micromachines

Article Robust Design of an Optical Micromachine for an Ophthalmic Application †

Ingo Sieber *, Thomas Martin and Ulrich Gengenbach

Karlsruhe Institute of Technology, Institute for Applied Computer Science, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany; [email protected] (T.M.); [email protected] (U.G.) * Correspondence: [email protected]; Tel.: +49-721-608-25746 † This paper is an extended version of our paper published in Optical System Alignment, Tolerancing, and Verification VIII, San Diego, CA, USA, 17–21 August 2014

Academic Editors: Pei-Cheng Ku and Jaeyoun (Jay) Kim Received: 14 March 2016; Accepted: 3 May 2016; Published: 6 May 2016

Abstract: This article describes an approach to the robust design of an optical micromachine consisting of a freeform optics, an amplification linkage, and an actuator. The robust design approach consists of monolithic integration principles to minimize assembly efforts and of an optimization of the functional components with respect to robustness against remaining assembly and manufacturing tolerances. The design approach presented involves the determination of the relevant tolerances arising from the domains manufacturing, assembly, and operation of the micromachine followed by a sensitivity analysis with the objective of identifying the worst offender. Subsequent to the above-described steps, an optimization of the functional design of the freeform optics with respect to a compensation of the effects of the tolerances is performed. The result leads to a robust design of the freeform optics and hence ensures a defined and optimal minimum performance of the micromachine in the presence of tolerances caused by the manufacturing processes and the operation of the micromachine. The micromachine under discussion is the tunable optics of an ophthalmic implant, an artificial accommodation system recently realized as a demonstration model at a scale of 2:1. The artificial accommodation system will be developed to replace the human crystalline lens in the case of a cataract.

Keywords: robust design; tolerance analysis; freeform optics; optical modeling and simulation; design optimization; ophthalmic implant

1. Introduction The design and realization of complex microoptical devices are challenging tasks since both fabrication tolerances of the individual optical components and assembly tolerances may severely impact the systems’ optical performance. There are two approaches to alleviating this challenge: Active alignment of the optical components and robust design. Active alignment of the optical components, on the one hand, is an expensive process and, on the other hand, often is not possible in miniaturized systems. The robust design approach is two-pronged. The first step on system level is to eliminate assembly steps as far as possible by applying monolithic integration principles. This implies selection of technologies that allow realization of the monolithic design. The technology selection in turn may have repercussions on the design of the optical components. With this step alone, a few assembly tolerances have been avoided. The second step on component level is to design the critical components in such a way that their optical performance becomes robust i.e., as invariant as possible against the remaining tolerances. This robust design approach shall be illustrated by the example of a demonstration model of an ophthalmic implant.

Micromachines 2016, 7, 85; doi:10.3390/mi7050085 www.mdpi.com/journal/micromachines Micromachines 2016, 7, 85 2 of 19 Micromachines 2016, 7, 85 2 of 19

The Artificial Accommodation System (AAS) is a highly integrated system i.e., an ophthalmic deviceThe designed Artificial for Accommodationrestoration of the Systemaccommodation (AAS) is ability a highly of integratedthe human systemeye [1]. i.e.It is, ana mechatronic ophthalmic microsystemdevice designed designed for restoration to be implanted of the accommodation into the capsular ability bag of of the the human human eye eye [2] [1]. and It is consists a mechatronic of five functionalmicrosystem units. designed The active to be implantedoptical unit into i.e. the, the capsular optical bagmicromachine, of the human is eyecomposed [2] and consistsof a freeform of five optics,functional an amplification units. The active linkage, optical and unit ani.e. actuator, the optical to form micromachine, a tunable optical is composed unit [3,4 of]. a freeformA further optics, unit addressesan amplification the sensing linkage, and and the an actuatorcontrol toof formthe aaccommodation tunable optical unit[5,6]. [3 ,The4]. A internal further unitand addressesexternal communicationthe sensing and of the the control implant of theis provided accommodation by an additional [5,6]. The unit internal [7]. The and units external are communicationintegrated in a housingof the implant [8] and is an provided energy supply by an additional [9] covers unitthe energy [7]. The demand units are of integratedthe implant. in a housing [8] and an energyThe supply components [9] covers of the the AAS energy are demandpackaged of in the a glass implant. housing with integrated optical surfaces. A demonstrationThe components model of of this the eye AAS implant are packaged was designed in a glass and housing realized withat a scale integrated of 2:1 [10]. optical The surfaces. optical subsystemA demonstration of the modeldemonstrator of this eye is implantcomposed was of designed the glass and housing realized with at a scaleintegrated of 2:1 [optics10]. The and optical the opticalsubsystem micromachine of the demonstrator [11]. Figure is composed 1 shows ofa theschematic glass housing representation with integrated of the opticsAAS andwith the its opticalmain components.micromachine The [11 ].position Figure 1of shows the implanted a schematic system representation within the of thecapsular AAS withbag itsof mainthe human components. eye is depictedThe position also. of the implanted system within the capsular bag of the human eye is depicted also.

FigFigureure 1. 1. SchematicSchematic representation representation of of the the AAS AAS and and its its position position within within the the capsular capsular bag bag of of the the human human eyeeye (inset (inset on on the the right right).).

TheThe set set-up-up of of the the AAS AAS is is realized realized in in a a modular modular way way so so that that different different types types of of active active optical optical subsystemsubsystem can can be be integrated integrated [12] [12].. For For an analysisalysis and and evaluation evaluation of of different different optical optical principles principles for for the the adjustmentadjustment of of refraction refraction power, power, an an a axiallyxially moveable moveable optics optics as as well well as as a a laterally laterally moveable moveable optics optics can can bebe mounted mounted in in the the housing. housing. The The optics optics integrated integrated in in the the housing housing are are designed designed for for use use of of an an a axiallyxially moveablemoveable lens. lens. To To be be able able to to analyze analyze and and evaluate evaluate laterally laterally moveable moveable optics optics with with only only one one housing housing design,design, an an inset lens is usedused toto provideprovide thethe necessary necessary basic basic refraction refraction power. power. The The inset inset lens lens is fixedis fixed by by an index-matched adhesive to the optics integrated in the rear window of the housing. The rear an index-matched adhesive to the optics integrated in the rear window of the housing. The rear surface surface curvature matches the curvature of the optics integrated. The mechanical interface for curvature matches the curvature of the optics integrated. The mechanical interface for mounting the mounting the different types of optics into the housing has been designed to allow for that different types of optics into the housing has been designed to allow for that modularity. The reference modularity. The reference for positioning of the optical subsystem is the rear wall of the glass for positioning of the optical subsystem is the rear wall of the glass housing. Both types of optical housing. Both types of optical subsystems are mounted to this surface by means of an ultraprecision subsystems are mounted to this surface by means of an ultraprecision milled mount. milled mount. The optical micromachine is of vital importance to the total system since the optical performance The optical micromachine is of vital importance to the total system since the optical performance of the AAS depends on its reliable function. For this reason the above-outlined two-pronged robust of the AAS depends on its reliable function. For this reason the above-outlined two-pronged robust design approach was applied. design approach was applied. The article is organized as follows: Section2 presents the optical subsystem of the AAS. Section3 The article is organized as follows: Section 2 presents the optical subsystem of the AAS. Section 3 covers the determination of the relevant tolerances of the optical subsystem and the tolerance analysis covers the determination of the relevant tolerances of the optical subsystem and the tolerance analysis of the optical micromachine based on a sensitivity approach. The optimization of the freeform optics of the optical micromachine based on a sensitivity approach. The optimization of the freeform optics with the objective of minimizing the impact of the tolerances on the systems’ performance as well as with the objective of minimizing the impact of the tolerances on the systems’ performance as well as a discussion of the results are presented in Section4. The paper is completed by a discussion of the a discussion of the results are presented in Section 4. The paper is completed by a discussion of the presented robust design approach. presented robust design approach.

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2. Optical Optical Subsystem Subsystem of of the the Demonstration Demonstration Model Model The components of the optical subsystem of the demonstration model are comprised of the optical micromachine, a glass housing with optics integrated, and an additional inset lens fixedfixed by an index-matchedindex-matched adhesive to the optics integrated in the rear windowwindow ofof thethe housing.housing.

2.1.2.1. Optical Micromachine The optical optical micromachine micromachine consists consists of ofthree three principal principal components components freeform freeform optics, optics, actuator, actuator, and amplificationand amplification linkage. linkage. The following The following subsections subsections present and present discuss and the discuss individual the individual principal principalcomponents components..

2.1.12.1.1.. Optics The optical concept discussed in this article works according to the principle invented in the 1960’s1960’s by Alvarez and Humphrey [[13].13]. According to this principle, in an optical system comprising of two cubic-typecubic-type lens parts the refraction power is varied by mutually shifting both lens parts laterally as shown in Figure 2 2.. AHAH lenseslenses werewere studiedstudied extensivelyextensively [[1414–16] and and proposed proposed for different different technical [[1717–19] as well as ophthalmological applicationsapplications [[20,21].20,21].

