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Article Enhancement of High-Resolution 3D Inkjet-Printing of Optical Freeform Surfaces Using Digital Twins

Ingo Sieber 1,* , Richard Thelen 2 and Ulrich Gengenbach 1

1 Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany; [email protected] 2 Institute of Microstructure Technology-KNMF, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany; [email protected] * Correspondence: [email protected]

Abstract: 3D-inkjet-printing is just beginning to take off in the optical field. Advantages of this technique include its fast and cost-efficient fabrication without tooling costs. However, there are still obstacles preventing 3D inkjet-printing from a broad usage in optics, e.g., insufficient form fidelity. In this article, we present the formulation of a digital twin by the enhancement of an optical model by integrating geometrical measurement data. This approach strengthens the high-precision 3D printing process to fulfil optical precision requirements. A process flow between the design of freeform components, fabrication by inkjet printing, the geometrical measurement of the fabricated optical surface, and the feedback of the measurement data into the simulation model was developed, and its interfaces were defined. The evaluation of the measurements allowed for the adaptation of the printing process to compensate for process errors and tolerances. Furthermore, the performance of the manufactured component was simulated and compared with the nominal performance, and the enhanced model could be used for sensitivity analysis. The method was applied to a highly complex helical surface that allowed for the adjustment of the optical power by rotation. We show



 that sensitivity analysis could be used to define acceptable tolerance budgets of the process.

Citation: Sieber, I.; Thelen, R.; Keywords: digital twin; modeling; simulation; additive manufacturing; 3D inkjet-printing; freeform Gengenbach, U. Enhancement of optics; varifocal optics High-Resolution 3D Inkjet-Printing of Optical Freeform Surfaces Using Digital Twins. Micromachines 2021, 12, 35. https://doi.org/10.3390/ 1. Introduction mi12010035 Additive manufacturing (AM) offers great freedom in design, a short lead-time, and Received: 30 November 2020 the possibility of functional integration [1]. AM technologies can be divided into seven Accepted: 25 December 2020 main categories [2,3], including 3D ink-jet printing. In 3D-ink-jet printing, ink droplets Published: 30 December 2020 are deposited and cured, layer by layer, by UV illumination. The ink may be polymer or hybrid polymer (e.g., Ormocer (R)) based. Moreover, metal or ceramic nanoparticle- Publisher’s Note: MDPI stays neu- enhanced polymer inks continuously increase the application range of 3D-ink-jet printing. tral with regard to jurisdictional clai- Ink rheological properties, piezo waveform, and nozzle shape are important parameters for ms in published maps and institutio- droplet generation. Upon droplet impact on the substrate or on previously printed layers nal affiliations. wetting behavior, droplet size, droplet overlap, droplet coalescence, and UV-curing strategy govern layer formation. A considerable amount of research has gone into developing an understanding of these complex interrelations between materials, equipment, and printed shapes. Though 3D printing was originally primarily used for prototyping, it has Copyright: © 2020 by the authors. Li- censee MDPI, Basel, Switzerland. evolved to be used for the custom mass production of functional parts [4,5] and is used in This article is an open access article several industrial fields, e.g., automotive, aerospace, and medical, as a well-established distributed under the terms and con- alternative to conventional manufacturing processes [6–9]. The potential of high-resolution ditions of the Creative Commons At- 3D printing to produce parts with a high degree of design freedom makes this technology tribution (CC BY) license (https:// interesting for optical applications where an increasing use of optical freeform surfaces creativecommons.org/licenses/by/ can be observed. Optical applications using freeform optical surfaces include automotive 4.0/). lighting [10,11], beam expanders [12], and ophthalmic implants [13–15]. The established

Micromachines 2021, 12, 35. https://doi.org/10.3390/mi12010035 https://www.mdpi.com/journal/micromachines Micromachines 2021, 12, 35 2 of 12

processes to manufacture freeform optical components from optical polymers are the direct diamond turning of the optical component [16] and the diamond turning of a mold insert for precision injection molding [17]. High-resolution 3D printing, with its enormous flexibility in terms of shape variation, opens up great potential for creating new approaches and solutions for optical systems [18]. The manufacturing of optical components and, especially, freeform surfaces has to meet high requirements, e.g., the high fidelity of the optical surfaces. Macroscopic imaging freeform optics pose great challenges for high-resolution 3D inkjet printing with regard to shape fidelity [19]. There are only few companies mastering the above-mentioned complex set of 3D-inkjet printing to a degree that optical surface can be realized. This process know-how is their competitive advantage and thus, understandably, not disclosed to the public. For the performance prediction of the manufactured freeform optics in an optical system, the use of measurement data of the manufactured parts to enhance the optical model with respect to realistic representation of the optical surfaces as manufactured is a useful strategy [20]. Detailed comparisons of nominal freeform surfaces with tactile measurements of freeform surfaces produced by 3D inkjet printing allow for a better understanding of the 3D inkjet printing process and the derivation of process improvement strategies [21]. Though the term digital twin is versatilely used, there is a basic and common understanding in scientific and industrial publications that a digital twin is a virtual equivalent of a real system [22]. It maps physical objects and does not only describe physical objects but also optimizes physical objects based on models [23]. This article presents the development of a digital twin on the basis of the above-described approach to adapt the high-resolution inkjet printing process to a respective geometry with the objective of increasing the shape fidelity of optical surfaces. Our work was based on an enhancement of the simulation models with data from surface measurements. This allows for the detection of discrepancies in shape fidelity and for the direct determination of the influence of these discrepancies on the optical performance of a freeform component. Hence, the printing process can be adjusted in order to reduce or increase material accumulation or to adjust curing strategies, thus leading to higher- performance components. The organization of the paper is as follows: in Section2, the digital twin and the process flow are developed. Section3 introduces the optical system used as an example. The evaluation of the measurements leading to a height map of the difference surface between measured and nominal surface is presented in Section4. Section5 is about the model enhancement and the usage of the digital twin in sensitivity analysis. A discussion of the achievements closes this article.

2. Concept of Generating and Using the Digital Twin The flow between the individual process steps of process parameter deviation and digital twin generation is shown in Figure1. The first step was the creation of an initial optical design. The arbitrary freeform optical surface was calculated with the help of mathematical software. The mathematical development tool Mathematica [24] served as a tool in our example. To perform an optical analysis of the complete system model, the mathematical freeform surface was input to an optical simulation tool. In our example, OpticStudio [25] was used for this purpose. In order to perform an automatic analysis of the optical properties of different variants, a communication between the two tools was implemented based on the WSTP (Wolfram Symbolic Transfer Protocol) extension of OpticStudio. A design optimization of the initial design was realized by integration of the Single-Objective Genetic Algorithm (SOGA) of the Sandia Dakota (Design Analysis Kit for Optimization and Terascale Applications) Box [26]. The evolutionary algorithm controlled the optimization process of the optical freeform surface with respect to the specific system requirements. The optimized optical freeform surface S0 was the initial design for the process flow and was input in the optical performance evaluation where performance analysis was carried out on the basis of optical simulations. The surface Si (where the index i denotes the number of passes of this loop; the initial surface therefore has the index 0) was then transferred into mechanical design, where an optical component Micromachines 2021, 12, x 3 of 12

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analysis was carried out on the basis of optical simulations. The surface Si (where the in- dex i denotes the number of passes of this loop; the initial surface therefore has the index 0) was then transferred into mechanical design, where an optical component Ci was gen- C was generated by adding a thickness and support structures, e.g., substrate or mounting eratedi by adding a thickness and support structures, e.g., substrate or mounting and andalignment alignment structures. structures. Manufactur Manufacturinging complex complex structures structures requires requires an adaption an adaption of the de- of the designsign to the to the design design rules rules of the of involved the involved proce processes,sses, e.g., the e.g., manufacturing the manufacturing of the optics, of the optics,or orthe the assembly assembly of the of theoptical optical parts parts to an to adjusted an adjusted optical optical system. system.

