# An interferometric force probe for beam diagnostics and the study of sputtering

**Part of the following topical collections:**

## Abstract

A novel probe for the measurement of forces exerted by a beam of ions and neutral

atoms on a small target is described. The force probe is intended for various applications: First, diagnostics of thruster plumes of electric spacecraft propulsion engines, in particular the determination of the spatial distribution of the momentum in

an exhaust beam, and second, the study of sputtering of solid targets by means of the related momentum transfer. The instrument makes use of a sensitive cantilever whose elastic deflection is measured interferometrically along two axes and enables that way a simultaneous measurement of two independent components of the force vector. The present contribution focuses on setup and calibration of the probe. Example measurements with obliquely irradiating ion beams are performed.

## Keywords

Sputtering Momentum transfer Force measurement Beam diagnostics Electric spacecraft propulsion## Introduction

Diagnostics for exhaust plumes of electric spacecraft propulsion engines are often based on electric probes like Faraday cups [1] and retarding field analyzers [2]. Spectroscopic techniques like optical emission spectroscopy [3], laser absorption spectroscopy [4], and laser-induced fluorescence [5] are also applied, but less frequently. Recently, force measurements were performed in the beam of an industrial broad-beam ion source that is very similar to the plume of gridded ion thrusters [6]. The advantage of the force measuring method over the standard electric techniques is that the latter are sensitive only for the charged species. This is an important issue because thruster plumes often contain a non-negligible amount of energetic neutral atoms [7]. Energetic neutral atoms originate from charge-exchange collisions with the gas behind the exhaust, which unintentionally escapes from the thruster, and with residual gas in the test chamber [8, 9]. In this contribution, we describe in detail a novel technique for the measurement of forces exerted by a thruster plume on a small test surface.

Recently, force measurements have also been applied for the study of sputtering [10, 11, 12]. When sputtering occurs, not only momentum carried by the impinging beam particles is transferred to the irradiated surface, but momentum also leaves the surface together with, firstly, the sputtered target atoms and, secondly, re-emitted beam particles. The measured force is an average, just like the sputter yield is an average, but it contains different information. The force is a vectorial quantity, and the involved particles are differently weighted in the averaging integral according to their directions, speeds, and masses. Therefore, force measurements provide information about sputtering beyond the scalar sputter yield, and comparisons of measured forces with computer simulations would be a stronger test for numerical models than comparisons of sputter yields.

Another field of application of force measuring techniques emerged recently in the context of plasma–wall interaction. The forces exerted by low-temperature low-pressure plasmas on the surface of solid boundaries are comparable to the forces that occur in the aforementioned thrust measurements and sputtering experiments [13, 14]. A method similar to the technique presented in this article was recently applied for a precise measurement of the “plasma pressure” by means of a force probe that was integrated into a plane plasma boundary [15].

In comparison with our formerly reported force measurements that used electromagnetically compensated pendula [6, 11, 16, 17], the here presented probe is based on a sensitive cantilever whose deflection is measured interferometrically along two axes.

Scanning a cross section of a thruster plume yields vectorial information about the momentum distribution in the plume. In comparison with a thrust balance, the thrust vector could be calculated from the measured force profiles: Integration over the entire cross section yields the thrust vector, and not only the scalar thrust component defined by the orientation of the thrust balance. On the other hand, the measured “differential thrust” at a specific point in the plume, i.e. the spatially resolved momentum flux density, is valuable information for improvements of the efficiency of a thruster.

A further application of force probes is the experimental study of momentum transfer from energetic ions and atoms to solid targets, at which sputtering not necessarily occurs. For example, air molecules interact with surfaces of space vehicles, especially of satellites and small objects in low Earth orbits. A force probe with targets of different materials mounted on a satellite could provide a direct measurement of material specific drag data when the angle of approach of the target surface is varied. From such empirical data, drag coefficients of satellites could be calculated with high accuracy [18].

The article is arranged as follows. A detailed description of the force probe is given in the “Description of the probe” section. The following “Calibration and errors” section is dedicated to the calibration of the probe and the accuracy and temperature stability of the method. The “Example of a vectorial force measurement” section presents, as an example, a force vector measurement in that an ion beam impinges at variable angles of incidence at the target. Finally, an Appendix contains, as a supplement to the “Calibration and errors” section, a theoretical treatment of the eigenmodes of the probe that is useful for the understanding of the mechanical noise.

## Description of the probe

_{2}O

_{3}ceramic tube that serves as the 172 mm long cantilever (see Fig. 1). The target will be exposed to a particle beam, for example in a vacuum chamber for testing of electric space propulsion systems, while the cantilever itself is protected from the particles. When a force acts on the target, the tube is elastically bent (not more than a few micrometers), so that each point of the cantilever is minimally displaced from its initial position. In terms of the theory of elasticity, the cantilever is essentially a hollow cylindrical beam with one fixed end. The displacement of a specific part of the cantilever serves as the quantity to be measured; the movements occur mainly in the plane perpendicular to the cantilever.

The tube geometry, i.e. a hollow cylinder, was chosen with regard to the elastic properties that are in this case the same for all bending directions; moreover, this geometry allows to feed a wire through the cantilever. Such a wire is connected to the aluminum target holder and can be used for biasing and measurements of the current taken by the conductive target.