Figure 2. AlvarezAlvarez-Humphrey-Humphrey optics. optics. Variation Variation of of the the refraction refraction power power by by means means of of a shift of the Alvarez-HumphreyAlvarez-Humphrey surfaces perpendicularperpendicular toto thethe opticaloptical axisaxis ((zz-axis-axis inin thethe figure).figure).

The opposing opposing surfaces surfaces are are conjugated conjugated and and of a ofcubic a cubicshape. shape. Parameters Parameters for designing for designing Alvarez- Alvarez-HumphreyHumphrey (AH) lenses (AH) are lenses the arelateral the lateralmovement movement v of thev of lens the lensparts parts and andthe the“form” “form” parameter parameter A Adeterminingdetermining the the shape shape of of the the lens lens surface. surface. The The polynomial polynomial description description of of the the surface surface sag sag is is given given by Equation (1): 2 3 zpx, yqz( “x,yA) =pp Ay ((´y −v vq)xx2 +` 1/31{(3yp y− v´)3) vq q (1(1)) Lateral movement of lens parts is advantageous in all applications, where the space available along the optical optical axis axis is is limited, limited, as as is is true true for for the the demonstration demonstration model model at hand. at hand. Requirements Requirements of the of opticsthe optics and andits itsdesign design are are presented presented in in[22]. [22 ].To To compensate compensate for for aberration aberration effects effects and and fulfill fulfill the requirements ofof thethe demonstrator’s demonstrator’s specifications, specifications, the the description description of theof the AH AH surfaces surfaces is expanded is expanded and andis represented is represented by a by polynomial a polynomial of fourth of fourth order order (Equation (Equation (2)). (2)). The The parameters parameters used used are are depicted depicted in Tablein Table1. The 1. centerThe center thicknesses thicknesses of the individualof the individual lens parts lens are parts 500 µ m,are material500 µm, used material was polymethyl used was polymethylmetacrylate metacrylate (PMMA) with (PMMA) a refractive with indexa refractive of nD =index 1.4905. of nD = 1.4905.

z(x,y4) = a1x4 +3 a2x3y + a23x22y2 + a4xy33 + a5y4 4+ a6x3 + 3a7x2y + a28xy2 + a9y32 + a10x2 +3 a11xy + 2 zpx,yq = a1x ` a2x y ` a3x y ` a4xy ` a5y ` a6x ` a7x y ` a8xy ` a9y ` a10x ` (2) a12y2 2+ a13x + a14y (2) a11xy ` a12y ` a13x ` a14y

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TableTable 1. 1. ParametersParameters of the freeformfreeform surfaces. surfaces.

a1 a2 a3 a4 a5 a6 a7 a a a a a a a −13.18 × 10−6 −3.182 × 10−5 −1.483 × 10−4 0.004 5.14 × 105 −5 −1.91 × 106−5 0.02 7 ´3.18 ˆ 10´6 ´3.18 ˆ 10´5 ´1.48 ˆ 10´4 0.00 5.14 ˆ 10´5 ´1.91 ˆ 10´5 0.02 a8 a9 a10 a11 a12 a13 a14 a86.77 × 10−5 6.40a9 × 10−3 1.34a10 × 10−4 1.24a ×11 10−4 −8.92 ×a 1210−5 −1.68 × 10a13−5 −1.24 × 10a−414 6.77 ˆ 10´5 6.40 ˆ 10´3 1.34 ˆ 10´4 1.24 ˆ 10´4 ´8.92 ˆ 10´5 ´1.68 ˆ 10´5 ´1.24 ˆ 10´4 The performance of AH optics depends crucially on a proper alignment of the two lens parts [23].The Alignment performance of ofboth AH identical optics depends lens parts crucially to onthe a linkage proper alignment in x- and of y the-direction two lens is parts realized [23]. by two mountingAlignment ofpins both and identical an alignment lens parts surface to the linkage (see Figure in x- and 3). They-direction mating is features realized byof twothe mountingamplification linkagepins are and two an alignmentcylindrical surface holes (seeand Figure the respective3). The mating planar features alignment of the surface amplification on the linkage output are stage. two cylindrical holes and the respective planar alignment surface on the output stage. Alignment in Alignment in z-direction is accomplished by the two planar contact surfaces. The alignment surfaces z-direction is accomplished by the two planar contact surfaces. The alignment surfaces are enlarged are enlarged to reduce rotation/tilt around an axis through both pins. The position of the mounting to reduce rotation/tilt around an axis through both pins. The position of the mounting pins on the pins lenson the parts lens is subject parts tois highsubject tolerances to high due tolerances to shrinkage due during to shrinkage the injection during molding the process. injection In ordermolding process.to avoid In order over-determining to avoid over- bydetermining mating two by cylindrical mating two pins cylindrical with two cylindricalpins with two holes cylindrical a statically holes a staticallydeterminate determinate pin arrangement pin arrangement was selected. was Aselected. cylindrical A cylindrical mounting pinmounting defines thepin xdefines,y position the x,y positionof the of lens the partlens in part the plane,in the while plane, the while second the pin second with an pin elliptical with an shape elliptical defines shape the remaining defines the remainingrotational rotational degree of degree freedom. of Thisfreedom. way positional This way tolerances positional between tolerances the cylindrical between holes the cylindrical in the silicon holes in theamplification silicon amplification linkage and linkage the injection and the molded injection pins m ofolded the lenspins parts of the can lens be parts compensated. can be compensated.

FigureFigure 3. Computer 3. Computer Aided Aided Design Design (CAD (CAD)) model model of one ofof bothboth identical identical lens lens parts parts of theof the AH AH optics optics withwith alignment alignment structures structures indicated indicated [1]. [1].

The Thefreeform freeform optics optics was was manufactured manufactured using using microinjectionmicroinjection molding, molding, which which is ais replication a replication processprocess with with the thepotential potential for for mass mass fabrication. fabrication. This This process process also also allows allows for monolithicmonolithic integration integration of adjustmentof adjustment and alignment and alignment structures structures and and optical optical surfaces surfaces toto optimizeoptimize positioning positioning accuracy accuracy in in assembly.assembly. To Toprepare prepare manufacture, manufacture, aa set set of of simulation simulation experiments experiments as well as as well process as process optimization optimization were wereconducted conducted to demonstrateto demonstrate the feasibility the feasibility of freeform of surface freeform fabrication surface by fabrication means of a microinjection by means of a microinjectionmolding process molding [24]. process One result [24]. of One these result simulations of these was simulations that all cavities, was includingthat all cavities, the mounting including the mountingpins of the pins integrated of the fixing integrated structures, fixing and structures, the thinnest and part the of thinnest the lens with part a of thickness the lens of with 100 µam thickness can of 100be µm completely can be filledcompletely [25]. On filled the basis [25]. of On this the result, basis the of molding this result, tool was the fabricatedmolding andtool the was freeform fabricated optics was molded under consideration of the optimized parameter set. and the freeform optics was molded under consideration of the optimized parameter set. Adaptation of the refraction power of the optics to the accommodation demand is done by a Adaptation of the refraction power of the optics to the accommodation demand is done by a piezoelectric stack actuator and a silicon linkage for transmission and amplification of the actuation piezoelectricstroke [26 stack]. actuator and a silicon linkage for transmission and amplification of the actuation stroke [26].

2.1.2. Actuator The displacement-specific change of the refractive power of a lateral-shift optics can be designed in a wide range. However, for cost-effective fabrication, a trade-off between a small lens displacement to be provided by the actuator and high robustness of the optics against deviations of shape and

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2.1.2. Actuator The displacement-specific change of the refractive power of a lateral-shift optics can be designed

Micromachinesin a wide range. 2016, 7 However,, 85 for cost-effective fabrication, a trade-off between a small lens displacement5 of 19 to be provided by the actuator and high robustness of the optics against deviations of shape and position of the optical surfaces must be made. A relative displacement difference of both lens parts of 360 µmµm with the ideal actuator rest position being at 81 µmµm relative lens displacement allows for a chachangenge ofof 4.6 4.6 dpt dpt in opticalin optical power power and was and found was to found be a reasonable to be a compromisereasonable forcompromise the demonstration for the demonstrationmodel [27]. The model requirements [27]. The ofrequirements the actuator of with the respectactuator to with cycle respect lifetime, to cycle power lifetime, consumption, power consumption,fail-safe behavior, fail-safe response behavior, time, response and space time, can and be found space incan [4 ].be The found initial in [ actuator4]. The initial geometry actuator was geometryoptimized was to fitoptimized into a micromachine to fit into a micromachine with a cylindrical with a cylindrical design space. design As space. the availability As the availability of this ofactuator this actuator became became an issue, an anissue, oversized an oversized commercial commercial actuator actuator had to had be selected. to be selected. Hence, Hence, the scaled the scaledcylindrical cylindrical design design space spac of thee of micromachine the micromachine had tohad be to violated be violated locally locally and and the the design design had had to beto beadapted adapted accordingly. accordingly. The finalThe final actuator actuator for the for demonstration the demonstration model model was a piezoelectric was a piezoelectric stack actuator stack (Noliacactuator A/S, (Noliac NAC2001-H06, A/S, NAC2001 Noliac,-H06 Kvistgaard,, Noliac, Kvistgaard Denmark), Denmark with a nominal) with a stroke nominal of 4.9 strokeµm ofofwhich 4.9 µm a ofmaximum which a ofmaximum 2.9 µm are of used2.9 µm during are used operation during [ 26operation]. [26].