FigureFigure 1. DigitalDigital twin twin and and process process flow. flow.

A way to to approach approach such such adaptations adaptations and and the the final final design design for formanufacture manufacture were were de- de- scribedscribed inin [[27].27]. TheThe finalizedfinalized CADCAD modelmodel containedcontained bothboth opticaloptical andand mechanicalmechanical structuresstruc- intures one in comprehensive one comprehensive model model and and described described the the optical optical component componentC Ci ithat that waswas digitallydigi- inputtally input to the to printing the printing process. process. Up toUp this to this transfer, transfer, the the optical optical component component was was described de- entirelyscribed entirely digitally. digitally. After the After manufacturing the manufactur process,ing process, a real part a real existed. part existed. The printing The print- process tooking process place on took a high place resolution on a high 3D resolution inkjet printer 3D inkjet and wasprinter carried and outwas by carried the Dutch out by company the Dutch company Luximprint [28]. The result of the printing process was a real optical part Luximprint [28]. The result of the printing process was a real optical part Pi where the opti- calPi where surface the and optical the mechanicalsurface and alignmentthe mechanical structures alignment were structures manufactured were inmanufactured a single process stepin a [single29]. In process the next step step [29]. of digital In the twinnext generation,step of digital measurements twin generation, were measurements carried out on the were carried out on the basis of the finished optical parts Pi. The characterization of the basis of the finished optical parts Pi. The characterization of the optical properties of the optical properties of the printed optical component were conducted both in a laboratory printed optical component were conducted both in a laboratory setup to analyze the optical setup to analyze the optical performance and by tactile surface measurement, using a performance and by tactile surface measurement, using a Dektak V220 SI profilometer, Dektak V220 SI profilometer, to analyze the dimensional accuracy of the printing process. to analyze the dimensional accuracy of the printing process. The resulting data Di of the The resulting data Di of the measurement were again digital descriptions, now of the sur- measurement were again digital descriptions, now of the surface of the real manufactured face of the real manufactured part. On the basis of the measurement results, three different part. On the basis of the measurement results, three different surfaces were generated: the surfaces were generated: the surface SR of the real manufactured part, which was input surface S of the real manufactured part, which was input into the optical performance into the opticalR performance evaluation to simulatively analyze the systems performance evaluation to simulatively analyze the systems performance of the manufactured part; the of the manufactured part; the surface Si,R that was an adaptation of the nominal design to surfacedifferentS i,Rtolerancethat was budgets an adaptation of the manufacturing of the nominal process design and to was different also input tolerance to the budgetsoptical of theperformance manufacturing evaluation process for andthe wassensitivity also input analysis to the of opticala manufacturing performance process evaluation with un- for the sensitivityknown tolerance analysis distribution of a manufacturing (see also Section process 5); withand the unknown surface toleranceSi-R that was distribution the differ- (see alsoence Sectionbetween5); the and nominal the surface surfaceSi-R Sthati and was the the measured difference surface between SR. On the the nominal basis of surface the Si anddifference the measured surface S surfacei-R, a spatiallySR. On resolved the basis height of the map difference Mi-R of surface the differencesSi-R, a spatially between resolved tar- heightget and map actualM geometryi-R of the was differences created betweenby comparing target the and measured actual geometrydata with the was nominal created by comparing the measured data with the nominal data of the freeform surfaces. This map was used to derive feedback to the printing process and to determine at which positions adjustments to material deposition were necessary. With this information, the printing pro- cess could be adjusted to the respective freeform surface by adapting printing parameters (printing speed, drop size, UV-illumination control for curing). Micromachines 2021, 12, x 4 of 12

data of the freeform surfaces. This map was used to derive feedback to the printing pro- Micromachines 2021, 12, 35 4 of 12 cess and to determine at which positions adjustments to material deposition were neces- sary. With this information, the printing process could be adjusted to the respective freeform surface by adapting printing parameters (printing speed, drop size, UV-illumi- nationIn control Figure 2for, exemplary curing). results of the individual process steps are shown to illustrate the flowIn Figure diagram 2, exemplary in Figure 1results. The initialof the indi opticsvidual design process is depicted steps are in shown superelevated to illustrate repre- sentation,the flow diagram and the in optical Figure performance 1. The initial evaluationoptics design is presentedis depictedby in asuperelevated raytracing section repre- and asentation, map of the and point the optical spread performance function. In evaluati the mechanicalon is presented design, by thea raytracing underlying section substrate and is a map of the point spread function. In the mechanical design, the underlying substrate is shown, as are the alignment and centering structures of the optical component. A detailed shown, as are the alignment and centering structures of the optical component. A detailed discussion of the individual alignment structures can be found in [29]. The high-resolution discussion of the individual alignment structures can be found in [29]. The high-resolution 3D ink-jet printed optics part acted as input to measurements. Imaging and surface mea- 3D ink-jet printed optics part acted as input to measurements. Imaging and surface meas- surements at the laboratory setup were conducted. Here, the measurement and imaging urements at the laboratory setup were conducted. Here, the measurement and imaging results of the tactile measurement are shown. The evaluation step shows the resulting results of the tactile measurement are shown. The evaluation step shows the resulting measured surface, the adapted surface, and the height map. On the basis of these data, measured surface, the adapted surface, and the height map. On the basis of these data, adjustmentsadjustments toto thethe printingprinting process,process, asas wellwell asas inputinput inin thethe optics optics design design process process to to enhance en- thehance systems the systems model, model, were derived.were derived. Both waysBoth ways led to led creation to creation of a digitalof a digital twin twin that allowedthat usallowed to simulatively us to simulatively evaluate evaluate the system the performancesystem performance of the manufactured of the manufactured parts and parts derive fabricationand derive fabrication process improvements. process improvements.

FigureFigure 2. SimplifiedSimplified visual visual illustration illustration of of digital digital twin twin and and process process flow. flow.

3.3. Optical System System Used Used as as Example Example The example example used used to toillustrate illustrate creation creation and andapplication application of the ofdigital the digitaltwin was twin a var- was a varifocalifocal optics optics consisting consisting of two of twooptical optical freefor freeformm surfaces surfaces that were that werehelical helical in shape in shape with a with aradius radius of of curvature curvature changing changing azimuthally azimuthally [30]. [30 The]. The freeform freeform surfaces surfaces are aredescribed described by by EquationEquation (1) [31]. [31].

