Design of an electrostatic balance mechanism to measure optical power of 100 kW
11 Design of an electrostatic balance mechanism tomeasure optical power of 100 kW
Lorenz Keck ∗ , Gordon Shaw ∗ , René Theska § , Stephan Schlamminger ∗ ∗ Physical MeasurementLaboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899 USA § Precision Engineering Group, Technische Universität Ilmenau, 98693 Ilmenau, Germany
Abstract —A new instrument is required to ac-commodate the need for increased portability andaccuracy in laser power measurement above
100 W .Reflection and absorption of laser light providea measurable force from photon momentum ex-change that is directly proportional to laser power,which can be measured with an electrostatic bal-ance traceable to the SI. We aim for a relativeuncertainty of − with coverage factor k = 2 .For this purpose, we have designed a monolithicparallelogram 4-bar linkage incorporating elas-tic circular notch flexure hinges. The design isoptimized to address the main factors drivingforce measurement uncertainty from the balancemechanism: corner loading errors, balance stiff-ness, stress in the flexure hinges, sensitivity tovibration, and sensitivity to thermal gradients.Parasitic rotations in the free end of the 4-barlinkage during arcuate motion are constrained bymachining tolerances. An analytical model showsthis affects the force measurement less than 0.01percent. Incorporating an inverted pendulum re-duces the stiffness of the system without undulyincreasing tilt sensitivity. Finite element modelingof the flexures is used to determine the hingeorientation that minimizes stress which is there-fore expected to minimize hysteresis. Thermaleffects are mitigated using an external enclosure tominimize temperature gradients, although a quan-titative analysis of this effect is not carried out.These analyses show the optimized mechanism isexpected to contribute less than × − relativeuncertainty in the final laser power measurement. Index Terms —Electrostatic force balance, laserpower, balance, flexure mechanism.
Manuscript received December 1, 2020; Cor-responding author: S. Schlamminger (email:[email protected])
I. Introduction
Primary measurements of laser power rely on ei-ther the effect of absorbed laser power or the forcetransmitted in reflection, see [17] for a recent review.Instruments that use absorption suffer three criticaldisadvantages. First and foremost, the laser beam isno longer available after it has been absorbed. Thisnecessitates substitution or beam splitting processesfor calibration of secondary detectors and severelylimits in-situ use for industrial applications such as,e. g., laser welding. Second, every absorber scattersand reflects some light. Hence, it is difficult to capturethe entirety of the incident light at relative uncer-tainties smaller than 1 × − . This is expected tobe especially critical for laser powers above 1 kW.Third, absorption of multiple kilowatts of laser powergenerates a large amount of heat. Although flowingwater calorimeter systems capable of handling thesethermal loads have been developed [17] they arebulky and difficult to operate.In contrast to that, custom dielectric coating stacksthat have total optical loss lower than 1 × − arecommercially available. Thus it is, in principle, pos-sible to build a system that can measure the powerof a multi-kilowatt laser at relative uncertainty of1 × − or better with k = 2, using the photonpressure force from reflection of laser light.The optical characteristics of the mirror can be de-scribed by the specular reflectance R , the absorbance A , and the transmittance T . The effect of diffuse re-flectance is not considered in this work. Their sum isunity, i. e., R + A + T = 1. Using these coefficients [11],the photon pressure force, throughout the text alsoreferred to as the external force, on the mirror is givenby F ext = P cos αc (2 R + A ) . (1) a r X i v : . [ phy s i c s . i n s - d e t ] F e b Here, P denotes the power of the laser beam and α the angle of incidence relative to surface normal.A 100 kW light beam ( α = 0) normally incident ona perfect mirror ( R = 1) produces a force of 667 µN.According to the specifications above, the total allow-able force uncertainty is 667 nN at k = 2. Since we donot have an 100 kW laser at our disposal, a beam mul-tiplier, the High Amplification Laser-pressure Optic(HALO) has been constructed [1]. The HALO uses a10 kW laser and 14 reflections to produce a normalforce on the order of 667 µN. Other work describesmulti-reflection measurements at lower power [10],[16]. In this article, we describe the design of themechanical components of an electrostatic balance formeasuring laser power of up to 100 kW. We designedthe electrostatic balance to be compatible with theHALO, but it can also be used to measure a singlelaser beam application. This manuscript reuses somecontent from thesis [6] with permission. II. Theory of the electrostatic balance