2.1.32.1.3.. Amplification Amplification Linkage µ To provide the maximum relative lens part displacement difference of 279 µmm calculated from µ the actuator rest position ( i.e.,, a stroke of about 140 µmm per lens part), a displacementdisplacement amplificationamplification ratio of at least 48.1 is required. For reasons of miniaturization and and reduction reduction of of assembly assembly processes, processes, a monolithic design with flexure flexure hinges was selected. The The planarity of of the linka linkagege enables a Deep Reactive Ion Etching (DRIE) process for fabrication of the linkage in single-crystalsingle-crystal silicon [4]. [4]. Figure 44 depicts the linkage as designed (left) and the linkage manufactured by DRIE (right) fulfilling fulfilling the transmission requirementsrequirements [ 26[26].]. The The trapezoid-shaped trapezoid-shaped structure structure in the in upperthe upper part ofpart the of linkage the linkage serves servesas a mount as a formount the stackfor the actuator. stack actuator. Self-aligning Self-aligning structures structures to mount to the mount lens partsthe lens as well parts as as limit well stop as limitstructures stop structures for overload forprotection overload protection were integrated were integrated in the output in the stages output of the stages linkage. of the The linkage. minimum The minimumstructure width structure in the width silicon in the linkages silicon is linkages 24 µm. The is 24 linkage μm. The is enclosedlinkage is in enclosed a solid frame in a solid that servesframe thatboth serves as interface both as to interface the subsystem to the subsystem mount and mount as handling and as handling frame during framesubsystem during subsystem assembly. assembly.

(a) (b)

FigureFigure 4. 4. DesignDesign of the of amplification the amplification linkage linkage (a) and realized (a) and silicon realized linkages silicon manufactured linkages manufactured by DRIE (b). by DRIE (b). 2.1.4. Set-up of the Optical Micromachine 2.1.4.Assembly Set-up of theof the Optical freeform Micromachine optics, the piezo stack actuator, and the silicon linkage yields the opticalAssembly micromachine. of the freeform Figure 5 optics, shows the two piezo photographs stack actuator, of the and optical the siliconmicromachine linkage yields in two the different optical actuationmicromachine. states. Figure On the5 shows left hand two photographsside, the piezo of thestack optical actuator micromachine is contracted in two leading different to an actuation inward movementstates. On theof the left lens hand parts side, and the piezothe accommodated stack actuator state, is contracted while on leading the right to an the inward actuator movement is in an expandedof the lens partsstate andleading the accommodatedto an outward state, movement while on of the the right two the lens actuator parts isand in an therefore expanded to state the disaccommodated state. Also shown in Figure 5 is the mount, which is the mechanical interface to the glass housing.

Micromachines 2016, 7, 85 6 of 19 leading to an outward movement of the two lens parts and therefore to the disaccommodated state. MicromachinesAlso shown 201 in6 Figure, 7, 85 5 is the mount, which is the mechanical interface to the glass housing. 6 of 19

Piezoelectric stack actuator

Freeform optics

Silicon amplification linkage

Mount

Figure 5. 5. OpticalOptical micromachine micromachine and and its itscomponents: components: Tunable Tunable freeform freeform optics, optics, silicon silicon linkage, linkage, and piezoelectricand piezoelectric stack stack actuator actuator [1]. [1Left]. Left: contracted: contracted actuator actuator and and accommodated accommodated state, state, rightright:: expanded expanded actuator and disaccommodated state of the optics.

2.22.2.. Additional Additional Optical Components As mentioned above, the AAS features a modular concept, where housing as well as optical surfaces integrated in the housing housing will will be be used used for for different different optical optical principles. principles. Originally, Originally, the housing housing was designed for an optical micromachine using an axially mov moveableeable lens. Hence, Hence, the the rear rear-side-side optics is a negativenegative asphericaspheric lens. lens. Combining Combining this this rear-side rear-side optics optics with with an an AH AH optics, optics, an asphericalan aspherical inset inset lens lenshas tohas be to designed be designed to add to aadd positive a positive refraction refraction power power to the to negative the negative lens oflens the of rear the side.rear side. Theinset The insetlens islens monolithically is monolithically integrated integrated with the with diamond-lathed the diamond- mountlathed thatmount forms that the forms interface the interface between betweenactive optics active subsystem optics subsystem and glass and housing. glass housing. Since theSince rear the surface rear surface of the of asphere the asphere has tohas match to match the thecurvature curvature of the of opticsthe optics integrated integrated in the in housing, the housing, only theonly aspheric the aspheric front surfacefront surface remains remains as free as design free designparameter parameter for beam for modification. beam modification. The inset The lens inset was lens manufactured was manufactured from PMMA from with PMMA a refractive with a refractiveindex of n Dindex= 1.4905. of nD = 1.4905. The sag of of the the aspheric aspheric front front surface surface of of the the inset inset lens lens is is given given by by Equation Equation (3 (3),), with with r2 r=2 x=2 + x y22,+ the y2, conicthe conic constant constant k, andk, andthe curvature the curvature c (reciprocalc (reciprocal value value of radius). of radius). The center The center thickness thickness of the of inset the insetlens islens 1.6 is mm. 1.6 mm.

2 = c r + + 2 + 3+ 4+ +5 +6 +7 8 (3) z “ 1 + 1 (12+ ) ` β1 r ` β2 r ` β3 r ` β4 r ` β5 r ` β6 r ` β7 r ` β8 r (3) ` ´ p ` q 2 2 2 3 4 5 6 7 8 1 1 1𝑐𝑐 𝑟𝑟 k c r 1 2 3 4 5 6 7 8 𝑧𝑧 2 2 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 𝛽𝛽 𝑟𝑟 The parameters� used− are𝑘𝑘 depicted𝑐𝑐 𝑟𝑟 in Table 2. The parametersa used are depicted in Table2.

Table 2. Parameters of the aspheric front surface of the inset lens. Table 2. Parameters of the aspheric front surface of the inset lens.

c k β1 β2 β3 β4 β5 β6 β7 β8 c0.738 k −1.647 β1 4.17 × 10−4β 2 −0.328 β3−0.025 β0.0794 −0.039β5 0.010β 6 −1.47 × 10−3β 7 8.91 × 10−5 β8 0.738 ´1.647 4.17 ˆ 10´4 ´0.328 ´0.025 0.079 ´0.039 0.010 ´1.47 ˆ 10´3 8.91 ˆ 10´5 The optical surface integrated in the front half shell of the housing is also an asphere with a centerThe thickness optical surfaceof 250 µm, integrated whose in surface the front sag half is shelldescribed of the by housing Equation is also (4) an. The asphere material with used a center for precisionthickness molding of 250 µm, of whosethe half surface shells was sag isthe described Schott glass by Equation BF33 with (4). a refraction The material index used of fornD = precision 1.4714. molding of the half shells was the Schott glass BF33 with a refraction index of nD = 1.4714. = + + + (4) 1 + 1 (12+ ) 𝑐𝑐c𝑟𝑟 r2 2 4 6 z “ ` β2 r2 ` β4 r4 ` β 6 r6 (4) 𝑧𝑧 2 2 𝛽𝛽2 𝑟𝑟 𝛽𝛽4 𝑟𝑟 𝛽𝛽6 𝑟𝑟 The parameters used are shown1 `in� Table1−´ p 13.` 𝑘𝑘kq𝑐𝑐c 𝑟𝑟r a TheTable parameters 3. Parameters used of the are aspheric shown surface in Table of 3the. optics integrated in the front half shell of the housing.

c k β2 β4 β6 0.054 −2.444 0.015 1.08 × 10−3 −4.76 × 10−5

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Table 3. Parameters of the aspheric surface of the optics integrated in the front half shell of the housing.

c k β2 β4 β6 ´3 ´5 Micromachines 2016, 7, 85 0.054 ´2.444 0.015 1.08 ˆ 10 ´4.76 ˆ 10 7 of 19 Micromachines 2016, 7, 85 7 of 19

As a a substitute substitute of ofthe the human human cornea cornea a commercially a commercially available available converging converging lens was lens used was i.e. used, the NT As47 -a873 substitute lens supplied of the human by Edmund cornea Opticsa commercially (Barrington available, NJ, USA)converging with anlens effective was used focal i.e., thelength of NTi.e. 47, the-873 NT lens 47-873 supplied lens suppliedby Edmund by EdmundOptics (Barrington Optics (Barrington,, NJ, USA) NJ, with USA) an effective with an effectivefocal length focal of length 22.5 mm [28]. 22.5of 22.5mm mm[28]. [28]. Figure 6 shows a sectional view of the optical model of the AAS with the remaining geometrical FigureFigure 6 6shows shows a sectional a sectional view view of the of the optical optical model model of the of AAS the AAS with withthe remaining the remaining geometrical geometrical parameters given. The boundary between the inset lens and the optics integrated in the housing is parametersparameters given. given. The The boundary boundary between between the inset the inset lens lensand andthe optics the optics integrated integrated in the inhousing the housing is is shown in green. shownshown in in green. green.