𝑧𝑟, 𝛼A= 𝑟 𝛼 𝐴: form function; 𝑟 = q𝑥 +𝑦 ; α: azimuth (1) z(r, α) = r2α A : form function; r = x2 + y2 ; α : azimuth (1) The curvature of the2 lens bodies depends on the form factor A and the azimuth α, henceThe featuring curvature a discontinuity of the lens at bodies the transition depends from on theα = 2 form π to factorα = 0. TheA and left theof Figure azimuth 3 α, henceshows featuringa helical surface a discontinuity calculated at with thetransition Equation from(1): theα =surface 2 π to curvatureα = 0. The varied left of fromFigure 3 showsconvex ato helical concave, surface starting calculated with the withdiscontinuity Equation (transition (1): the surfacebetween curvature 2 π to 0) following varied from convexthe azimuth to concave, clockwise. starting The withprofiles the of discontinuity the surface parallel (transition to the between x-axis, 2alongπ to 0)the following blue the azimuth clockwise. The profiles of the surface parallel to the x-axis, along the blue intersection (Figure3, mid column) and parallel to the y-axis along the red intersection path (Figure3, right) illustrate the surface sag, which indicates the displacement of the p surface along the optical axis (z-axis) of the surface at distance r = x2 + y2 from the axis. Since discontinuity could seriously affect imaging quality, it has to be obscured by a diaphragm to prevent any disturbing effects in imaging, e.g., scattering. A combination of Micromachines 2021, 12, x 5 of 12

Micromachines 2021, 12, x 5 of 12

intersection (Figure 3, mid column) and parallel to the y-axis along the red intersection pathintersection (Figure 3,(Figure right) 3,illustrate mid column) the surface and parall sag, elwhich to the indicates y-axis along the displacementthe red intersection of the surfacepath (Figure along 3,the right) optical illustrate axis (z-axis) the surface of the sag, surface which at indicates distance the𝑟= displacement𝑥 +𝑦 from of thethe Micromachines 2021, 12, 35 5 of 12 axis.surface Since along discontinuity the optical could axis seriously(z-axis) of affect the surfaceimaging at quality, distance it has𝑟= to 𝑥be +𝑦 obscured from bythe a diaphragmaxis. Since todiscontinuity prevent any could disturbing seriously effects affect in imaging,imaging quality, e.g., scattering. it has to Abe combination obscured by ofa twodiaphragm lens parts to featuringprevent any such disturbing helical surfaces effects in arranged imaging, in e.g., sequence scattering. perpendicular A combination to the of opticaltwotwo lenslens axis partsparts would featuring possess such a constant helical helical surfaces surfacesrefractive arranged arranged power inover in sequence sequence the complete perpendicular perpendicular aperture. to to Thethe the refractionopticaloptical axis power would could possess be atuned constant continuously refrac refractivetive powerby power mutually over over the the rotating complete complete the aperture. aperture.lens bodies The The aroundrefractionrefraction the poweropticalpower could axiscould (see be be tuned Figure tuned continuously 4).continuously This optical by mutuallyby system mutually is rotating multifocal, rotating the lens withthe bodies lenstwo sectorsbodies around withthearound opticaldifferent the axisoptical individually (see axis Figure (see tuna 4Figure).ble This refraction4). optical This optical system powers system is [21,27]. multifocal, is multifocal, with twowith sectorstwo sectors with differentwith different individually individually tunable tuna refractionble refraction powers powers [21,27 [21,27].].

Figure 3. Left: Superelevated geometry of a lens surface with lines of intersection parallel to the x-axis (blue) and parallel Figure 3. Left: Superelevated geometry of a lens surface with lines of intersection parallel to the x-axis (blue) and parallel to to Figurethe y-axis 3. Left: (red). Superelevated Middle: Profile geometry parallel of to a lensx-axis surface along with the bluelines intersection of intersection path. para Forllel po tositive the x-axis abscissae, (blue) the and ordinates parallel aretheto zero the y-axis y-axisand (red). the (red). Middle:curve Middle: directly Profile Profile superimposes parallel parallel to x-axisto x-axisthe alongx-axis. along the Right: th bluee blue Profile intersection intersection parallel path. path.to Fory-axis For positive poalongsitive abscissae,the abscissae, red intersection the the ordinates ordinates path are [21].zeroare zero and theand curve the curve directly directly superimposes superimposes the x-axis. the x-axis. Right: Right: Profile Profile parallel parallel to y-axis to y-axis along along the red the intersection red intersection path [path21]. [21].

Figure 4. Rotation optics, consisting of two lens bodies with surfaces of azimuthal curvature depen- FigureFigure 4. 4. Rotation Rotation optics, optics, consisting consisting of of two two lens lens bodi bodieses withwith surfacessurfaces of azimuthal curvature de-de- pendencedencependence arranged arranged arranged coaxially coaxially coaxially to theto to the the optical optical optical axis axis axis and and and their theirtheir plane planeplane surfaces surfacessurfaces facing facing each each other other [27 [27].[27].]. Freeform surfaces like this with an azimuthal change of curvature and a discontinuity FreeformFreeform surfaces surfaces like like this this with with an an azimuthal azimuthal changechange of curvature and a discontinu- place very high demands on the manufacturing processes. ityity place place very very high high demands demands on on the the manufacturing manufacturing processes.processes. 4. Evaluation of the Surface Measurements 4.4. Evaluation Evaluation of of the the Surface Surface Measurements Measurements The first four steps in the process flow (see Figure1), namely initial design, optical The first four steps in the process flow (see Figure 1), namely initial design, optical performanceThe first evaluation four steps [27in ],the mechanical process flow design (see [29 Figure], and 1), printing namely [21 initial], have design, been described optical performance evaluation [27], mechanical design [29], and printing [21], have been de- elsewhereperformance in detail.evaluation In this [27], article, mechanical we focus design on analyzing[29], and printing surface measurements[21], have been of de- the scribed elsewhere in detail. In this article, we focus on analyzing surface measurements freeformscribed elsewhere surface and in detail. on the In derivation this article, of we the focus height on differenceanalyzing surface map between measurements nominal of the freeform surface and on the derivation of the height difference map between nom- andof the printed freeform geometry surface to and determine on the derivation at which positionsof the height adjustments difference of map material between deposition nom- inal and printed geometry to determine at which positions adjustments of material depo- ininal the and printing printed process geometry were to determine required. Thisat which information positions could adjustments be further of material used to depo- derive sition in the printing process were required. This information could be further used to processsition in parameters the printing to process directly were adjust required. the printing This processinformation to a respective could be further freeform used surface to by adapting, e.g., printing speed, drop size, and UV-illumination control for curing. Tactile surface measurements were conducted with a Dektak V220 SI profilometer. The measuring window had a dimension of 10.5 mm × 10.5 mm and encompassed the entire optical aperture (see Figure5a). In the measurement window, 210 profile lines with 5250 measuring points each were recorded. The path of the measuring tip was chosen to be Micromachines 2021, 12, x 6 of 12 Micromachines 2021, 12, x 6 of 12

