in the front view of the entrance port, the angle between parting line and adjacent surfaces is not 90°, but 87°. The acceptance-vs.-incident-angle diagrams compare the effects of 1° and 2° draft angles to the case of ideal optic (100% reflective coating, no draft angle) and of no draft angle.

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Next, we analyse the impact of the fillet radii. When using pocket micro-milling for the mold fabrication, the filled radius will be determined by the physical size of the milling tool. In Fig. 6, the lightguide edges on the opposite side with respect to the parting line have been rounded with a fillet radius of 0.375 mm. Given the small size of the optics, this conservative value already has a strong impact on the performance: the acceptance vs. angle diagrams in Fig. 6 show that the horizontal acceptance is severely compromised for a fillet radius of 0.375 mm. Considering smaller fillet radii, we observe that the horizontal acceptance is acceptable when the fillet radius gets as small as 0.030 mm. However, a 0.030-mm fillet radius is extremely hard to achieve in practice. This consideration indicates that pocket milling is not a suitable fabrication route and that a different approach for the construction of the mold is required, leading to the 3D puzzle approach that will be discussed later. In the following simulations we will use a null fillet radius (it will be discussed in Section 4 how this can be achieved).

 

Fig. 6. Simulated effects of different fillet radii on the acceptance angle, for a lightguide with 1° draft angle and Ag coating. The red parts in the lightguide are edges which have been filleted with a 0.375 mm radius. In the acceptance vs. angle diagrams, we considered three examples of fillet radius, from a standard 0.375 mm down to an extreme value of 0.030 mm.

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We now estimate the effects of surface finish. To this aim, we assigned a scattering pattern to all the surfaces of the lightguide. Since this is an a priori estimation, we considered Lambertian scattering for the fraction of the reflected light that is scattered. This fraction, called Total Integrated Scatter (TIS), is found via the formula TIS = 1 – exp(-(4πσ n cos θ/λ)²), where σ is the root-mean-square (rms) surface roughness, λ is the light wavelength in vacuum, n is the index of refraction of the material and θ is the incident angle of light rays on the surface (see [20,21]). A reference value for the incident angle θ is found via ray-tracing simulations: the lightguide’s entrance port is illuminated with the constant-radiance source mentioned in the second paragraph of Section 3. The incident angles for all the light-surface interactions occurring inside the optic are recorded: their average value is 61°, with a standard deviation of about 16°. Since TIS decreases when the incident angle increases, we choose a conservative approach and set 45° as the reference value for θ. For the rms surface roughness, we consider here values of 20, 15 and 10 nm, corresponding to a TIS of 9%, 5% and 2% respectively. In Fig. 7 the acceptance for these three situations is compared with the case of perfectly smooth surfaces in the optic. It appears that a roughness higher than 15 nm has a severe effect on the acceptance angle, while at 10 nm the situation improves. Consequently, we will target a σ equal or less than 10 nm during the fabrication. Since a surface finishing step after injection molding is not viable due to the complex geometry of the part, optical quality of the mold inserts is requested. Notice that using a Lambertian scatter pattern represents a worst case scenario, so we expect the actual performance of the optic to be equal to or better than what appears in Fig. 7.

 

Fig. 7. Simulated impact of rms surface roughness on the acceptance angle, for an Ag-coated lightguide with 1° draft angle and null fillet radius. The scatter pattern is Lambertian.

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Finally, we analyze the effect of some geometric modifications in view of easing injection molding. In Fig. 4 we presented the theoretical design and indicated two “unused” surfaces which have no optical function and should ideally be absorption coated. The lightguide shown in Fig. 8 is the final design that will be used for molding (“mold” version). In this solid, the applied draft angle is about 0.5°. Note that two curved surfaces (in red) replace the previous planar unused surfaces. These modifications avoid sharp edges which would complicate the molding process. We also noticed that depositing an absorptive layer on these surfaces, while keeping the Ag high-reflection coating on the adjacent optical facets, would make the coating process extremely difficult. Since this is not beneficial for a mass-production product, we examined the option in which all the optic’s lateral surfaces are reflection-coated (even the “unused” ones, but except for the entrance and exit ports). In Fig. 8, we analyze the acceptance angle for the mold version of the optic with uniform Ag coating on all the lateral surfaces. As a reference for comparison, we consider the 10-nm-roughness optic which was used in the simulations of Fig. 7. In the mold version with uniform coating the acceptance patterns tend to be slightly wider and present some tails at higher incident angles (well beyond the desired ±45°). These tails, on average, are below 0.2, which is about 25% of the central acceptance values. Given the advantages brought to the production process by uniform coating, we accept these imperfections in performance.

 

Fig. 8. Performance comparison between a “theoretical” version of the optic, with planar and absorption-coated unused surfaces (differential coating), and the “mold” version with curved unused surfaces and uniform reflective coating. Both optics have null fillet radius and 10-nm surface roughness.

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To conclude this section, we summarize the final requisites for the fabrication parameters: draft angle <1° (fixed to 0.5° in the mold version of the optic), fillet radius <0.030 mm, surface roughness <10 nm. The fillet radius limit, in particular, prohibits a pocket milling approach as explained earlier. To meet this limit, we developed a special design methodology for the mold, which will be detailed in the next section.