ImageImage plane plane

FigureFigure 6. Optical 6. Optical model model of the of AAS. theAAS. Figure 6. Optical model of the AAS. 2.3. Set-up and Model of the Optical Subsystem 2.32.3.. Set Set-up-up and and Model Model of of the the Optical Optical Subsystem The optical subsystem of the demonstrator is composed of a glass housing with integrated The optical subsystem of the demonstrator is composed of a glass housing with integrated optics, optics,The the insetoptical lens subsystem and the optical of the micromachine. demonstrator The is housing composed consists of aof glass two half housing shells containingwith integrated the inset lens and the optical micromachine. The housing consists of two half shells containing a aoptics, shaped the entrance inset lens and and exit thewindow optical providing micromachine. the basic The refraction housing power consists of theof two optical half system shells containing[11]. shaped entrance and exit window providing the basic refraction power of the optical system [11]. Aa precision shaped entrance molding andprocess exit was window chosen providing for fabricatio then basic of the refraction entrance andpower exit of window the optical of the system glass [11]. A precision molding process was chosen for fabrication of the entrance and exit window of the glass housing.A precision Figure molding 7 shows process a photograph was chosen of the precisionfor fabricatio-moldedn of rearthe entrancehalf shell and anexit integrated window opticsof the glass withhousing. an aperture Figure Figure of77 shows shows6 mm. aaFigure photograph 8 shows ofof the the optical precision-moldedprecision micromachine-molded rearrear assembled halfhalf shellshell on andand its anan mount integrated in the opticsoptics rearwith half an s hellaperture of the of glass 6 mm. housing. Figure 88 showsshows thethe opticaloptical micromachinemicromachine assembledassembled onon itsits mountmount inin thethe rearrear half s shellhell of the glass housing.

Figure 7. Precision-molded rear window of the demonstrator with its integrated optics (optical aperture of 6 mm) [1]. Figure 7. Precision-molded rear window of the demonstrator with its integrated optics (optical Figure 7. Precision-molded rear window of the demonstrator with its integrated optics (optical aperture aperture of 6 mm) [1]. of 6 mm) [1].

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Micromachines 2016, 7, 85 8 of 19

MicromachinesSilicon 2016, 7, amplification85 8 of 19 linkage Piezoelectric stack actuator Silicon amplification linkage Piezoelectric stack actuator

Freeform optics

Freeform optics Rear window of the housing Mount

Rear window of the housingFigure 8. Optical micromachine assembled to the rear half shell of Mountthe glass housing. Figure 8. Optical micromachine assembled to the rear half shell of the glass housing. To beFigure able 8.to Optical conduct micromachine a tolerance assembled analysis tosimulation, the rear half a shell model of the of glassthe entirehousing. optics has to be Togenerated. be able to Figure conduct 9 shows a tolerance the optical analysis model of simulation, the subsystem a modelwith some of thecalculated entire rays optics in the has case to be generated.of Tofour Figure be fields able9 showsspanningto conduct the half opticala toleranceof the model field analysis angle of the of simulation, subsystemthe foveal viewa with model of some the of human the calculated entire eye opticsand rays a sketchhas in the to of casebe the of four fieldsgenerated.housing. spanning Figure The optical half9 shows ofmodel the the consistsfieldoptical angle modelof the of opticalof the the foveal subsystemsubsystem view withas of defined the some human calculatedabove eyeand andraysan additional ain sketchthe case lens of the of repfourresenting fields spanning the cornea half of of the the human field angle eye. of the foveal view of the human eye and a sketch of the housing. The optical model consists of the optical subsystem as defined above and an additional lens housing. The optical model consists of the optical subsystem as defined above and an additional lens representing the cornea of the human eye. representing the cornea of the human eye. Housing Optics integrated into the housing

Housing Optics integrated into the housing

Image plane

Optical micromachine Image plane Cornea lens Optical micromachine Cornea lens Figure 9. Model of the demonstrator optics of the AAS: Cornea lens, sketch of the housing with the integrated optics, the optical micromachine, and the image plane [29]. FigureFigure 9. Model 9. Model of theof the demonstrator demonstrator optics optics ofof thethe AAS: Cornea Cornea lens, lens, sketch sketch of the of thehousing housing with withthe the 3. Tolerance Analysis integratedintegrated optics, optics, the the optical optical micromachine, micromachine, and and thethe image plane plane [29]. [29 ]. The above-defined model was used to conduct a sensitivity analysis with respect to the 3. Tolerance Analysis 3. Tolerancemanufacturing Analysis and operational tolerances. The first step in this approach is to determine the relevant tolerances.The above -defined model was used to conduct a sensitivity analysis with respect to the Themanufacturing above-defined and operational model wastolerances. used The to conductfirst step in a this sensitivity approach analysisis to determine with the respect relevant to the manufacturingtolerances.3.1. Determination and operational of the Relevant tolerances. Tolerances The first step in this approach is to determine the relevant tolerances.Tuning of the refraction power of the demonstration model is achieved by a lateral shift of the 3.1. Determination of the Relevant Tolerances two AH lens parts relative to each other in x-direction. Consequently, the focus in tolerance analysis 3.1. Determination of the Relevant Tolerances of Tuningthe optical of the subsystem refraction lies power on the of AH the optics. demonstration To obtain model a good is optical achieved performance, by a lateral two shift points of the are Tuningtwocritical, AH oflens namely the parts refraction manufacturingrelative to power each accurother of theacy in demonstration xof-direction. the optical Consequently, freeform model surfaces is achievedthe focus and in a by propertolerance a lateral alignment analysis shift of of the two AHof the lens freeformoptical parts subsystem surfaces relative in tolies the each on static the other AH as well inoptics.x -direction.as inTo the obtain dynamic Consequently, a good state. optical performance, the focus in two tolerance points analysisare of thecritical, optical Effectsnamely subsystem of manufacturing manufacturing lies on the accur tolerances AHacy optics. of ofthe theTo optical freeform obtain freeform a optics good surfaces opticalwere investigated and performance, a proper and alignment are two described points of are thein freeform [22]. The surfaces results inof thethis static tolerance as well analysis as in the show dynamic only negligible state. differences in the performance of critical, namely manufacturing accuracy of the optical freeform surfaces and a proper alignment of the theEffects freeform of manufacturing surfaces manufactured tolerances by of means the freeform of the microinjection optics were investigated molding process and are(represented described by freeformin their[22] surfaces. Themetrology results in the data) of staticthis and tolerance as the well nominal asanalysis in surfaces. the show dynamic only negligible state. differences in the performance of Effectsthe freeformBesides of manufacturing surfaces the manufacturing manufactured tolerances accuracy by means of theof ofthe freeform the freeform microinjection optics surfaces, were molding a proper investigated process alignment (represented and of arethe surfaces described by in [22their].to The metrologyeach results other of data)is this of fundamentaland tolerance the nominal analysis importance. surfaces. show A only positioning negligible error differences of the two in lens the parts performance may have of a the freeformsevere surfacesBesides impact the manufactured manufacturingon the performance by accuracy means of the ofof the total freeform microinjection system surfaces,[30]. The molding atunable proper process opticsalignment is (representedguided of the and surfaces shifted by their metrologyto each data) other and is of the fundamental nominal surfaces. importance. A positioning error of the two lens parts may have a Besidessevere impact the manufacturing on the performance accuracy of the of total the system freeform [30]. surfaces, The tunable a proper optics alignmentis guided and of theshifted surfaces to each other is of fundamental importance. A positioning error of the two lens parts may have a Micromachines 2016, 7, 85 9 of 19 severe impact on the performance of the total system [30]. The tunable optics is guided and shifted by a deformation of the monolithic silicon linkage. Hence, it is of great interest to determine the impact of mispositioning induced by assembling the lens parts and the silicon linkage. Mispositioning of the lens parts relative to the nominal positions can have three different origins. The first of which are the Micromachines 2016, 7, 85 9 of 19 manufacturing tolerances. Fabrication of the silicon linkage as well as of the positioning structures and the opticalby a surfacedeformation integrated of the monolithic in the freeform silicon opticslinkage. is Hence, subject it tois tolerances.of great interest Then to tolerancesdetermine the related to the assemblyimpact processof mispositioning have to be induced taken into by account.assembling Here the position lens parts errors and in joiningthe silicon the lenslinkage. parts and the linkageMispositioning and in joining of the lens the parts linkage relative with to thethe mountnominal andpositions the housing can have three play different a crucial origins. role and The have to first of which are the manufacturing tolerances. Fabrication of the silicon linkage as well as of the be considered. Last but not least operational influences can lead to tolerances. Operational influences positioning structures and the optical surface integrated in the freeform optics is subject to tolerances. are primarilyThen toler causedances related by gravitational to the assembly forces process on the have lens to be suspensions. taken into account. Furthermore Here position imperfections errors of the deformationin joining the kinematics lens parts and of thethe linkage linkage and lead in joining to a mispositioning the linkage with ofthe the mount lens and parts the relativehousing to the nominalplay positions. a crucial role and have to be considered. Last but not least operational influences can lead to Thesetolerances. tolerances Operational and influences influences causeare primarily a positioning caused error by gravitational of the freeform forces surfaces. on the Fourlens main types ofsuspensions. positioning Furthermore errors were imperfections identified of (see the Figuredeformation 10). Akinematics decentration of the oflinkage the total lead opticsto a with mispositioning of the lens parts relative to the nominal positions. respect to the glass housing has been significantly reduced due to monolithic integration of the inset These tolerances and influences cause a positioning error of the freeform surfaces. Four main lens withtypes the of mount positioning for the errors micromachine. were identified The (see remaining Figure 10). tolerancesA decentration result of the from total the optics assembly with of the mountrespect into the to the housing, glass housing which has is facilitatedbeen significantly by the reduced self-alignment due to monolithic of the integration inset lens of in the the inset matching concavelens shape with ofthe the mount rear for glass the micromachine. housing. In the The next remaining assembly tolerances step the result optical from the micromachine assembly of the in turn is alignedmount on the into mount the housing, by specifically which is designedfacilitated by matching the self-alignment alignment of structuresthe inset lens on in the the upper matching rim of the mountconcave and in theshape handling of the rear frame glass of housing. the amplification In the next assembly linkage. step The the DRIE optical process micromachine for fabrication in turn of the is aligned on the mount by specifically designed matching alignment structures on the upper rim of linkages does not yield perfectly vertical sidewalls. Thus the intended cylindrical holes turned out the mount and in the handling frame of the amplification linkage. The DRIE process for fabrication to haveof a the slight linkages conicity. does not The yield diameter perfectly tolerance vertical sidewalls. of the cylindrical Thus the intended holes oncylindrical the wafer holes front turned side was 800 ˘ 2outµm, to whilehave a theslight backside conicity. tolerance The diameter of the tolerance hole was of the 800 cylindrical + 40 µm. holes In order on the to wafer allow front insertion side of the mountingwas pins800 ± of 2 µm, the while lens into the backside the silicon tolerance linkage of the during hole was assembly 800 + 40 a µm. 15 µ Inm order clearance to allow between insertion nominal hole diameterof the mounting and nominal pins of the pin lens diameter into the wassilicon selected linkage yieldingduring assembly a nominal a 15 µm mounting clearance pin between diameter of 795 µm.nominal Shrinkage hole diameter of PMMA and in nominal injection pin molding diameter iswas typically selectedin yielding the range a nominal of 0.1%–0.8% mounting and pin within diameter of 795 µm. Shrinkage of PMMA in injection molding is typically in the range of 0.1%–0.8% that range severely influenced by injection molding process parameters (holding, packing pressure, and within that range severely influenced by injection molding process parameters (holding, packing mold cooling).pressure, mold On the cooling). basis ofOnthese the basis individual of these individual tolerances tolerances the remaining the remaining position position errors errors have been estimatedhave (see been Table estimated4). (see Table 4).