derive process parameters to directly adjust the printing process to a respective freeform derive process parameters to directly adjust the printing process to a respective freeform surface by adapting, e.g., printing speed, drop size, and UV-illumination control for cur- surface by adapting, e.g., printing speed, drop size, and UV-illumination control for cur- ing. ing. Tactile surface measurements were conducted with a Dektak V220 SI profilometer. Tactile surface measurements were conducted with a Dektak V220 SI profilometer. Micromachines 2021, 12, 35 The measuring window had a dimension of 10.5 mm × 10.5 mm and encompassed6 of the 12 The measuring window had a dimension of 10.5 mm × 10.5 mm and encompassed the entire optical aperture (see Figure 5a). In the measurement window, 210 profile lines with entire optical aperture (see Figure 5a). In the measurement window, 210 profile lines with 5250 measuring points each were recorded. The path of the measuring tip was chosen to 5250 measuring points each were recorded. The path of the measuring tip was chosen to be parallel to the surface discontinuity of the rotation optics. Figure 5b shows a superele- beparallel parallel to theto the surface surface discontinuity discontinuity of the of the rotation rotation optics. optics. Figure Figure5b shows 5b shows a superelevated a superele- vated 3D-plot of the 210 measurement profiles, each containing 5250 data points. vated3D-plot 3D-plot of the of 210 the measurement 210 measurement profiles, profiles, each containing each containing 5250 data 5250 points. data points.

Figure 5. (a) Measuring window and (b) superelevated 3D-plot of the 210 measurement profiles. FigureFigure 5. ((aa)) Measuring Measuring window window and and ( (bb)) superelevated superelevated 3D-plot 3D-plot of the 210 measurement profiles.profiles. Application-specificApplication-specific manufacturing tolerances could be derived from thethe differencedifference Application-specific manufacturing tolerances could be derived from the difference surface Si-Ri-R ofof the nominal freeform surface and the measurement data.data. Figure6 6aa shows shows surface Si-R of the nominal freeform surface and the measurement data. Figure 6a shows the overlayedoverlayed plots plots of of the the nominal nominal freeform freeform surface surface (blue) (blue) and and the measurementthe measurement data (red).data the overlayed plots of the nominal freeform surface (blue) and the measurement data It(red). is obvious It is obvious that too that little too materiallittle material was printedwas printed in the in edge the edge areas areas of the of aperture the aperture and and too (red). It is obvious that too little material was printed in the edge areas of the aperture and muchtoo much material material was was printed printed in the in central the central area andarea near and thenear discontinuity. the discontinuity. The penetration The pene- too much material was printed in the central area and near the discontinuity. The pene- surfacestration surfaces of the measurement of the measurement data are data clearly are seen clearly by theseen red by islands the red in islands the blue in nominalthe blue tration surfaces of the measurement data are clearly seen by the red islands in the blue surface,nominal indicatingsurface, indicating a surplus a of surplus material. of ma Theterial. difference The difference between measuredbetween measured and nominal and nominal surface, indicating a surplus of material. The difference between measured and surfacenominal is surface depicted is indepicted Figure 6inb. Figure Regarding 6b. Regarding the nominal the aperture, nominal indicatedaperture, byindicated the black by nominal surface is depicted in Figure 6b. Regarding the nominal aperture, indicated by ring,the black it is obviousring, it is that obvious the printed that the optics printed did not optics entirely did not fill theentirely aperture. fill the The aperture. discrepancy The the black ring, it is obvious that the printed optics did not entirely fill the aperture. The betweendiscrepancy manufactured between manufactured and nominal surfaceand nomina was evaluatedl surface was bya evaluated comparison by ofa comparison the profiles. discrepancy between manufactured and nominal surface was evaluated by a comparison Figureof the profiles.7 shows Figure selected 7 shows profiles selected spanning prof theiles entire spanning measurement the entire window.measurement window. of the profiles. Figure 7 shows selected profiles spanning the entire measurement window.

FigureFigure 6.6. (a) Overlay of nominal surface (blue) and measured surfacesurface (red);(red); ((b)) difference surface.surface. TheThe blackblack circlecircle indicatesindicates Figure 6. (a) Overlay of nominal surface (blue) and measured surface (red); (b) difference surface. The black circle indicates the nominal aperture. the nominal aperture.

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The figure shows nominal profiles (blue), measured profiles (red), and difference profiles (green) inside the limits of the nominal aperture (perpendicular solid black lines). Consid- ering Figure 7, it is apparent that the printed surface did not extend over the nominal lens aperture: the red curves depicted in Figure 7 dropped to substrate level far inside the nominal aperture. The evaluation showed a lens aperture reduced by about 20%. This shape deviation at the edge of the aperture could be corrected directly by adjusting the allowance in the process data. The compensation of the lateral shrinkage for such struc- tures could lead to a design aperture of 125% of the nominal value [21]. Furthermore, strong form deviations could be observed, and these were most pronounced at the transi- tion from convex to concave. These form deviations must be corrected by an adapted con- trol of the high-resolution 3D printing process. To get information of shape deviation over the entire optical aperture, the difference surface is represented as a spatially resolved height map Mi-R (Figure 8). The scale of the height map directly shows the tolerance range with which the respective design could be manufactured by the specific process. Regard- ing the given case, the tolerance of the surface accuracy was in the range of ±50 µm. How- ever, the height map also quantitatively shows the difference between nominal and meas- ured surface at every surface position. This directly gives information regarding in which areas more and in which areas less material must be deposited; hence, this was a quanti- tative measure of the surface accuracy of the printing process. The quantitative infor- Micromachines 2021, 12, 35 mation of the height map could be used to adjust the parameters of the high-resolution 7 of 12 printing by adapting printing speed, drop size, and UV-illumination control for curing and to define an allowance for compensation.

FigureFigure 7. Measured 7. Measured profile profile (red), (red), nominal nominal profile profile (blue), (blue), andand differencedifference profile profile (green) (green) inside inside the thenominal nominal aperture aperture (solid (solid black and perpendicular lines). Insets: position of the respective profile on the surface. black and perpendicular lines). Insets: position of the respective profile on the surface.

The figure shows nominal profiles (blue), measured profiles (red), and difference profiles (green) inside the limits of the nominal aperture (perpendicular solid black lines). Considering Figure7, it is apparent that the printed surface did not extend over the nominal lens aperture: the red curves depicted in Figure7 dropped to substrate level far inside the nominal aperture. The evaluation showed a lens aperture reduced by about 20%. This shape deviation at the edge of the aperture could be corrected directly by adjusting the allowance in the process data. The compensation of the lateral shrinkage for such structures could lead to a design aperture of 125% of the nominal value [21]. Furthermore, strong form deviations could be observed, and these were most pronounced at the transition from convex to concave. These form deviations must be corrected by an adapted control of the high-resolution 3D printing process. To get information of shape deviation over the entire optical aperture, the difference surface is represented as a spatially resolved height map Mi-R (Figure8). The scale of the height map directly shows the tolerance range with which the respective design could be manufactured by the specific process. Regarding the given case, the tolerance of the surface accuracy was in the range of ±50 µm. However, the height map also quantitatively shows the difference between nominal and measured surface at every surface position. This directly gives information regarding in which areas more and in which areas less material must be deposited; hence, this was a quantitative measure of the surface accuracy of the printing process. The quantitative information of the height map could be used to adjust the parameters of the high-resolution printing by adapting printing speed, drop size, and UV-illumination control for curing and to define an allowance for compensation. MicromachinesMicromachines2021 2021, 12, 12,, 35 x 8 of 12 8 of 12

FigureFigure 8. 8.HeightHeight map map representation representation of the of difference the difference surface. surface.