Electrostatic force balances have been used suc-cessfully in mass metrology [10], [12], and, morerecently, to measure the force exerted by light forpower levels up to 3 W [11].In force mode, an external force is compensatedby the electrostatic force between two capacitor elec-trodes. A feedback system adjusts the voltage appliedto the capacitor to hold a movable electrode at anominal position based on input from a displace-ment measuring device, usually an interferometer.The electrostatic force generated with the capacitiveactuator depends on two quantities; the square of thepotential difference, V , and the capacitance gradientd C/ d x , and is described with F el = −
12 d C d x V . (2)The capacitance gradient, d C/ d x is not known apriori. It is obtained by measuring capacitance atfixed electrode positions, fitting a polynomial to themeasured data and then calculating the derivative ofthe polynomial about the nominal operating position.Excursions of a few tenths of a millimeter are requiredto measure the capacitance gradient with sufficientaccuracy [13]. Gradients on the order of 1 pF mm − are achieved with concentric cylindrical capacitors asin electrostatic force balances at NIST [12], [13].Measurements performed with the electrostaticbalance are directly traceable to the SI, as revised in 2019 [14]. The measurement may therefore beconsidered a primary reference for force, as no otherreference, i. e., a force traceable to mass in a gravita-tional field, is required for calibration. More detailsto the traceability path of measured force to h areoutlined in [12] and [17]. III. Conceptual design of a new balancemechanism
The design objective is to achieve linear translationof the payload (mirror plus electrode) over ± .
25 mmwith minimal parasitic rotation. Furthermore, themechanism should be as simple as possible for easeof use, manufacturing, and uncertainty analysis. Aplanar parallelogram linkage, see Fig. 1, is a suitablesolution for this application.The mechanism has four pivots. The two backpivots connect two rotating bars (referred to asswings) to the frame of the balance and the two frontpivots connect the coupler to the rotation bars. Twodimensions must be chosen: the vertical separation ofthe pivots, i. e., the length of the coupler, a , and thelength of the swings, b .Fig. 1: Drawing of the mechanism with attachedinner capacitor electrode and mirror. The couplerwith length a is shown on the left. The swings, withlength b determine the distance between the fixedback pivots and the movable front pivots. ˜ L indicatesan offset of the center axis of the mirror to the centeraxis of the capacitor. A. Sizing the linkage
As shown in Fig. 1 the center of the mirror is hori-zontally offset from the center axis of the capacitor toprevent heating of sensitive components by transmit-ted light [11]. The variable ˜ L denotes the horizontaloffset between the application points of the externalforce due to the laser and the electrostatic force. Thislever arm increases the measurement sensitivity tocoupler rotations. In an ideal parallelogram linkage,where the four pivots are at the corner of a perfect parallelogram, the coupler will not rotate. In reality,a perfect parallelogram is impossible to achieve dueto machining tolerances, ∆.To examine the effect of machining tolerances onthe rotation angle of the coupler, φ z , a worst caseis assumed. The horizontal distance at the top is2∆ longer than the one at the bottom. Further,the coupler length is 2∆ longer than the verticalseparation of the back pivots, see Fig. 2.Fig. 2: The geometric relations for an assumed im-perfect parallelogram linkage with b - length of theswings, a - length of the coupler, ∆ - manufacturingtolerances. A worst case scenario is assumed: theopposite linkages differ by 2∆. The dashed line z isone diagonal. Here, ψ a , ψ b are calculated as functionof γ . The parasitic rotation of the coupler is φ z .A and B display fixed back pivots while A and Bare moveable hinges. The red lines show the linkageat the nominal zero position and the black linesexaggerate a deflected state of the linkage.The squared length of the diagonal shown in Fig. 2is given by z = ( a + ∆) + ( b + ∆) − a + ∆)( b + ∆) sin ( γ ) , (3)where the rotation angle of the top swing sin ( γ ) = x/ ( b + ∆) ≈ γ . With the length of the diagonal,the angles around point B can be obtained with thecosine rule. They are ψ a = arccos (cid:18) ( a + ∆) + z − ( b + ∆) z ( a + ∆) (cid:19) , (4) and ψ b = arccos (cid:18) ( b − ∆) + z − ( a − ∆) z ( b − ∆) (cid:19) . (5)The rotation angle of the coupler is φ z ( x ) =arcsin (cid:18) ( b + ∆) cos ( γ ) − ( b − ∆) sin ( ψ a + ψ b )( a − ∆) (cid:19) . (6)To minimize coupler rotation the first derivative,d φ z / d x , should be zero at the nominal zero position( γ = 0). This is the case for the perfect geometry,∆ = 0.