(a) (b)

(c) (d)

Figure 10. Different types of position errors identified for the AH optics. Figure 10. Different types of position errors identified for the AH optics.

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Table 4. Position errors due to a decentration of the total optics. Table 4. Position errors due to a decentration of the total optics.

x-Directionx-Direction yy-Direction-Direction zz-Direction-Direction ´−46–µ52m–+52 µm µm ´45−45µ–m–+4848 µmµ m ´50−µ50m–+50–50 µmµm

TheThe lateral lateral position position error error means means a relative a relative displacement displacement of both of both lens lens parts parts perpendicular perpendicular to the to functionalthe functional shifting shifting direction. direction. This is Thiscaused is by caused assembly by assembly tolerances tolerances between lens between parts lensand linkage parts and as welllinkage as asby well manufacturing as by manufacturing tolerances tolerances of the lens of the parts lens partsand is and estimated is estimated to be to bein inthe the range range of of −´2525–25µm–+25 µm. Theµm. wedge The wedge error erroris caused is caused by a by relative a relative rotation rotation of ofboth both lens lens parts parts around around an an axis axis perpendicularperpendicular to the the optical optical axis axis and and the the shift shift axis, axis, due due to to assembly assembly tolerances tolerances between between lens lens parts parts and and linkagelinkage and and is is e estimatedstimated to to be be in in the the range range of −´1.24°1.24˝––+1.241.24°. ˝. TheThe cause cause of of the the orientation orientation error error is is a a relative relative rotation rotation of of both both lens lens parts parts around around an an axis axis parallel parallel toto the the optical optical axis. axis. In In [31], [31 ],it itwas was shown shown that that the the orientation orientation error error may may have have a severe a severe impact impact on the on system’sthe system’s performance. performance. The orientation The orientation error error can be can reduced be reduced drastically drastically by a byproper a proper design design of the of alignmentthe alignment structures. structures. In our In case, our case,it is considered it is considered to be to negligible be negligible compared compared to manufacturing to manufacturing and assemblyand assembly tolerances. tolerances. Operational Operational influences influences caused caused by bygravitational gravitational forc forceses and and the the imperfect deformationaldeformational kinematicskinematics of of the the linkage linkage remain remain as error as error sources. sources. Mechanical Mechanical simulations simulations of the actuator of the actuatorsystem resulted system resulted in an orientation in an orientation error range error of range´0.003 of˝–+0.012 −0.003°˝–.0.012°.

3.23.2.. Sensitivity Sensitivity Analysis Analysis of of the the Optical Optical Micromachine Micromachine ForFor investigating investigating the the influence influence of of the the individual individual tolerances tolerances on on an an evaluation evaluation criterion criterion and and to to find find thethe worst worst offenders, offenders, a a sensitivity sensitivity analysis analysis was was conducted conducted.. The The modulation modulation transfer transfer function function (MTF), (MTF), a widelya widely accepted accepted measure measure for for optical optical imaging imaging quality, quality, was was us useded as as evaluation evaluation criterion criterion [31]. [31]. Figure Figure 1111 showsshows the the MTF MTF of of the the nominal nominal (error (error-free)-free) configuration configuration for for three three different different adjustments adjustments of of refraction refraction power,power, in particular −´0.0.88 dpt dpt (far (far vision, vision, shown shown in in red), red), 1.5 1.5 dpt dpt (intermediate (intermediate vision, vision, shown shown in in blue), blue), andand 3.0 3.0 dpt dpt (near vision, shown in green). The The MTF MTF was was evaluated evaluated up up to to a a limit of 100 cycl./mm. NoticeableNoticeable is is the the decrease decrease of the the MTF MTF in in case case of of the the far far vision vision adjustment. adjustment. This This is is due due to to the the hyperopic hyperopic designdesign of of the the optics optics to to be be able able to to adapt adapt the the refraction refraction power power range range of of the the optics optics to to the the patient’s patient’s demand demand afterafter implantation. implantation.

FigureFigure 11. 11. MTFMTF of of the the nominal nominal configuration configuration and and three three different different refraction refraction power power adjustments: adjustments: −´0.80.8 dpt dpt (red), (red), 1.5 1.5 dpt dpt (blue), (blue), and and 3.0 3.0 dpt dpt (green). (green).

ForFor better better clarity, clarity, the following presentation presentation is is limited limited to to the the MTF MTF of of the the central central field field and and an an adjustmentadjustment of of the the accommodation accommodation state state of of 1.5 1.5 dpt. dpt. In In this this sensitivity sensitivity analysis analysis the the worst worst case case values values of theof theindividual individual tolerances tolerances were were chosen chosen to tocalculate calculate the the MTF. MTF. The The simulations simulations were were done done with with the the opticaloptical simulation tooltool OpticStudioOpticStudio 15.515.5 (ZEMAX (ZEMAX LLC, LLC, Kirkland, Kirkland, WA, WA USA), USA [32) [32].]. Figure Figure 12 depicts12 depicts the thesimulation simulation results results of the of MTF the for MTF the fourfor the main four position main errorsposition identified errors inidentified a worst-case in a configuration worst-case configuration (drawn in blue). The error of decentration of the total optics was calculated for the three

Micromachines 2016, 7, 85 11 of 19

Micromachines 2016, 7, 85 11 of 19 (drawn in blue). The error of decentration of the total optics was calculated for the three directions x, y, directionsand z. The x MTF, y, and of the z. nominalThe MTF (error-free) of the nominal configuration (error- isfree) the configuration reference for the is tolerancethe reference analysis for andthe toleranceis depicted analysis as a red and curve is depicted in the following as a red curve graphs. in the following graphs.