5.5. Model Model Enhancement Enhancement and and Digital Digital Twin Twin ToTo be be able able to toanalyze analyze the theprinted printed parts parts by means by means of optical of opticalsimulation, simulation, an enhance- an enhance- mentment of of the the optical optical model model was was implemented. implemented. The model Themodel enhancement enhancement was conducted was conducted by by integratingintegrating the the measuring measuring surface surface data dataSR intoSR theinto simulation the simulation model [20]. model Hence, [20]. a Hence,perfor- a perfor- mancemance evaluation evaluation of the of theprinted printed part partwas enable was enabledd in the indigital the digitalspace. This space. approach This approach also also enabledenabled a acomparison comparison of the of theprinted printed parts parts with the with nominal the nominal surface surfacedescription. description. Table 1 Table1 shows a comparison of imaging analysis between the nominal surface S0 and the printed shows a comparison of imaging analysis between the nominal surface S0 and the printed optics (represented by the measuring surface data SR). The MTF (modulation transfer optics (represented by the measuring surface data SR). The MTF (modulation transfer function), a well-established measure of optical image quality, and the geometrical imag- function), a well-established measure of optical image quality, and the geometrical imaging ing of a bar pattern are presented, each for three adjustments of the varifocal optics: 1, 2, of a bar pattern are presented, each for three adjustments of the varifocal optics: 1, 2, and 3 dpt (where dpt is the unit symbol for diopter which is the unit of optical power: 1 diopterand 3 dpt= 1 m (where−1). The first dpt row is the shows unit the symbol simulation for diopterresults of whichthe nominal is the surface, unit of and optical the power: −1 second1 diopter row = shows 1 m the). Thesimulation first row results shows for the optical simulation surface results reconstructed of the nominal from meas- surface, and urementthe second data. row As shown shows above, the simulation the achieved results dimensional for the accuracy optical corresponded surface reconstructed to a tol- from erancemeasurement of ±50 µm. data. The first As shownrow of Table above, 1 sh theows achieved the performance dimensional of the nominal accuracy rotation corresponded optics.to a tolerance The MTF, of depicted±50 µm. up Theto a spatial first row frequency of Table of1 100 shows lp/mm, the is performance smooth and resulted of the nominal inrotation a value optics.of around The 0.4 MTF, at 100 depicted lp/mm for up each to a of spatial the three frequency adjustments. of 100 This lp/mm, value could is smooth and beresulted seen as inthe a contrast value of with around which 0.4 details at 100 wi lp/mmth a spatial for each frequency of the of three 100 lp/mm adjustments. could be This value resolvedcould be by seen the optical as the system. contrast Results with whichare presented details underneath with a spatial the MTF frequency of the simu- of 100 lp/mm latedcould imaging. be resolved The clear by and the sharp optical imaging system. of the Results bar pattern are presented confirms the underneath good imaging the MTF of behaviorthe simulated of the nominal imaging. system. The clear The andsecond sharp row imagingshows the of same the baranalysis pattern for confirmsthe meas- the good ured data representing the manufactured part. The MTF collapsed at around 11 lp/mm imaging behavior of the nominal system. The second row shows the same analysis for (please take notice of the different scale of the spatial frequency axis-in case of the meas- uredthe measureddata; the MTF data is depicted representing only up the to manufactured50 lp/mm). The part.geometric The image MTF analysis collapsed also at around reflects11 lp/mm the MTF:(please the harsh take collapse notice of of theMTF different led to a blurred scale image of the of spatial the bar frequencypattern. Ghost axis-in case reflections,of the measured resulting data; from thenon-matchi MTF isng depicted surfaces, only could up also to be 50 observed. lp/mm). The geometric image analysis also reflects the MTF: the harsh collapse of MTF led to a blurred image of the bar pattern. Ghost reflections, resulting from non-matching surfaces, could also be observed.

MicromachinesMicromachinesMicromachines 2021, 202112, x, 202112, x, 12, x 8 of 128 of 128 of 12

FigureFigure 8. HeightFigure 8. Height map8. Height maprepresentation maprepresentation representation of the of difference the of difference the difference surface. surface. surface.