60 80 100 120 140 . × ( a + b ) / mm − − . × ( a − b ) / mm . . . . φ z , x × m . . φ z , x × m φ z , x × m Fig. 3: The figure of merit, φ z , x = d φ z / d x , for theoptimization of the parallelogram sides a and b . Forthis figure, ∆ = 5 µm is used.Hence, d φ z / d x is a good figure of merit to investi-gate the coupler rotation. Fig. 3 shows the derivativeas a function of the average ( a + b ) / a − b ) / a = b . Hence, a = b is a good choice for the design.Since the coupler rotation decreases monotonicallyas a + b increases , the requirement for a compactinstrument dictates the choice of 100 mm for a and b . For the chosen geometry a coupler rotation ofd φ z / d x = 1 µrad mm − is obtained. The distance tothe apparent center of rotation is the reciprocal ofthis value, L = (d φ z / d x ) − = 1 km. L constrains the systematic and statistical un-certainty caused by the horizontal difference in theapplication points of the external force and the com-pensating force, ˜ L in Fig. 1. With respect to therotation point, the electrostatic force F el producesa torque of L · F el , while the external force F ext produces a torque ( L ± ˜ L ) · F ext . The sign is positiveif the rotation point is to the right of the coupler. Inequilibrium, both torques must be equal and, hence,the relative force difference is F el /F ext − ± ˜ L/L .This expression describes the corner loading error.With typical values of ˜ L = 90 mm ± L =1 km, the relative corner loading error is 9 × − ± × − which shows that this error contribution isorders of magnitude below the required 1 × − . B. Analytical description
In this section, the balance is analyzed using theLagrange equations of the second kind. This will yieldthe stiffness, the eigenfrequency, and the conditionfor the equilibrium position of the mechanism. Thefunctional components (masses, springs, and pivots)are shown in Fig. 4. All connecting bars are assumedto be perfectly rigid and all damping is neglected. Asingle hinge has a torsion stiffness of κ s , as indicatedby the subscript s for single. The x axis of the coor-dinate system is aligned with gravity, the metrologyframe is inclined by φ from the x axis, and therotating links are deflected by γ from the metrologyframe. The two masses m h are offset by h and h along the negative x direction from the back pivotsin a non-deflected system, i. e., for γ = φ = 0. Thesetwo masses m h and the compensation spring labelled k b can be used to adjust the mechanism stiffness [2],[9]. Besides the mechanical stiffness k b the zero length λ and the extended length λ are the importantphysical parameters for the spring. The masses m p1 and m p2 are counterweights and compensate themasses m a (coupler), m M (mirror), and m E (capac-itor electrode). Here, a E / b E and a M / b M denote thevertical/horizontal distances from the center of thecoupler to the electrode and the mirror, respectively.The symbols a and b without indices abbreviate thelengths of the parallelogram, similar to Fig. 2. Thesymbol e captures the length of the extension of the upper or lower swing to the right of the back pivots,to the counterweights.Fig. 4: Rigid body model of the mechanism with itsattachments.The differential equation for γ can be obtainedfrom the Lagrange equation assuming small angles γ and φ . It is derived in appendix A and the resultcan be written in the following form, J ¨ γ + κγ = − J φ ¨ φ + κ φ φ − N eq , (7)where the coefficients are given by J = b m E + b m M + b m a + b m b bem b e m b
6+ 2 e m p + h m h + h m h , (8) J φ = J + ah m h − ah m h bb e m E + bb m m M (9) κ = 4 κ s − gm h ( h + h ) − l bk b (cid:18) − λ b + l (cid:19) , (10) κ φ = gm h ( h + h ) , and (11) N eq = g ( − bm E − bm M − bm a + 2 em p ) . (12)The imaginary eigenvalues of the homogeneouspart of the differential equation provide the eigen-frequency ω due to deflections γ . It is ω = κJ . (13) The torque N eq determines the equilibrium positionaccording to γ = N eq /κ . The nominal zero position( γ = 0 for φ = 0) can be obtained according tothe equilibrium condition in Eq. 12 by adjusting thecounterweights such that 2 em p = b ( m E + m M + m a ).Then, the load is distributed equally to the upper andlower pivots. The masses on both sides of the xz planegenerate equal and opposite torques, minimizing theeffects of external vertical acceleration. By choosing b = e , the equilibrium position of the balance remainslargely unchanged with temperature change, sincethermal expansion in both lengths b and e wouldcause the lever arms to expand symmetrically, andthus the equilibrium condition stays stable. Never-theless, a temperature gradient within the materialcould lead to asymmetric thermal expansion, butsince the chosen Aluminum alloy has a high thermalconductivity, this effect is considered negligible.