The interferometric displacement sensor is a commercial product (attoFPS 3010) from the Attocube Systems AG, Germany [20, 21], which applies the frequency-modulation continuous wave (FMCW) interferometer method [22]. It uses the path between a plane end of a single-mode fiber and a mirror as cavity, where a lens (focal length 8.18 mm, diameter 9 mm) is placed between fiber and mirror. The lens in the sensor head has a distance of its focal length from the fiber end in the connector (known as FC/PC connector), so that the light from the fiber leaves the sensor head as a thin beam of parallel (collimated) rays with a diameter less than 2 mm. The mirrors have a distance of approximately 6 mm from the respective lenses.

Approximatly 4% [20] of the 1550 nm laser light (150 *μ*W) from a tunable distributed feedback (DFB) laser are reflected backwards at the plane end of the fiber and serve as the reference wave, while most of the light leaves the fiber and reaches the mirror. The light reflected at the mirror could directly re-enter the fiber through the same plane end and finally interfere with the reference wave at the detector outside the vacuum chamber at the other end of the wave guide. However, this single-path technique has two disadvantages: first, the mirror has to be aligned very precisely in order to receive and reflect the light rays perpendicularly, and second, the wave coming from the mirror has a much higher intensity than the reference wave. In such a case, the difference between constructive and destructive interference is only weekly pronounced (low signal contrast) and might not be detectable. For this reason, the wave from the mirror is attenuated by the following ‘trick’: The mirror is slightly misaligned by a small rotation (<1°) of the cantilever about its axis, so that the returning light is not focused on the fiber core, but rather with a small lateral shift at the zirconia ferrule of the connector. (This can be done by turning the stainless steel cylinder with fine thread at the fixed end of the cantilever, when a locking screw is loosened.) That way, strayed and significantly attenuated light is reflected and collimated by the lens, passes the cavity once again in both directions, and is focused this time right at the fiber core, where it finally re-enters. This patented ‘self-aligning’ dual-path technique was described and investigated in detail by Thurner, Braun, and Karrai [20].

In order to overcome the typical drawbacks of a conventional Fabry-Pérot interferometer, i.e. low sensitivity sensing spots at the interference extrema and the ambiguity of the displacement direction, the interferometer modulates the laser wavelength (about ±10^{−3} wavelengths) periodically with a frequency of 12.5 MHz. This fast tuning is accomplished in the DFB laser by means of a modulation of the current that powers the laser diode [21, 22, 23]. The use of optical fibers and vacuum feedthroughs allows to keep the cavity length short and to have only non-electric parts in the probe itself, while the electronic parts are placed outside the vacuum chambers in which the probe is applied.

Now, we come back to our mechanical setup. The masses of the mirror unit, the target with its holder, and the ceramic tube give rise to undesired mechanical vibrations. In particular, the pumps of vacuum chambers always excite eigenmodes; but also an abrupt change of the force that is to be measured would cause a ballistic overshoot beyond the equilibrium position and subsequent weakly damped oscillations.

*μ*m and ±1.2

*μ*m, respectively. In general, the amplitudes are changing on a time scale of seconds. We attribute this to irregular exitation amplitudes from the environment and an unavoidable coupling between several modes of the entire setup.

The frequency analysis (twice the absolute values of the single-sided discrete Fourier transforms divided by the number of samples) of the 10 s long time series of the displacements in Fig. 3c exhibits one distinct peak at *f*_{0}=(11.1±0.1) Hz, and no significant amplitudes are found beyond 25 Hz.

*m*

_{1}=3.85×10

^{−4}kg and

*m*

_{2}=1.78×10

^{−3}kg (calculated from the dimensions of the parts) are at the positions

*x*

_{1}=78 mm and

*x*

_{2}=172 mm measured from the fixed end, and the linear mass density of the vibrating ceramic beam is

*m*=2.25×10

^{−3}kg m

^{−1}(for the density of 3825 kg m

^{−3}given by the manufacturer). Furthermore, we assume the modulus of elasticity

*E*=3.09×10

^{11}Pa for the Al

_{2}O

_{3}ceramic at room temperature [24] and use \(I = \frac {\pi }{4} \left ({r_{1}^{4}} - {r_{2}^{4}}\right)\) as the second moment of area of the cantilever cross-section for the direction of the bending. Note that

*I*is independent of the bending direction because of the rotational symmetry of the cross section. The model results in a first mode at a frequency of

*f*

_{1}=10.6 Hz, and subsequent eigenmodes at

*f*

_{2}=124.7 Hz,

*f*

_{3}=588.6 Hz, and

*f*

_{4}=1269 Hz. Figure 4 shows the corresponding modes. The

*n*=1 mode is an in-phase oscillation, at which the mirror unit and the target unit move always in the same direction. The

*n*=2 mode corresponds to an anti-phase oscillation of the mirrors and the target. However, the modes

*n*>1 cannot be found in the measured spectra. The good approximation of the calculated

*f*

_{1}frequency with the measured natural frequency

*f*

_{0}identifies this one as a mode 1 vibration.