5. Model5. Model5. Enhancement Model Enhancement Enhancement and andDigital andDigital Twin Digital Twin Twin To beTo able beTo ableto be analyze ableto analyze to theanalyze printedthe printedthe parts printed parts by partsmeans by means by of means optical of optical of simulation, optical simulation, simulation, an enhance- an enhance- an enhance- mentment of thement of optical the of optical the model optical model was model wasimplemented. wasimplemented. implemented. The modelThe modelThe enhancement model enhancement enhancement was wasconducted wasconducted conducted by by by integratingintegratingintegrating the measuring the measuring the measuring surface surface data surface dataSR into dataSR intothe SR simulation intothe simulation the simulation model model [20]. model [20].Hence, [20].Hence, a Hence,perfor- a perfor- a perfor- mancemance evaluationmance evaluation evaluation of the of printed the of printed the part printed partwas partwasenable wasenabled inenable dthe in digital dthe in digital the space. digital space. This space. Thisapproach Thisapproach approach also also also enabledenabled aenabled comparison a comparison a comparison of the of printedthe of printedthe parts printed parts with parts with the with nominalthe nominalthe nominalsurface surface description. surface description. description. Table Table 1 Table 1 1 showsshows a comparisonshows a comparison a comparison of imaging of imaging of imaginganalysis analysis betweenanalysis between betweenthe nominal the nominal the nominalsurface surface S surface0 and S0 andthe S0 printed andthe printed the printed opticsoptics (representedoptics (represented (represented by the by measuringthe by measuringthe measuring surface surface datasurface dataSR). dataTheSR). STheMTFR). TheMTF (modulation MTF (modulation (modulation transfer transfer transfer function),function),function), a well-established a well-established a well-established measure measure measureof optical of optical of image optical image quality, image quality, and quality, theand geometrical theand geometrical the geometrical imag- imag- imag- ing ofing a barofing a patternbarof a patternbar are pattern presented, are presented, are presented, each each for threeeach for three foradjustments three adjustments adjustments of the of varifocal the of varifocal the varifocaloptics: optics: 1, optics: 2, 1, 2, 1, 2, and and3 dpt and3 (wheredpt 3 (wheredpt dpt (where isdpt the isdpt unitthe is unitsymbolthe unitsymbol for symbol diopter for diopter for which diopter which is whichthe is unitthe is unitofthe optical unitof optical of power: optical power: 1 power: 1 1 diopterdiopter =diopter 1 m =− 1). m The=− 1). m firstThe−1). firstrowThe firstrowshows rowshows the shows simulation the simulation the simulation results results of re thesults of nominal the of nominal the surface,nominal surface, and surface, theand theand the secondsecond rowsecond rowshows rowshows the shows simulation the simulation the simulation results results for results the for optical the for optical the surface optical surface reconstructed surface reconstructed reconstructed from from meas- from meas- meas- urementurement urementdata. data. As showndata. As shown As above, shown above, the above, achieved the achieved the achieved dimensional dimensional dimensional accuracy accuracy accuracycorresponded corresponded corresponded to a tol-to a tol-to a tol- eranceerance oferance ±50 of µm.±50 of µm.The±50 µm.firstThe firstTherow firstrowof Table rowof Table 1of sh Table ows1 sh owsthe1 sh performance owsthe performance the performance of the of nominal the of nominal the nominalrotation rotation rotation optics.optics. Theoptics. MTF,The MTF,The depicted MTF, depicted depictedup to up a spatialto up a spatialto afr spatialequency frequency fr equencyof 100 of lp/mm,100 of lp/mm,100 is lp/mm, smooth is smooth is and smooth resultedand resultedand resulted in a valuein a valuein of a aroundvalue of around of 0.4 around at 0.4 100 at 0.4 lp/mm100 at lp/mm100 for lp/mm each for each offor the each of three the of three the adjustments. three adjustments. adjustments. This Thisvalue Thisvalue could value could could be seenbe seen asbe the seen as contrast the as contrast the withcontrast with which with which details which details wi detailsth wi a spatialth wi a spatialth afrequency spatial frequency frequency of 100 of lp/mm100 of lp/mm100 could lp/mm could be could be be resolvedresolved resolvedby the by opticalthe by opticalthe system. optical system. Resultssystem. Results are Results presentedare presentedare presented underneath underneath underneath the MTFthe MTF theof theMTF of simu-the of simu-the simu- latedlated imaging.lated imaging. imaging.The clearThe clearThe and clear sharpand sharpand imaging sharp imaging imagingof the of bar the of pattern bar the pattern bar confirms pattern confirms confirmsthe good the good theimaging good imaging imaging behaviorbehaviorbehavior of the of nominalthe of nominalthe nominalsystem. system. Thesystem. Thesecond Thesecond row second rowshows rowshows the shows samethe samethe analysis same analysis foranalysis the for meas-the for meas-the meas- uredured dataured datarepresenting datarepresenting representing the manufacturedthe manufacturedthe manufactured part. pa Thert. pa TheMTFrt. TheMTF collapsed MTF collapsed collapsed at around at around at 11 around lp/mm 11 lp/mm 11 lp/mm Micromachines 2021, 12, 35 (please(please take(please takenotice takenotice of noticethe of different the of different the differentscale scale of the scale of spatial the of spatial the frequency spatial frequency frequency axis-in axis-in case axis-in caseof the caseof meas- the of meas- the meas-9 of 12 uredured data;ured data; the data;MTF the MTF theis depicted MTF is depicted is depictedonly only up toonly up 50 to uplp/mm). 50 to lp/mm). 50 Thelp/mm). Thegeometric Thegeometric geometric image image analysis image analysis alsoanalysis also also reflectsreflects thereflects MTF: the MTF: the the MTF: harsh the harsh the collapse harsh collapse ofcollapse MTF of MTF le ofd toMTF le ad blurredto le ad blurredto a image blurred image of imagethe of bar the of pattern. bar the pattern. bar Ghost pattern. Ghost Ghost reflections,reflections,reflections, resulting resulting resulting from from non-matchi from non-matchi non-matching surfaces,ng surfaces,ng surfaces,could could also could alsobe observed. alsobe observed. be observed. Table 1. The modulation transfer function (MTF) and geometric image analysis of three different adjustments of the rotation opticsTableTable (1, 1. The 2,Table and1. modulation The 31. modulationdpt)The modulation for transfer the transfer nominal function transfer function surface (MTF function (MTF) and (first (MTF) geometricand row)) geometricand and geometric image the image measuredanalysis image analysis of analysis datathr ofee thr (seconddifferent ofee thr differentee row).adjustments different adjustments adjustments of the of rota- the of rota- the rota- tion opticstion opticstion (1, 2,optics (1,and 2, 3(1,and dpt) 2, 3and fordpt) 3the fordpt) nominal the for nominal the surfac nominal surface (first surface (firstrow)e (first row)and therow)and measured theand measured the measured data data(second data(second row). (second row). row). 1 dpt 2 dpt 3 dpt 1 dpt1 dpt1 dpt 2 dpt 2 dpt 2 dpt 3 dpt 3 dpt 3 dpt

MicromachinesMicromachines 2021Micromachines, 202112, x, 12, x 2021, 12, x 9 of 129 of 12 9 of 12 Micromachines Micromachines MicromachinesMicromachinesMicromachines 20212021 Micromachines,, 202112, 2021x, 202112,, 12x, , 12 2021x , x , 12, x 9 of 12 9 of9 of 12 912 of 1299 of of 12 12