[9].The torsional stiffness κ given in Eq. 10 can beconverted to a linear stiffness K of the coupler movingin x with [9]: K = κb . (14)It can be seen that the stiffness decreases with 1 /b .Hence, to obtain the necessary small linear stiffness, b should be as large as possible, but, as mentionedabove, the installation space limits b to a maximumof 100 mm. C. Monolithic design
A compliant mechanism is a key part for the real-ization of precision balance instrumentation. Flexurehinges need no lubrication, are stick-slip free, shownegligible hysteresis and provide highly reproduciblemotion. [5], [8], [18]A monolithic design also has several advantages.Fabricating the functional parts of the mechanismin one setup maintains small machining tolerancesyielding two major benefits: (1) The rotation axes ofthe four pivots are parallel, and (2) the machiningtolerances ∆ are small. Furthermore, no assemblyis required, saving time and eliminating a source ofpotential variation from the model. Hence, nearlyidentical copies can be made. The lack of fastenersalso reduces excess mass.Many different possible contours for hinges exist. Adetailed overview can be found in [7]. They differ inthree functional properties, (1) the torsional stiffness,(2) the stability of the rotation axis under deflection, and (3) the maximum admissible deflection. A smallrotational stiffness and a good stability of the axis ofrotation are both important in the current design.The linear stiffness is given by a combination ofEq. 10 and Eq. 14. Here, a stiffness of 0 . − isdesired. For the chosen b = 100 mm and no stiffnesscompensation, the rotational stiffness of one pivotshould be κ s = 2 . × − N · m. Generally, a circulargeometry has moderate bending stiffness, is easy tomanufacture, and the precision of rotation is highcompared to other geometries [3]. The latter isbecause the compliant part of the hinge performingthe rotation is concentrated in the very center of thehinge geometry. For simplicity, the following circularhinge contour was chosen for each flexure hinge:radius 2 . .
05 mm andwidth of 10 mm. These dimensions can be obtainedwith high speed milling or wire electrical dischargemachining.The rotational stiffness of this hinge design is κ s =0 .
018 N · m, as calculated with non-linear equationsof large deflections assuming a pure moment loadinga hinge [4]. This is about two orders of magnitudelarger than desired, but the final stiffness of themechanism can be adjusted with the masses m h . Thefinal design keeps the inverted pendulum to reducestiffness, but omits the spring to reduce tempera-ture sensitivity which would have affected the elasticmodulus , and, hence, the equilibrium position of thebalance. The maximum admissible angle of deflectionof the chosen hinge under a pure moment load is ≈ ±
87 mrad and provides more than an order ofmagnitude more than the required ± . ± .
25 mm with the chosen linkagedimensions. Aluminum 7075-T6 was chosen as thematerial for the monolithic mechanism due to itshigh yield strength (503 MPa), low elastic modulus(72 GPa), and good machineability.
IV. Detailed design by use of finite elementanalysis
With the design analysis above as a starting point,the mechanism can be refined to its final form.This section describes an optimization for robustness,functionality, portability, and machinability using fi-nite element methods. (a) all hingesoriented in y : (b) oriented in x and y : (c) oriented intension:110 . . . E = 72 GPa and a Poisson’s ratio of ν = 0 .
33 were used. The maximum stress arises inall configurations in the center of each back hinge.