FigureFigure 12. MTFsMTFs for for the the different different types of position tolerances identified identified (blue) (blue) in in comparison comparison to to the the referencereference (red) (red).. ( (aa)) Decentration Decentration in in xx;;( (bb)) Decentration Decentration in in yy;;( (cc)) Decentration Decentration in in zz;;( (dd)) Lateral Lateral position position ererror;ror; ( (ee)) Wedge Wedge error; (f) Orientation error.error.

Evaluating the effects of the different position errors on systems performance leads to the Evaluating the effects of the different position errors on systems performance leads to the following results. following results. A comparison of the decentration of the total optics in x-, y-, and z-direction, respectively A comparison of the decentration of the total optics in x-, y-, and z-direction, respectively (Figure 12a–c) with the reference does not reveal any relevant impact of this position error on the (Figure 12a–c) with the reference does not reveal any relevant impact of this position error on the imaging quality of the optical subsystem. Hence, it may be concluded that the optical subsystem is imaging quality of the optical subsystem. Hence, it may be concluded that the optical subsystem is robust with respect to a decentration of the complete optics in the specified range. robust with respect to a decentration of the complete optics in the specified range. Consideration of the lateral position error (Figure 12d) reveals that the displacement of the two Consideration of the lateral position error (Figure 12d) reveals that the displacement of the two lens parts against each other perpendicular to the functional shifting direction has a severe impact on lens parts against each other perpendicular to the functional shifting direction has a severe impact on the imaging quality of the optical subsystem. The MTF drops steeply and meets the x-axis at the imaging quality of the optical subsystem. The MTF drops steeply and meets the x-axis at 45 cycles 45 cycles per mm. A spatial resolution above this value is not possible anymore i.e., the lateral position per mm. A spatial resolution above this value is not possible anymore i.e., the lateral position error in error in its worst case configuration limits the spatial resolution drastically from 0.42 at its worst case configuration limits the spatial resolution drastically from 0.42 at 100 cycl./mm to 0.0 at 100 cycl./mm to 0.0 at 45 cycl./mm. 45 cycl./mm. Tilting both surfaces perpendicular to the optical axes, as described by the wedge error (Figure 12e), Tilting both surfaces perpendicular to the optical axes, as described by the wedge error (Figure 12e), leads to a steep decrease of the MTF curve in the lower frequency range below 10 cycl./mm. Above leads to a steep decrease of the MTF curve in the lower frequency range below 10 cycl./mm. Above 10 cycl./mm, the slope of the decrease is only small and results in an MTF value of 0.2 at a spatial 10 cycl./mm, the slope of the decrease is only small and results in an MTF value of 0.2 at a spatial frequency of 100 cycl./mm. frequency of 100 cycl./mm. In the worst case configuration of the orientation error, the axis of rotation lies parallel to the In the worst case configuration of the orientation error, the axis of rotation lies parallel to the optical axis at the lower edge of the optics (as shown in Figure 10). The rotation leads to a lateral optical axis at the lower edge of the optics (as shown in Figure 10). The rotation leads to a lateral displacement of all points on the surface with a distance to the axis of rotation. At the upper edge of the optics, the lateral displacement amounts to 1.62 µm. No relevant impact of the orientation error on the imaging quality can be seen (Figure 12f).

Micromachines 2016, 7, 85 12 of 19 displacementMicromachines 201 of6, all7, 85 points on the surface with a distance to the axis of rotation. At the upper edge12 of of19 the optics, the lateral displacement amounts to 1.62 µm. No relevant impact of the orientation error on One result of the tolerance analysis performed is that the set-up of the demonstrator of the AAS the imaging quality can be seen (Figure 12f). is robust with respect to a decentration of the total optics. The impact of the orientation error is One result of the tolerance analysis performed is that the set-up of the demonstrator of the AAS negligible due to precautions taken in the design of the alignment structures. Comparison of the is robust with respect to a decentration of the total optics. The impact of the orientation error is position errors of the individual freeform surfaces yields the following findings: Wedge error and negligible due to precautions taken in the design of the alignment structures. Comparison of the lateral position error will have very severe impacts on the imaging quality of the demonstrator. In position errors of the individual freeform surfaces yields the following findings: Wedge error and the case of the lateral position error the resolution of spatial frequencies is limited to 45 cycl./mm. lateral position error will have very severe impacts on the imaging quality of the demonstrator. In the Lateral displacement along an axis perpendicular to the shift axis of the AH freeform optics was case of the lateral position error the resolution of spatial frequencies is limited to 45 cycl./mm. Lateral identified to be the worst offender in this sensitivity analysis. This lateral displacement is the core of displacement along an axis perpendicular to the shift axis of the AH freeform optics was identified to the lateral position error. An optimization of the design of the freeform optics in terms of robustness be the worst offender in this sensitivity analysis. This lateral displacement is the core of the lateral to reduce the impact of the lateral position error on the system’s performance was conducted and is position error. An optimization of the design of the freeform optics in terms of robustness to reduce presented in the following section. the impact of the lateral position error on the system’s performance was conducted and is presented in the following section. 4. Optimization and Results 4. OptimizationThe optimization and Results with respect to a robust design was conducted with an actively damped least squaresThe algorithm optimization provided with respect by the to op atical robust simulation design was software conducted Optic withStudio an actively15.5. The damped local search least squaresalgorithm algorithm is capable provided of optimizing by the optical a merit simulation function software composed OpticStudio of weighted 15.5. The target local values. search algorithmOptimization is capable parameters of optimizing i.e., parameters a merit function to be composedchanged ofbyweighted the algorithm target values.are the Optimization parameters parametersdescribing i.e.the, parametersshapes of the to beindividual changed byfreeform the algorithm surfaces. are Using the parameters this approach, describing the shapes the shapes of the of thefreeform individual surfaces freeform can be surfaces. altered independently Using this approach, of each the other, shapes hence of the both freeform surfaces surfaces are not can mandatorily be altered independentlyconjugated anymore. of each Only other, by hence this measure both surfaces and an are additional not mandatorily extension conjugated of the polynomial anymore. description Only by thisrelevant measure imp androvements an additional could be extension reached. of Additionally, the polynomial the description parameters relevant of the aspheric improvements front surface could beof reached.the inset Additionally,lens described the by parameters Equation of3 are the used aspheric as parameters front surface for of optimization. the inset lens The described modular by Equationconcept of 3 the are optical used as set parameters-up is not afflicted for optimization. by this. As The a criterion modular to concept control of the the optimization optical set-up the is MTF not afflictedwas calculated by this. Asat adifferent criterion sampling to control points the optimization with individual the MTF weighting was calculated factors. at different Geometrical sampling and pointsmanufacturing with individual restrictions weighting were factors.used to Geometrical formulate the and boundary manufacturing conditions restrictions to ensure were that used the to formulateoptimization the results boundary in a conditionsreasonable to design. ensure that the optimization results in a reasonable design. FigureFigure 1313 showsshows thethe MTFsMTFs ofof thethe individualindividual positionposition tolerancestolerances forfor threethree differentdifferent adjustmentsadjustments ofof refraction power: power: −0.8´0.8 dpt dpt (red), (red), 1.5 dpt 1.5 (blue), dpt (blue), and 3.0 and dpt 3.0 (green) dpt (green)using the using optimized the optimized freeform freeformsurfaces. surfaces.

FigureFigure 13.13. ContCont..

Micromachines 2016, 7, 85 13 of 19 Micromachines 2016, 7, 85 13 of 19 Micromachines 2016, 7, 85 13 of 19

FigureFigure 13. 13.MTF MTF forfor the different different types types of ofposition position tolerances tolerances identified identified and the and robust the robustdesign designfor three for threeFiguredifferent different 13. adjustmentsMTF adjustments for the differentof refraction of refraction types ofpower: position power: −0.8 tolerances´ dpt0.8 dpt(red), identified (red), 1.5 1.5dpt and dpt(blue), the (blue), robust and and 3.0design 3.0dpt dpt for (green). three (green). (a)different( Decentrationa) Decentration adjustments in inx;( x;b ( )bof Decentration) Decentrationrefraction power: in iny y;(; (cc ))− DecentrationDece0.8 dptntration (red), in in 1.5zz;;( ( dddpt)) Lateral Lateral (blue), position position and 3.0 error; error; dpt (e ) ((green). eWedge) Wedge error;(error;a) Decentration (f) ( Orientationf) Orientation in x error.; (error.b) Decentration in y; (c) Decentration in z; (d) Lateral position error; (e) Wedge error; (f) Orientation error. Figure 13 illustrates that the optimized robust design performs relatively even between the distinctFigureFigure adjustments 13 13illustrates illustrates of thatrefraction that the the optimized power. optimized The robust onlyrobust design exception design performs isperforms the relativelydecentration relatively even in even betweeny, where between thethe distinctMTF the adjustmentsdistinctfor the intermediateadjustments of refraction visionof refraction power. experiences The power. only a strongThe exception only degradation exception is the decentration iscompared the decentration to inthey two, where in othery, where the adjustments. MTF the MTF for the intermediateforThe the even intermediate performance vision experiences vision is owedexperiences a strongto the degradationafact strong that degradation the comparedsurfaces compared of to both the twoAH to the otherlens two parts adjustments. other were adjustments. optimized The even performanceTheindependently even performance is owed of each to the other.is owed fact that to the fact surfaces that ofthe both surfaces AH lensof both parts AH were lens optimized parts were independently optimized ofindependently eachAgain, other. for of better each clarity,other. Figure 14 only shows the MTFs in the case of the intermediate vision (1.5Again, dpt).Again, Figure for for better better 14 depicts clarity, clarity, the Figure Figure individual 1414 onlyonly position showsshows tolerances the MTFs MTFs using in in the the the case case optimized of of the the intermediate surface intermediate geometries vision vision (1.5(1.5(green) dpt). dpt). Figurein Figure comparison 14 14 depicts depicts to the the the original individual individual design positionposition as shown tolerances in Figure using using 12 (blue). the the optimized optimized The reference surface surface for geometries evaluation geometries (green)(green)of the in robust in comparison comparison design tois to the the nominal original original (error design design-free) asas shown MTF, whichin in Figure Figure is depicted 12 12 (blue). (blue). in The Thethe reference graphs reference as for a for evaluationred evaluation curve. ofof the the robust robust design design is is the the nominal nominal (error-free) (error-free) MTF,MTF, whichwhich is depicted in in the the graphs graphs as as a ared red curve. curve.