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Table targets.2 Tabl showsto determinee 2 Tabl theshows esimulation-based 2 theshowsacceptable simulation-based the simulation-based manufacturing optical opticalanal- tolerances opticalanal- anal- in order ysisallows ofallows a Using forrotationallows a for sensitivity the a opticsfor sensitivity digital a sensitivity co analysisnstrained twin analysis toanalysis to with determine predict to adetermine manufacturing to determine optical acceptable acceptable performance acceptable tolerance manufacturing manufacturing of manufacturing ±10 for µm. differenttolerances A tolerances compar- tolerances tolerance in order in order in regimes order ysis of aysis torotation achieveof a rotation optics specific co opticsnstrained performance constrained with a targets.manufacturingwith a manufacturing Table 2 tolerance shows tolerance the of ±10simulation-based µm. of ±10 A compar-µm. 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A±10 thecompar-evaluation µm. opticalA ofcompar- theA compar- analysisof the ison of the simulation results of the imaging quality of the adapted model (Table 2) with twoisonof tolerance aisonof rotationtwo the isonof tolerance simulation the tworegimes: of opticssimulation the tolerance regimes:simulation ±50 constrainedresults µm regimes: results ±50 asof results manufacturµmthe of ±50 withasimaging the ofmanufacturµm imaging the a ased manufacturing imagingqualitymanufactur and quality ed±10 ofand qualityµm theed ±10of as adapted andthe toleranceadapted. µmof adapted ±10the as adapted.µmadaptedmodel Regarding ofas model ±adapted. (Table 10 Regardingmodel µ(Table them. 2) Regarding (Tablewith A 2) comparisonthe with 2) with the that of the measured data (row 2 in Table 1) allows for the performance evaluation of the MTF,that that ofanMTF, the improvement thatof measured theanMTF, of improvement measured the an measured improvementwasdata mainly data(row was data(row 2 mainlyseen in was(row Table2 inin mainly seen thTable2 e1)in 1 in allowsTabledpt seen1)th eadjustment.allows 1 in1) fordpt thallows thee adjustment.for 1 dptperformance the forThe adjustment. performance the improvement Theperformance improvement evaluation The evaluation improvementof theevaluation of of the the of the of of the the of the simulationtwo tolerance results regimes: of the imaging ±50 µm as quality manufactur of theed adapted and ±10 modelµm as adapted. (Table2) Regarding with that the MTFtwo fortwotoleranceMTF the twotolerance 2for andMTF tolerance theregimes: 3 2fordpt regimes:and the adjustments 3regimes:±50 2 dpt and µm ±50adjustments 3 dpt asµm±50 are manufacturadjustments as µmnot manufactur areas eye-catching. manufacturnoted areas andeye-catching. noted asand±10ed Regardineye-catching. µmand±10 asRegardinµm±10 gadapted. the asµm Regardin imageadapted. asg the adapted.Regarding analysis imageg Regardingthe analysis image Regarding the analysis the the of the measuredMTF, an data improvement (row 2 in Table was mainly1) allows seen for in thethe 1 performance dpt adjustment. evaluation The improvement of the two of the underneathMTF,MTF, anunderneath MTF,improvement an theunderneath improvement MTF,an theimprovement the MTF, was improvementthe thewasmainlyMTF, improvement wasmainly the seen mainly improvementin seen imagin th seen inine quality1 imagth dpt ine in1 eth adjustment. dptquality imageis 1 more adjustment.dpte quality is adjustment.obvious. more The is obvious. improvementThemore The improvementThe detailsobvious. The improvement detailsin of The the ofdetailsin the of thein tolerance regimes:MTF for ±the50 2 µandm as3 dpt manufactured adjustments are and not± as10 eye-catching.µm as adapted. Regardin Regardingg the image the MTF,analysis theMTF barMTF for thepattern theMTF barfor 2 thethepattern andarefor bar 2 resolved,the 3and dptpattern are2 3and adjustmentsresolved,dpt the 3are adjustments dpt contrast resolved, adjustments the arecontrast is notthe higher,are contrastas not isare eye-catching. higher, and as not eye-catching. isthe as higher, and eye-catching.ghost the Regardin andimages ghost Regardin the images areghostRegarding the reduced.g images image arethe g reduced.image the analysisare image reduced.analysis analysis an improvementunderneath was mainlythe MTF, seen the inimprovement the 1 dpt adjustment. in image quality The is improvement more obvious. of The the details MTF in underneathunderneathunderneath the MTF,the MTF,the the MTF, improvementthe improvementthe improvement in imag in image inquality image qualitye is quality more is more obvious.is more obvious. obvious.The Thedetails Thedetails in details in in Table 2.Table Modulation 2.Table Modulation transfer 2. Modulation transferfunctionfor the transfer function (MTF) 2 andthe functionan (MTF) 3dbar geometric dpt patternan (MTF)d adjustments geometric imagean ared geometric resolved,analysis image are analysis ofimage notthethree contrastanalysis as ofdi fferentthree eye-catching. of diis adjustmentsthreefferent higher, di adjustmentsfferent and Regardingof theadjustmentsthe rotation ghostof the rotation theimages of the image rotation are reduced. analysis the barthe patternbarthe patternbar are pattern resolved, are resolved, are resolved, the contrast the contrast the iscontrast higher, is higher, is and higher, theand ghost theand ghost the images ghost images are images reduced. are reduced. are reduced. optics (1,optics 2, and (1,optics 3 2,dpt) and (1, for 3 2, dpt)the and underneathnominal for 3 dpt)the nominal forsurface the thenominal cosurfacenstraint MTF, cosurface nstraintwith the co improvementa manufactur nstraintwith a manufactur withing a tolerancemanufactur ining image tolerance of ing±10 quality tolerance µm. of ±10 is µm. moreof ±10 µm. obvious. The details in the Table 2. Modulation transfer function (MTF) and geometric image analysis of three different adjustments of the rotation TableTable 2. ModulationTable 2. Modulation1 dpt 2. Modulation 1transfer dpt transfer 1bar function dpt transfer pattern function (MTF) function are(MTF) an resolved,d(MTF) geometricand geometric 2an dptd thegeometric image 2 contrastdpt image analysis 2image dptanalysis is of analysis higher, three of three di offferent and three different the adjustmentsdifferent ghost3 dptadjustments adjustments 3images dpt of the 3of rotationdpt arethe of rotation reduced. the rotation optics (1, 2, and 3 dpt) for the nominal surface constraint with a manufacturing tolerance of ±10 µm. opticsoptics (1, 2,optics (1,and 2, 3(1,and dpt) 2, 3and fordpt) 3the fordpt) nominal the for nominal the surface nominal surface co surfacenstraint constraint co withnstraint witha manufactur witha manufactur a manufacturing toleranceing toleranceing oftolerance ±10 of µm. ±10 of µm. ±10 µm. 1 dpt 2 dpt 3 dpt 1 dpt1 dpt1 dpt 2 dpt 2 dpt 2 dpt 3 dpt 3 dpt 3 dpt

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TheThe enhancedenhanced modelmodel onon thethe basisbasis ofof surfacesurface SSi,Ri,Ri,R couldcould bebe furtherfurther usedused asas aa digitaldigital twintwin toto predictpredict systemsystem performanceperformance duedue toto realisrealistictic andand application-specificapplication-specific manufacturingmanufacturing properties.properties. AsAs defineddefined inin SectionSection 2,2, thethe surfacesurface SSi,Ri,Ri,R isis anan adaptationadaptation ofof thethe nominalnominal surfacesurface toto differentdifferent tolerancetolerance budgetsbudgets ofof thethe manufactmanufacturinguring process.process. TheThe approachapproach toto thisthis isis toto confineconfine thethe differencedifference betweenbetween nominalnominal andand measuredmeasured surfacesurface toto aa specificspecific tolerancetolerance range.range. ByBy addingadding thisthis confinedconfined deviationdeviation spspatiallyatially resolvedresolved toto thethe nominalnominal surface,surface, aa modelmodel ofof aa manufacturedmanufactured opticsoptics waswas generatedgenerated forfor thesethese specificspecific manufacturingmanufacturing toler-toler- ances.ances. Hence,Hence, basedbased onon measurementmeasurement data,data, aa newnew opticaloptical surfacesurface waswas syntheticallysynthetically cre-cre- ated.ated. InIn thisthis way,way, thethe effecteffect ofof differentdifferent prprocessocess parametersparameters andand processingprocessing strategiesstrategies onon componentcomponent performanceperformance couldcould bebe studiedstudied inin didigitalgital spacespace withoutwithout physicallyphysically manufactur-manufactur- inging aa largelarge numbernumber ofof parts.parts. UsingUsing thethe digitaldigital twintwin toto predictpredict opticaloptical performanceperformance forfor differentdifferent tolerancetolerance regimesregimes allowsallows forfor aa sensitivitysensitivity analysisanalysis toto determinedetermine acceptableacceptable manufacturingmanufacturing tolerancestolerances inin orderorder toto achieveachieve specificspecific performanceperformance targets.targets. TablTablee 22 showsshows thethe simulation-basedsimulation-based opticaloptical anal-anal- ysisysis ofof aa rotationrotation opticsoptics coconstrainednstrained withwith aa manufacturingmanufacturing tolerancetolerance ofof ±10±10 µm.µm. AA compar-compar- isonison ofof thethe simulationsimulation resultsresults ofof thethe imagingimaging qualityquality ofof thethe adaptedadapted modelmodel (Table(Table 2)2) withwith thatthat ofof thethe measuredmeasured datadata (row(row 22 inin TableTable 1)1) allowsallows forfor thethe performanceperformance evaluationevaluation ofof thethe twotwo tolerancetolerance regimes:regimes: ±50±50 µmµm asas manufacturmanufactureded andand ±10±10 µmµm asas adapted.adapted. RegardingRegarding thethe MTF,MTF, anan improvementimprovement waswas mainlymainly seenseen inin ththee 11 dptdpt adjustment.adjustment. TheThe improvementimprovement ofof thethe Micromachines 2021, 12, 35 MTFMTF forfor thethe 22 andand 33 dptdpt adjustmentsadjustments areare notnot asas eye-catching.eye-catching. RegardinRegardingg thethe imageimage analysisanalysis10 of 12 underneathunderneath thethe MTF,MTF, thethe improvementimprovement inin imagimagee qualityquality isis moremore obvious.obvious. TheThe detailsdetails inin thethe barbar patternpattern areare resolved,resolved, thethe contrastcontrast isis higher,higher, andand thethe ghostghost imagesimages areare reduced.reduced. Table 2. Modulation transfer function (MTF) and geometric image analysis of three different adjustments of the rotation TableTable 2.2. ModulationModulation transfertransfer functionfunction (MTF)(MTF) anandd geometricgeometric imageimage analysisanalysis ofof threethree didifferentfferent adjustmentsadjustments ofof thethe rotationrotation optics (1, 2, and 3 dpt) for the nominal surface constraint with a manufacturing tolerance of ±10 µm. opticsoptics (1,(1, 2,2, andand 33 dpt)dpt) forfor thethe nominalnominal surfacesurface coconstraintnstraint withwith aa manufacturmanufacturinging tolerancetolerance ofof ±10±10 µm.µm.