A. Optimization of hinge orientation
Three potential hinge orientations in the mecha-nism are investigated, see Fig. 5. In (a) all hinges areoriented along y , in (b) the two back hinges pointalong x and the front hinges along y . Finally, in (c)the hinges are oriented along the force vectors suchthat all hinges are in tension in the nominal zero posi-tion. For a mathematical derivation of the orientationof the hinge force vectors, see section 5 in [6]. Forall three cases, a simplified design is studied usingfinite element simulation in ANSYS Workbench .Two analyses are performed to determine the stressin the hinges and stiffness of the mechanism.For each geometry a finite element model is calcu-lated in each analysis. Both simulations use quadraticelements and a nonlinear solver.For the stiffness analysis masses and the grav-itational vector are excluded from the simulation.Hence, the only forces in the stiffness simulation aregenerated by the hinges.In order to obtain the stiffness of the mechanism K , the coupler in the model is displaced vertically by0 . Certain commercial equipment, instruments, and materialsare identified in this paper in order to specify the experimentalprocedure adequately. Such identification is not intended to im-ply recommendation or endorsement by the National Instituteof Standards and Technology, nor is it intended to imply thatthe materials or equipment identified are necessarily the bestavailable for the purpose. tic part of the mechanism stiffness is K ≈ . − .This result is remarkable. The admixture of appliedtransversal force and torque on a given hinge underdeflection changes with orientation and so does itsstiffness leading one to expect a larger variationin stiffness. In this design, the effect is negligible.Similarly, adding the gravitational load to the hingesdoes not change their torsional stiffness significantly.For the hinge stress analysis, see Fig. 6, thegravitational vector and masses are included in thesimulation. All masses are modelled as points andthe locations of the centers of mass of the swingscoincide with the back pivots as assumed in the rigidbody model. The mechanism is considered with anequilibrium at the nominal zero position ( N eq = 0)without applying further external displacements. Theresult of the calculation is the maximum equivalentstress that occurs in the hinges of the mechanism.Fig. 6: Mass placement used for the hinge stressanalysis via the finite element method. Variation (b)in Fig. 5 serves here as representation. The values forthe masses and lengths are derived from preliminarydesigns and investigations of the functional compo-nents included in the design of the balance and aregiven in Table. I.In the hinge stress analysis, the differences inthe calculated outcome is tremendous. For the twoextreme cases it differs by more than an order ofmagnitude, 110 . . loaded along the force vector, and the loading is intension rather than compression. The latter can leadto buckling in these ultra thin notch flexures. B. Complete three dimensional model
With the design choices described above a 3D CADmodel is generated, see Fig. 7. The whole mechanismis built from an aluminum block measuring 241 mm,146 mm, and 40 mm in length, height, and width.The moving part of the mechanism consists of twoplanar structures each 5 mm thick and spaced 30 mmapart. They are connected at four locations withconnectors and move as one. The defining features inboth planes are machined in a single fixtured positionand are therefore nominally identical. The front andback plates protect the moving parts of the mech-anism. With the chosen approach, the attachmentsand counterweights can be mounted at the plane ofsymmetry between the two mounting plates. Thisconfiguration eliminates parasitic rotations about they axis while preserving monolithic machinability.The moving parts, the two swings and the couplerare separated from the plates by a 4 mm wide channelthat is milled through both plates simultaneously.The channel runs not completely thru but is in-terrupted by the four hinges and sixteen sacrificialbridges. After milling, an electrical wire dischargemachine is used to precisely contour the hinges.The bridges block the motion of the mechanism andprovide sufficient stability for all machining steps,and are carefully removed at the end of the machin-ing process. The center of the mechanism is solidexcept for a few through holes, providing thermaland mechanical stability. All four connectors havetapped holes. To the front connectors the mirrorand capacitor can be mounted. Trim masses can beattached to the two back connectors, see section V.Two additional plates not shown in the drawing canbe bolted on to the front and back plane. Six trans-portation safety pins through these plates immobilizethe moving part of the mechanism during transport,see Fig. 9.