Figure 14. MTF for the different types of position tolerances identified and the robust design (green) FigureFigurein comparison 14. 14.MTF MTF for with for the the that different different of the original types types of of design position position as tolerancesshown tolerances in Figure identifiedidentified 12 (blue), andand the theand robustrobust the MTF design of the (green) error- in comparisoninfree comparison configuration with with that (red). that of the of (a originalthe) Decentration original design design in as x shown;as (b shown) Decentra in Figurein Figuretion 12 in 12(blue), y ;(blue), (c) Decentration and and the the MTF MTF in of zof; the ( thed) error-freeLateral error- configurationfreeposition configuration error; (red). (e) Wedge(red). (a) Decentration (a error;) Decentration (f) Orientation in xin;( xb; )( berror. Decentration) Decentra tion in iny y;(; (cc)) Decentration in in z;z (;(d)d Lateral) Lateral positionposition error; error; (e ()e Wedge) Wedge error; error; ( f()f) Orientation Orientation error.error.

Micromachines 2016, 7, 85 14 of 19

It can be stated in general that the severe impacts of the position errors lateral position error (Figure 14d) and wedge error (Figure 14e) are fixed by the robust design. This improvement is at the expense of a performance degradation in the case of the decentration error in y (Figure 14b). In the following, a comparison of the robust design (green) with the original design (blue) is conducted for the individual position errors. Concerning the decentration of the total optics in x-, y-, and z-direction, respectively (Figure 14a–c), the reference MTF coincides with the MTF resulting from the original design (see also Figure 12 a–c). A slight improvement of the MTF resulting from the robust design can be stated in the frequency range 10 cycl./mm to up to 70 cycl./mm for the decentration errors in x and z in comparison with the reference. The MTF curves intersect at 70 cycl./mm. For frequencies above 70 cycl./mm, the robust design achieves marginally lower values compared with the original design (and the reference). In the case of a decentration in y, the robust design is more sensitive, the course of its MTF curve is smooth and without steep drops but runs below that of the original design over the complete frequency range shown. The lateral position error (Figure 14d) had the most severe impact on the optical performance for the original design. The improvement of the robust design approach is most obvious in this case. The collapse of the MTF at 45 cycl./mm is fixed and the resulting MTF curve is smooth in the higher frequencies. Nevertheless, the MTF curve resulting from the robust design runs below the reference curve beyond 40 cycl./mm. Concerning the wedge error (Figure 14e), the steep decrease in the low-frequency range is obliterated by the robust design. In the higher frequency range, both curves approach each other. Again, the MTF resulting from the robust design runs below the reference curve for frequency values beyond 10 cycl./mm. In the case of the orientation error (Figure 14f), the reference MTF coincides with the MTF resulting from the original design (see also Figure 12f). The course of the curves resembles that of the decentration errors in x and z: The performance of the robustly designed optics is comparable to that of the original design (and the error-free design), with slightly higher values below and marginally lower values above 70 cycl./mm. As expected, the optimization with respect to a robust design results in two different descriptions of the freeform surfaces violating the conjugation of the originally designed AH optics. Furthermore, parameters of higher order of the polynomial equation describing the freeform surfaces received significant values. The mathematical description of the sag of the optimized freeform surfaces as well as the parameters of the individual surfaces are given in Appendix A. The robust freeform surfaces are described by a seventh order polynomial equation using 35 parameters instead of the fourth order form described by Equation 2. The parameters of the optimized aspheric front surface of the inset lens are also depicted in Appendix A. The differences to the original design as given in Table1 seem to be marginal but their impact on the performance is nevertheless significant. Another point to be considered in sensitivity analysis is the deviation in position of the image due to prismatic effects caused by tilting the surfaces. It must be ensured that the image will not be formed outside the fovea centralis with a diameter of approximately 1.5 mm. The image position was examined by tracing the chief ray through the optical system. Figure 15 shows the position of the chief ray in the image plane in dependence of the individual position tolerances of the freeform optics. Calculations of the chief ray positions were performed for three different adjustments of the refraction power: Far vision (´0.8 dpt) depicted by red symbols, intermediate distance (1.5 dpt) shown in blue, and near vision (3 dpt) labeled by green symbols. Micromachines 2016, 7, 85 15 of 19 Micromachines 2016, 7, 85 15 of 19

Figure 15.15. ChiefChief ray ray positions positions for for the the individual individual position position errors errors of the of freeform the freeform optics foroptics three for different three differentadjustments adjustments of refraction of refraction power: ´ power:0.8 dpt − (red),0.8 dpt 1.5 (red), dpt (blue), 1.5 dpt and (blu 3.0e), dpt and (green). 3.0 dpt (green).

Figure 15 shows that the prismatic effect of a tilt of the AH optics has no relevant impact on the Figure 15 shows that the prismatic effect of a tilt of the AH optics has no relevant impact on the image position on the fovea centralis. The major effect on the image position is caused by the wedge image position on the fovea centralis. The major effect on the image position is caused by the wedge error with an adjustment of the refraction power of 3.0 dpt. In this case, the chief ray suffers a error with an adjustment of the refraction power of 3.0 dpt. In this case, the chief ray suffers a deviation deviation in position of only 4 µm in x- and 7.1 µm in y-direction which ensures an image forming in position of only 4 µm in x- and 7.1 µm in y-direction which ensures an image forming well inside well inside the fovea centralis. The weak influence of the position errors on the image position is the fovea centralis. The weak influence of the position errors on the image position is considered to be considered to be due to the small thickness of the AH-parts of only 500 µm. due to the small thickness of the AH-parts of only 500 µm.

5. Discussion Discussion It is crucial in system integration to ensure the specifiedspecified performance of the overall subsystem composed of individual components. This This is is due due to to the fact that each individual comp componentonent has its own deviation from the nominal values values and and that that the the assembly assembly process process itself itself is is subject subject to to tolerances. tolerances. Furthermore the operation of the system can affect the systems’systems’ performance.performance. Applying monolithic design approaches is a firstfirst step towardstowards reducingreducing assemblyassembly tolerances.tolerances. To To make make sure that the manufactured subsystem fulfills fulfills its specification specification under operational conditions, a tolerance analysis can be carried out by means of a simulation tool. If the tolerance-afflictedtolerance-afflicted system fails the system’s specification,specification, a robust design approach with respect to an optimization optimization of the the functional functional surfaces surfaces may may lead toto a a design design fulfilling fulfilling the the specification specification even even in the in presence the presence of fabrication of fabrication and operational and operational tolerances. tolerances.The robust designThe robust approach design presented approach in presented this article in consists this article of three consists stages. of three The first stages. step The is a tolerancefirst step isanalysis: a tolerance Therelevant analysis: tolerances The relevant andtheir tolerances effects onand the their positioning effects on of thethe opticalpositioning components of the haveoptical to componentsbe determined. have A subsequentto be determined. sensitivity A subsequent analysis gives sensitivity information analysis about gives the impact information of the individualabout the impacttolerances of onthe theindividual system’s tolerances performance. on Thethe nextsystem’s step consistsperformance. of an optimizationThe next step with consists respect of toan a optimizationrobust design. with The respect choice to of a the robust optimization design. The parameters choice of isthe of optimization crucial importance parameters to the is successof crucial of importancethe approach. to the The success optimization of the parametersapproach. The should optimization describe theparamete shapers of should the optics describe to be the modified. shape ofThe the optimization optics to be criteriamodified. must The represent optimization the system’scriteria must specification, represent andthe system’s restrictions specification, can be used and to restrictionsensure the manufacturability can be used to ensure of the the optimized manufacturability design. The of approach the optimized is finalized design. by a The validation approach of the is finalizedresults by by a comparativea validation of tolerance the results analysis. by a comparative tolerance analysis. This conceptconcept was was applied applied to theto opticalthe optical micromachine micromachine of a demonstration of a demonstration model of anmodel ophthalmic of an ophthalmicsystem. Design system. optimization Design optimization was conducted was using conducted the shapes using of thethe varifocal shapes of freeform the varifocal surfaces freeform as well surfacesas the shape as well of the as aspherical the shape inset of lensthe ofaspherical the rear housing inset lens as optimization of the rear parameters.housing as Thisoptimization approach parameters.ensures the This modular approach concept ensures of the the housing modular where concept the of front the housing and rear where half-shells the front can and be rear used half for- shells can be used for different active optical principles. Since modification of the freeform surfaces of both AH lens parts was allowed independently in optimization, the robust design results in