11 dptdpt 2 2 dpt dpt 3 3 dptdpt

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The sensitivity analysis of anticipated manufacturing tolerances of ±10 µm resulted The sensitivity analysis of anticipated manufacturing tolerances of ±10 µm resulted in the expected system performance and thus allowed for the definition of an acceptable in the expected system performance and thus allowed for the definition of an acceptable range of manufacturing tolerances depending on system performance. range of manufacturing tolerances depending on system performance.

6.6. Discussion Discussion TheThe article article demonstrates a a proc processess flow flow with the aim of enhancing high resolution 3D3D inkjet inkjet printing printing to to manufacture manufacture optical optical freeform freeform surfaces surfaces by by means means of of a a digital digital twin. twin. BesidesBesides its its obvious advantages like fast and cost-efficientcost-efficient fabrication, 3D-inkjet printing offersoffers the the possibility possibility to to manufacture manufacture almost arbitrary shapes. To To enable the inkjet printing process to manufacture freeform freeform surfaces surfaces in in optical optical quality, quality, we pr proposedoposed a process flow flow betweenbetween optical performanceperformance evaluation,evaluation, mechanics mechanics design, design, high high resolution resolution inkjet-printing, inkjet-print- ing,measurement, measurement, and and anevaluation an evaluation and and design design adaptation adaptation with with a feedback a feedback to beto be suited suited to tocontrol control the the printing printing process. process. Regarding Regarding the the example example of of rotation rotation opticsoptics inin thisthis article,article, we focusedfocused on on the the evaluation evaluation of of the the measurement measurement da datata and and the the creation creation of of a a digital digital twin. twin. First, First, thethe evaluation evaluation of of the the measured measured data data resulted resulted in in a a height height map map that that quantitatively quantitatively indicated indicated thethe difference difference between the printed surface shap shapee and the nominal shape. This This information information was available available in spatial spatial resolution resolution and, and, hence, hence, could could be be used used to to derive derive process process parameters, parameters, e.g.,e.g., printing printing speed, speed, drop drop size, size, and and UV-illumination UV-illumination control for curing for the specificspecific freeformfreeform surface, surface, to to control control the the printing proc processess so that material deposition and printing parameters could could be be adjusted. adjusted. Second, Second, the the enhanced enhanced model model was was used used to simulate to simulate the per- the formanceperformance of the ofthe manufactured manufactured part part and and compare compare it itto to the the performance performance of of the the nominal shape.shape. The The enhanced enhanced model model was was further further used used as asa digital a digital twin twin for forsensitivity sensitivity analysis analysis re- gardingregarding the the interaction interaction between between process process and and pa partrt when when manufacturing manufacturing tolerances tolerances are are not known.known. Additionally, thisthis approach approach allowed allowed for for the the definition definition of the of minimumthe minimum dimensional dimen- sionalaccuracy accuracy for a given for a surfacegiven surface shape shape to maintain to maintain the required the required system system performance. performance. TheThe surface surface roughness of of printed components could also have an impact on optical performance. A A measurement measurement of of surface surface roughness roughness could could be integrated into the process flow,flow, as shown in Figure 11,, inin thethe measurementmeasurement boxbox andand evaluatedevaluated byby meansmeans ofof aa digitaldigital twin.twin. The The consideration consideration of of the the surface surface roughness roughness of of the the printing printing process process with with the the aim aim of adjustingof adjusting the theprocess process parameters parameters to achieve to achieve a higher a higher surface surface quality quality is a concern is a concern of future of work.future work.

Author Contributions: I.S. developed the conceptualization of of the design approach and together with U.G. the underlyingunderlying methodology.methodology. Furthermore, Furthermore, he he performed performed the the optical optical simulation simulation and and formal for- malanalysis analysis as well as well as the as validation. the validation. He also He wrotealso wr theote original the original draft. draft. R.T. developedR.T. developed the conception the concep- of tionsurface of surface measurements measurements and was and responsible was responsible for its execution.for its execution. He also He did also proofreading did proofreading of the article.of the article.U.G. further U.G. further administrated administrate the projectd the andproject did and proofreading did proofreading of the article. of the All article. the authors All the read authors and read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the Helmholtz-Gemeinschaft (POF 3, Science and Technol- ogy of Nanosystems). Acknowledgments: This work was carried out with the support of the Karlsruhe Nano Micro Fa- cility (KNMF, www.knmf.kit.edu), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT, www.kit.edu). The authors would also like to acknowledge the support of mi- croworks GmbH, Germany in performing the tactile measurements as well as Marco De Visser and his team at Luximprint for data preprocessing and 3D printing of the optical structures. Further- more, we acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology.

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approved the final manuscript. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the Helmholtz-Gemeinschaft (POF 3, Science and Technology of Nanosystems). Acknowledgments: This work was carried out with the support of the Karlsruhe Nano Micro Facility (KNMF, www.knmf.kit.edu), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT, www.kit.edu). The authors would also like to acknowledge the support of microworks GmbH, Germany in performing the tactile measurements as well as Marco De Visser and his team at Luximprint for data preprocessing and 3D printing of the optical structures. Furthermore, we acknowledge support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology. Conflicts of Interest: The authors declare no conflict of interest.

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