V. Physical properties of the final design
With the final design, a second iteration of fi-nite element analysis was performed with all thecomponents necessary for laser power measurement.Expected values for the masses of the mirror, andthe inner electrode were assigned. The two identical Fig. 7: The upper image shows the CAD model ofthe designed mechanism. It consists of two separatemechanisms with a connector in between to preventcorner loading and provide stability to the structure.The lower image shows the geometry of the mecha-nism illustrating the boundary conditions for finiteelement simulation. Material density was consideredwith ρ = 2 . / cm for aluminum. Note that thecenter of mass of each rotating link was designed tocoincide with its pivot. Note also, that the weight ofthe coupler differs slightly from what was assumed inthe first simulation in Fig. 6. In the real CAD modelit is m a = 32 . m p = 171 . N eq = 0, seeEq. 12. These four masses are concentrated at points,while the masses of the coupler and the swings wereassumed to be distributed, see Fig. 7.The swing assembly consists of the upper/lowerswing and the upper/lower counter mass plus half the mass to the left of the front flexures, see Eq. 12. Thecenter of mass of the lower swing assembly coincidesvertically with the lower back pivot point and canbe finally adjusted horizontally. The opposite is truefor the upper swing assembly. Here, the center ofmass coincides horizontally with the upper back pivotpoint and can be finally adjusted vertically.With the lower mass, the restoring torque of thebalance is adjusted which allows to adjust the equi-librium position of the balance close to the nominalzero position of the linkage. With the upper massthe stiffness of the balance is adjusted. If the centerof mass of the upper swing assembly coincides withthe pivot ( h = 0), the restoring torque is providedby the pivots alone. By moving the center of mass up,the restoring torque is reduced by the gravitationalmoment of the mass. The drawback of this methodis that the gravitational moment changes with tilt ofthe balance frame. Hence, the tilt sensitivity increasesas the mass of the swing assembly moves away fromthe pivot point.A finite element simulation is performed with theupper mass in six positions to investigate the trade-off between stiffness reduction and tilt sensitivity.The results of these simulations and the calculationof the analytical model using Eqs. 10, 14 and 11 aredisplayed in the upper plot in Fig. 8. The data in blueindicated by the left axis show the linear stiffness ofthe mechanism as a function of h . Black data pointswith the scale on the right show the produced torqueon the coupler. The circles are calculated with finiteelement simulation, and the lines from analyticalequations. A good agreement indicates the validityof the analytical equations.The goal is to obtain a linear stiffness of the couplerof K ≤ . − . Hence, h ≥ .
12 mm.As indicated in the upper plot in Fig. 8 a large h increased the tilt sensitivity. Fig. 8 provides moredetailed information. The plots show both the sensi-tivity of the balance readout to ground tilt in open(upper plot) and closed (lower plot) loop. In openloop, the ground tilt causes a deflection of the couplerin vertical direction with respect to the metrologyframe. In closed loop, a restoring force generated bythe capacitor is necessary to maintain the couplerat the desired equilibrium position. In either case,a static tilt will drop out, since the external forceis modulated. Hence only a tilt that occurs on thesame time scale as the laser light modulation will contribute a bias to the measurement. We estimatesuch a modulation of the tilt is 1 nrad. Using h =44 .
12 mm, a spurious force of 0 .
76 nN will be indi-cated in this case, which has no significant impact tothe measurement result. . . . . K / ( N / m ) h / mm ( d x / d φ ) / ( m / r a d ) . . . . ( d F / d φ ) / ( N / r a d ) Fig. 8: The upper plot shows the analytical (solidlines) and finite element simulation (dots) results forthe mechanism stiffness (blue filled circle) and the er-ror force sensitivity due to ground tilt φ (black emptycircle) for different compensation mass positions h in closed loop. The lower plot shows the sensitivityof the excursion of the coupler in x due to groundtilt φ for different compensation mass positions h in open loop. Note that the excursion is measuredwith respect to the equilibrium in the nominal zeroposition of the coupler for φ = 0.In addition to the above, the dynamic behaviour ofthe balance is also important to consider. The eigen-frequencies of the device are obtained with a modalanalysis. A finite element simulation was carried outwith the system at equilibrium, h = 44 .
12 mm andthe mass distribution as shown in Fig. 7. For thefirst resonance in the system, the oscillation alongthe force measurement axis ( x ), the eigenfrequencyis f = 0 .