Micromachines 2016, 7, 85 16 of 19 different active optical principles. Since modification of the freeform surfaces of both AH lens parts was allowed independently in optimization, the robust design results in surfaces which are no longer conjugated. As a consequence for manufacturing, this means that two individual molding tools have to be fabricated instead of only one which can be used for both lens parts in the case of the originally designed optics. This significantly increases tooling and potential process costs. The nominal configuration was used as a reference for the evaluation of the robust design. However, it is in the nature of a tolerance optimization that the reference values cannot be reached for all tolerances under consideration. Therefore, a second criterion for evaluating the performance of the robust design plays an important role namely for avoiding the collapse of the performance. The robust design approach leads to important improvements in cases where a collapse of the imaging quality was observed already at very low spatial frequencies i.e., for a mutual position error of the freeform lens parts perpendicular to the shifting direction (lateral position error) and a mutual inclination of the lens parts, the wedge error. In both cases, the robust design results in MTF curves which are smooth over the entire range under consideration. This improvement comes with a loss in imaging quality in the case of a total decentration of the optics perpendicular to the shifting direction of the lens parts (decentration of the total optics in y). Nevertheless, the MTF curve is also smooth in its course and suffers no abrupt drop in the frequency range considered. Lateral position error and decentration of the total optics in y are not independent of each other: optimization of the one error with respect to a robust design leads to a design more sensitive to the other error. Both errors have to be balanced to find a design offering the best minimum performance in a worst-case configuration. Since a poor imaging quality in the lower frequency range is not acceptable in the case of the AAS, the elimination of the collapse in MTF caused by the lateral position error and the wedge error as well as a smooth course of the MTF curve in the frequency range is essential. The robust design approach presented ensures a certain performance in the presence of assembly and manufacturing tolerances and helps to avoid disproportional efforts in assembly and manufacturing. To investigate the influence of the individual tolerances on the image position, tracing of the chief ray through the optical system was performed. The examination of the chief ray positions as a function of the individual tolerances showed only negligible effects and proved that image forming is well inside the fovea centralis.

6. Conclusions System integration of complex microoptical systems is a challenging task the result of which is severely affected by yields on individual process steps or subsystems. In order to achieve a high yield on the final system, the yields from subsystem level down to individual process steps have to be maximized. At the same time, costs have to be kept in check, meaning that expensive measurement and active alignment procedures have to be reduced as much as possible. In this article a monolithic integration approach combined with design optimization for robustness were illustrated on the example of an ophthalmic implant. Fabrication and assembly tolerances of the optical micromachine were derived. The impact of the tolerances on the micromachine’s optical performance was analyzed by means of simulation. The simulation results indicated a severe impact on system’s performance due to the tolerances. In a subsequent step the freeform surfaces of the micromachine were optimized with respect to robustness to tolerances. The robust design ensures a satisfactory optical performance even in presence of fabrication and assembly tolerances. The article illustrates that the proposed two-pronged design approach leads to an optical subsystem that fulfills the specification while still achieving high yield with a reasonable effort.

Acknowledgments: The work presented in this paper was conducted under the KD OptiMi2 programme (grant no. 16SV5472K), subtopic B “Fabrication of housing shells & measurement data” funded by the German Federal Ministry of Education and Research. The authors would also like to acknowledge Allen Yi and Likai Li from the Ohio State University for their support in manufacturing of the freeform optics. Furthermore, we are grateful for the support by Deutsche Forschungsgemeinschaft and Open Access Publishing Fund of Karlsruhe Institute of Technology. Micromachines 2016, 7, 85 17 of 19

Author Contributions: Ingo Sieber conceived and designed the robust design approach and performed the optimization and analysis. Thomas Martin conceived and designed the actuation principle and the amplification linkage. Furthermore he determined the relevant tolerances and derived the position errors. Ulrich Gengenbach conceived the underlying integration concept of the Artificial Accommodation System. Moreover, these persons contributed the respective text passages. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A The surface sag of the optimized freeform surfaces is given by Equation (A1). The parameters of the individual surfaces are given in Tables A1 and A2. Numbering of the freeform surfaces is done from left to right following the beam path as shown in Figure9.

7 6 5 2 4 3 3 4 2 5 6 7 6 zpx,yq= a1x + a2x y + a3x y + a4x y + a5x y + a6x y + a7xy + a8y + a9x 5 4 2 3 3 2 4 5 6 5 4 1 3 2 + a10x y + a11x y +a12x y + a13x y + a14xy + a15y + a16x + a17x y + a18x y 2 3 4 5 4 3 2 2 3 4 3 (A1) + a19x y + a20xy +a21y + a22x + a23x y + a24x y + a25xy + a26y + a27x 2 2 3 2 2 + a28x y + a29xy + a30y + a31x +a32xy + a33y + a34x + a35y

Table A1. Parameters of freeform surface 1.

a1 a2 a3 a4 a5 a6 a7 a8 a9 4.71 ˆ 10´6 ´1.46 ˆ 10´5 ´5.47 ˆ 10´5 ´7.50 ˆ 10´6 1.50 ˆ 10´5 1.99 ˆ 10´6 ´8.93 ˆ 10´6 ´2.63 ˆ 10´6 3.16 ˆ 10´6

a10 a11 a12 a13 a14 a15 a16 a17 a18 2.78 ˆ 10´5 8.46 ˆ 10´6 ´2.36 ˆ 10´5 6.08 ˆ 10´5 1.93 ˆ 10´5 1.86 ˆ 10´5 1.18 ˆ 10´4 ´7.58 ˆ 10´5 2.03 ˆ 10´4

a19 a20 a21 a22 a23 a24 a25 a26 a27 ´1.70 ˆ 10´4 ´1.05 ˆ 10´5 ´5.04 ˆ 10´6 ´8.57 ˆ 10´5 ´2.55 ˆ 10´4 3.68 ˆ 10´5 7.20 ˆ 10´5 ´2.26 ˆ 10´4 ´3.27 ˆ 10´3

a28 a29 a30 a31 a32 a33 a34 a35 1.95 ˆ 10´2 5.32 ˆ 10´4 6.67 ˆ 10´3 1.16 ˆ 10´3 8.53 ˆ 10´4 2.51 ˆ 10´3 5.67 ˆ 10´5 6.28 ˆ 10´4

Table A2. Parameters of freeform surface 2.

a1 a2 a3 a4 a5 a6 a7 a8 a9 ´3.06 ˆ 10´6 ´1.43 ˆ 10´5 ´6.84 ˆ 10´5 1.21 ˆ 10´5 ´2.40 ˆ 10´6 2.39 ˆ 10´5 ´8.91 ˆ 10´6 1.01 ˆ 10´6 ´5.13 ˆ 10´5

a10 a11 a12 a13 a14 a15 a16 a17 a18 1.65 ˆ 10´5 ´1.24 ˆ 10´4 ´2.94 ˆ 10´5 ´8.11 ˆ 10´5 5.29 ˆ 10´6 ´2.98 ˆ 10´5 1.83 ˆ 10´4 ´1.21 ˆ 10´4 2.92 ˆ 10´4

a19 a20 a21 a22 a23 a24 a25 a26 a27 ´2.58 ˆ 10´4 2.53 ˆ 10´5 ´3.19 ˆ 10´5 2.93 ˆ 10´4 ´1.05 ˆ 10´4 5.03 ˆ 10´4 5.98 ˆ 10´5 1.75 ˆ 10´4 ´3.46 ˆ 10´3

a28 a29 a30 a31 a32 a33 a34 a35 1.98 ˆ 10´2 4.23 ˆ 10´4 6.74 ˆ 10´3 ´7.23 ˆ 10´4 9.25 ˆ 10´4 6.48 ˆ 10´4 4.16 ˆ 10´4 6.15 ˆ 10´4

The parameters of the optimized aspheric front surface of the inset lens are also depicted in Table A3.

Table A3. Parameters of the optimized aspheric front surface.

c k β1 β2 β3 β4 β5 β6 β7 β8 0.742 ´1.690 4.47 ˆ 10´4 ´0.328 ´0.025 0.079 ´0.039 0.010 ´1.47 ˆ 10´3 7.49 ˆ 10´5

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