4. Mold design via the 3D puzzle approach and fabrication of the component

The lightguide has six sides: the entrance and exit ports, the lateral sides (parallel to the x = 0 plane) and the front and back surfaces (perpendicular to the x = 0 plane). To achieve homogeneous optical quality and sharp edges between all of the optic’s side surfaces, we designed the mold through a puzzle approach: the main optical cavity in the moving platen of the mold is an assembly of separate parts, or “inserts”, that together form the non-flat side of the sender and the receiver. Notice that we aim for a mold containing both the sender and the receiver lightguides, such that one copy of each is being obtained in a single injection cycle. The flat lateral surface indicated as “parting line” in Fig. 5 is assigned to the fixed platen of the mold, which is an ultraprecision diamond milled flat surface with the optical quality requested by the application (rms roughness <10 nm). The entrance ports will be obtained by cutting out the sender and the receiver and polishing after assembly in a mechanical housing, so no dedicated surface is here foreseen. Each one of the remaining four surfaces is assigned to a dedicated surface of one of four so-called “inserts”. These inserts are first obtained from a special-grade aluminum block (see Fig. 9(a)), by pocket milling from one side; they are then removed from the block by micro-milling from the back side and subsequently secured to a single custom-designed holding structure, as shown in Fig. 9(b). For each insert, the “optical face”, corresponding to a surface of the lightguide, lies on one lateral side. This allows to ultraprecision diamond mill the optical face of each insert separately, to the desired level of optical quality (i.e. rms roughness <10nm). Finally, the four inserts are combined in a 3D arrangement to form the mold core, as shown in Fig. 10(a), where the optical faces of the inserts are colored in red. Adjacent facets of the optic are obtained by juxtaposing independent surfaces of the mold, so it is not necessary to define a rounding on their common edge: this way, the fillet radius is virtually zero. Notice in Fig. 10(b) that the optic protrudes beyond the entrance port and the extra section has no optical quality. The PMMA injection is done from the center of the core and then fills the cavity for the receiver lightguide at the same time as the cavity for the sender lightguide.

 

Fig. 9. (a) Two of the four inserts obtained by micro-milling a special-grade aluminum block. According to our nomenclature, they are inserts number 2 and 3. The other pieces visible on the block are part of the sender optic mold, not discussed here. (b) The four inserts (encircled in dashed lines) are fixed to a supporting structure for diamond milling of the surfaces with optical quality. The additional visible pieces refer to the sender optic mold.

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Fig. 10. (a) Exploded view of the mold: the four inserts are arranged in 3D such that each surface with optical quality forms a side of the lightguide. (b) A photograph of the final mold: the encircled part is the one relative to the lightguide discussed here.

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Finally, a microscope image of a molded sample, before the coating phase, is shown in Fig. 11(a). A Wittman-Battenfeld MicroPower 5 micro-molding machine was used for the injection molding. The receiver and sender components are joined to the central disk body. Metrological analysis on different areas of the lightguides gave an rms surface roughness in the range 3–10 nanometers, which meets the requested upper limit.

 

Fig. 11. (a) The full molded sample comprising both the receiver and the sender components, viewed through a microscope. (b) The same sample after the Ag coating has been applied.

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The Ag reflection coating is applied when the two parts are still connected to the central disk (see Fig. 11(b)). For normal incidence, the reflectance of this coating is 96.8% at 850nm. At present, the coating phase is in progress; special care and multiple iterations are required in this step due to the small size and the peculiar geometry of the components. After coating, the prototypes will enter validation tests aimed at verifying their optical efficiency and their performance with respect to acceptance angle.

5. Conclusion

In this paper, we have reviewed the design, tolerancing and fabrication of a miniature optical lightguide to be used in sensing applications. The main optical features of the optic are a 45° acceptance half-angle in two perpendicular directions and a 40° steering of the light beam travelling inside. These functionalities were achieved by combining two concentrators designed with the flow-line method from Nonimaging Optics. This approach for concentrator design provided a compact device which met the stringent geometrical constraints set by the available volume. The resulting lightguide has been further optimized to increase its robustness, in view of injection molding fabrication. By an accurate tolerance analysis of the theoretical design, we determined the target values for the main fabrication parameters (draft angle, fillet radius and surface finish). To minimize the fillet radius and maintain uniform optical quality (with rms roughness less than 10 nm) on all of the optic’s surfaces, we developed a specific approach for designing the mold. Different inserts are machined independently by ultraprecision milling and then accurately assembled together in 3D to form the mold. With this technique, it is not necessary to add a fillet radius to the mold. To our knowledge, this 3D-puzzle approach to mold manufacturing is not commonly applied to optical components with size in the millimeter range. Currently, the first prototypes of the lightguide have been fabricated and the optimization of the coating process is in progress.

A comprehensive set of validation measurements is planned as future work. The features which will be characterized and compared with simulations are the acceptance (both in the horizontal and vertical directions) and the optical efficiency when using extended non-collimated sources which match the acceptance angle of the optic. On the metrological side, an accurate inspection of the edges of the molded prototype will allow us to verify the effective draft angle and the edge sharpness generated by the fillet-radius-free mold design approach.