51 Hz. The next resonance at f = 19 Hz isthe out of plane bending of the swings. A total ofseven resonances occur below 100 Hz. Since the nextlowest frequency is more than an order of magnitudeaway from the most compliant mode, the influenceof the higher order modes on the measurement isbelieved to be negligible, or can easily be mitigatedwith appropriate filtering.A low angle view of the device with protective side plates installed is shown in Fig. 9. The side platesallow mounting, add thermal mass, and provide holesfor transport safety pins, see inset in Fig. 9. Eachsafety pin is a spring sleeve. It can be compressedand inserted in the assembly without applying aninsertion force. By engaging a screw in the sleeve,it expands and the mechanism is locked. Both sideplates were manufactured in one setup together withthe mechanism, ensuring that the holes for the safetypins are precisely aligned. Hence, the pins lock themechanism without additional forces and with zeroclearance.Fig. 9: Low angle view of the assembled mechanism.The inset on the bottom shows, not to scale, one ofthe total six transport safety pins. VI. Summary and Outlook
An electrostatic force balance mechanism was de-veloped to measure the photon pressuren force of a100 kW laser. The task required the measurement ofa 667 µN force with a relative uncertainty of 1 × − in air. Therefore, a portable monolithic parallelogramlinkage was designed. At first, the impact of ma-chining tolerances on the corner loading error wasinvestigated as a function of the two length in theparallelogram, a and b . It was found, that cornerloading is generally minimal when linkage dimensionsare equal, a = b , and decreases with increasing a + b . The largest possible size in this application is a = b = 100 mm and was chosen for the final design. Furthermore, the basic equation of motion wasderived according to the Lagrange equations of thesecond kind and results were confirmed with finiteelement simulation. With these, the static and dy-namic behavior of the system could be optimized byparameter variations. The most critical static prop-erties of the mechanism are its equilibrium conditionand the linear stiffness of the moving balance coupler.Two moveable masses can be used to adjust both theequilibrium position and the linear stiffness of thecoupler in the mechanism independently. In the finaldesign the stiffness of the mechanism in the measur-ing direction can be adjusted from K = 7 . − to below zero according to finite element simulation.Lowering the stiffness increases the sensitivity of thebalance to ground tilt. Analyzing this trade-off withboth finite element simulation and analytical model-ing allowed to estimate this impact, which was foundto be negligible for the measurement consideringa desired value for the mechanism stiffness at thecoupler of K = 0 . − .After manufacturing and system integration, ex-periments will be required to study especially the ef-fects of internal heating to the measurement readoutdue to absorbed laser light from the 100 kW laser.Also, air currents on the mirror due to the operationof the balance are expected to cause a significantamount of noise. In order to mitigate these problems,heat and draft shields will be installed in the finalsetup.Mechanically, hysteresis due to anelastic after-effects in the flexures might cause time-dependentrestoring forces, which would bias the measurement.Hysteresis is difficult to study theoretically due tolimited information provided by literature, but fornow, these anelastic forces were not considered tobe problematic, because the maximum stress in thehinges is within ranges suggested by Sydenham [15]to keep anelastic effects small. Further experimentalwork will be carried out to verify this.In conclusion, the mechanism described above ful-fills the criteria necessary for measurement of photonpressure force from high power laser systems, withrelative uncertainties below 1 × − . VII. Acknowledgements
This work has been done in close cooperation be-tween the Physical Measurement Laboratory at NISTand the Precision Engineering Group at Technische Universität Ilmenau. We also want to thank the NISTinternal reviewers Vincent Lee and David Newell forconstructive feedback.
Appendix A: Summary of the parametersused for the final design
The numerical values for the important mechanicalparameters of the final design are shown in Table I.The locations of the parameters are indicated inFig. 4. masses m a . m E . m M . m p . lengths a . b . e . horizontal distances between b M . b E . L . vertical distances between h . m p & pivot for K = 0 . − TABLE I: Overview of mechanical relevant parame-ters for the final design. The horizontal distances tothe coupler are measured to its center.
Appendix B: Derivation of the equation ofmotion
The equation of motion can be derived using theLagrange equations of the second kind. The La-grangian is given by L ( q i , ˙ q i , t ) = T − U, (15)where q i are the generalized coordinates, i = 1 ...n for n degrees of freedom.The parallelogram linkage has one degree of freedom, q = Θ. This generalized coordinate can be writtenas the sum of the tilt of metrology frame, φ , andthe deflection of the balance with respect to themetrology frame, γ , i. e., Θ = φ + γ .The position of nine point masses, as shown inFig. 4 can be written as products of lever arms andsines and cosines of the corresponding angles. Theyare, r mb1 = − a φ − a φ , (16) r mb2 = a φa φ , (17) r ma = b sin ( φ + γ ) − b cos ( φ + γ ) ! , (18) r mp1 = − a φ − e sin ( φ + γ ) − a φ + e cos ( φ + γ ) , (19) r mp2 = a φ − e sin ( φ + γ ) a φ + e cos ( φ + γ ) , (20) r mh1 = − a φ − h cos ( φ + γ ) − a φ − h sin ( φ + γ ) , (21) r mh2 = a φ − h cos ( φ + γ ) a φ − h sin ( φ + γ ) , (22) r mM = r ma + − a M φ + b M sin ( φ + γ ) − a M φ − b M cos ( φ + γ ) , and(23) r mE = r ma + a E φ + b E sin ( φ + γ ) a E φ − b E cos ( φ + γ ) . (24)Note, the first/second line of the vectors indicatethe x - /y -coordinate, with the x being positive in thedownward vertical direction. The total kinetic energy is the sum of the transla-tional ( t ) and rotational ( r ) energies, T = T t + T r .The kinetic energy of the two swings can be cap-tured by a single rotational term. The centers ofrotation are pivots A and B , respectively. Themoment of inertia is calculated around these centersof rotations considering the bars as rods with length b + e . It is J b = m b
12 ( b + e ) . (25)The motion of the coupler is described as a trans-lation of its center of mass with velocity ˙ r ma anda rotation about φ around its center of mass. Themoment of inertia of the coupler J a is J a = m a a . (26)Hence the sum of the translational and rotationalkinetic energies are T t = 12 m a ˙ r + 12 m p ˙ r + 12 m p ˙ r + 12 m h ˙ r + 12 m h ˙ r + 12 m M ˙ r + 12 m E ˙ r , and (27) T r = 12 J a ˙ φ + J b ( ˙ γ + ˙ φ ) . (28)The potential energy of the system is a sum of theenergy stored in the torsional stiff pivots ( k ) and thesum of the gravitational energies of the masses ( m ), U = U k + U m . It is U k = 4 12 κ s γ , and (29) U m = − g ( m b r mb1x + m b r mb2x + m a r max + m p r mp1x + m p r mp2x + m h r mh1x + m h r mh2x + m M r mMx + m E r mEx ) . (30)Consistent with the main text, the torsional stiffnessof a single flexure hinge is given by κ s . Only oneexternal moment needs to be considered. It arisesfrom the stiffness adjustment spring and is Q e = − F F (cid:16) ( l + b cos γ ) sin σ + b sin γ cos σ (cid:17) , (31) where σ is the angle between the orthogonal of themetrology frame and the spring force F F , i. e., σ = b sin γl + b cos γ , and (32) F F = − ( λ − λ ) k b , with (33) λ = p ( b sin γ ) + ( l + b cos γ ) . (34)With the Lagrangian in Eq. 15 and the previousconsiderations the equation of motion due to γ yieldsdd t (cid:18) ∂ L ∂ ˙ γ (cid:19) − ∂ L ∂γ = Q e . (35)Taking the derivatives and regrouping the expres-sions yields a compact result, J ¨ γ + κγ = − J φ ¨ φ + κ φ φ − N eq , (36)where the coefficients are given by J = b m E + b m M + b m a + b m b bem b e m b
6+ 2 e m p + h m h + h m h , (37) J φ = J + ah m h − ah m h bb e m E + bb m m M (38) κ = 4 κ s − gm h ( h + h ) − l bk b (cid:18) − λ b + l (cid:19) , (39) κ φ = gm h ( h + h ) , and (40) N eq = g ( − bm E − bm M − bm a + 2 em p ) . (41) References [1] Alexandra Artusio-Glimpse, Kyle Rogers, Paul Williams,and John Lehman. Halo – high amplification laser-pressure optic. Proc. in NEWRAD, Boulder CO, USA,2020.[2] Maximilian Darnieder, Markus Pabst, Thomas Fröhlich,Lena Zentner, and René Theska. Mechanical propertiesof an adjustable weighing cell prototype. euspen’s 19thInternational Conference & Exhibition, 2019.[3] F. Dirksen and R. Lammering. On mechanical proper-ties of planar flexure hinges of compliant mechanisms.Mechanical Sciences, 2(1):109–117, 2011.[4] Stefan Henning, Sebastian Linß, and Lena Zentner. de-tasflex – a computational design tool for the analysis ofvarious notch flexure hinges based on non-linear modeling.Mechanical Sciences, 9(2):389–404, 2018.2