The vibrations at the natural frequency *f*_{0} do not constitute a serious drawback for force measurements as long as the displacements caused by the force are not too small in comparison with the mechanical noise. Averaging over several periods of the natural frequency reduces the noise.

However, in order to keep the required sampling time for the averaging short, the probe is equipped with a damping mechanism. Figure 3b shows the essential parts of the eddy current damping unit. A pair of small cylindrical permanent magnets provides a magnetic field that penetrates a thin copper sheet attached to the back side of the target holder. The target together with the copper sheet oscillates perpendicularly to the field lines, so that eddy currents are generated in the sheet. The currents create a counteracting force (Lenz’s law) and dissipate energy stemming from the oscillation. Figure 3a shows the x-y plot of the measured displacements for the damped case (blue curve), too. Now, the oscillations are significantly reduced in both directions. The natural frequency *f*_{0} disappeared completely from the spectra shown in Fig. 3d, which indicates the efficient damping of the cantilever vibrations. In contrast, the amplitudes at other frequencies persist; the remaining frequencies can be understood as the complicated modes of the entire setup that result not only in vibrations of the lenses but also of the damping unit, which are unintentionally transmitted to the target via the eddy currents. We determined the time constants *τ* for the exponentially decaying amplitudes, i.e. decaying proportionally to exp(−*t*/*τ*) with the time *t*. To this end, we knocked carefully on the target with a thin wire and recorded the oscillations. We found *τ*=(6.0±0.5) s without eddy current damping, and *τ*^{′}=(0.3±0.1) s with the damping unit. The resulting quality factor \(Q = \frac {1}{2} \omega _{0} \tau \) for the cantilever as an oscillating system becomes *Q*=209±18 without additional damping and *Q*^{′}=10.5±3.5 for the intentionally damped system. This means that the damped cantilever looses its oscillation energy after only a few (≈*Q*^{′} / 2*π*) oscillations, but it is still underdamped (*Q*^{′}>1/2), i.e. abrupt displacements cause an overshooting. Inthe “Example of a vectorial force measurement” section, a measurement without damping will be shown in comparison with a corresponding measurement with the damping system.

The method requires a reference measurement without the force that is to be determined. The reference measurements are performed shortly before or after the main measurement with the force. This method ensures that temporal mechanical changes of the experimental setup are not misinterpreted as forces. For example, in electric propulsion test chambers, diagnostics are usually mounted on mechanical structures that are exposed to the truster plume. The structures are unintentionally heated and give rise to small displacements. Thus, the gravity vector could slightly be tilted relative to the probe, so that the cantilever with the masses of the target and mirror assemblies adopts a different equilibrium position with changed displacements *d*_{1} and *d*_{2}. Such drifts occur at time scales much longer than a measurement, which can be performed within a few seconds, and become negligible when reference measurements are performed shortly before or after the main measurement. A practical procedure will be described inthe “Example of a vectorial force measurement” section.

## Calibration and errors

For calibration, the instrument is laid on its rear side so that certified weights of 1, 2, 5, and 10 mg (±0.006 mg) can be put on the target surface. Combinations of up to four of the weights with the total mass *m* exert the force *F*=*m**g* on the target, where *g*=9.81 ms ^{−2} is the local gravitational acceleration. For the calibration, the damping unit is removed, because the cantilever is bent almost 3 mm due to the weight of the mirrors and the target holder; the target holder would rest on the damping unit.

The weights are put on the center of the target, which is possible within a radius of 3 mm. In our case, no special measures are taken for an exact horizontal orientation, since even a slope angle of the cantilever of 5 degrees would cause a reduction of the measured component of less than 0.4 % (cosine of the angle). Furthermore, a rotation about the cantilever axis would cause asymmetric displacements measured by the two sensors, which not affects the magnitude of the measured force (vector sum of the two components).

*k*=23.4 N m

^{−1}.

The error of *k*, i.e. the uncertainty of the slope of the regression line, is, due to the number of data points, smaller than the least significant specified digit and therefore negligible. However, the standard deviation of the individual measurements from the regression line is 0.09 *μ*m, corresponding to 2.1 *μ*N.

The following calculation can be used to estimate the order of magnitude of the calibration constant. The underlying theory of the deformation of solids can be found in many textbooks, e.g. [25]. The geometrical and material parameters of the cantilever are the same as in the “Description of the probe” section.

*S*=

*F*/

*δ*describes the deflection

*δ*of the free end under the action of the force

*F*at the free end as a special case of Hook’s law. The stiffness of the cantilever beam

*E*and the second moment of area

*I*of the cantilever cross-section, which both were already used in the “Description of the probe” section. Note that not the deflection at the free end, where the target is mounted, is measured but the deflection

*d*at the position

*x*

_{1}of the mirrors. The deflection at this specific position on the beam can also be calculated [25], and the deflection-to-force conversion factor

*k*=

*F*/

*d*becomes

Assuming again an elastic modulus *E*=3.09×10^{11} Pa for the Al_{2}O_{3} ceramic at room temperature [24], one would expect *k*=32.8 N m ^{−1}. This is the correct order of magnitude, but the measured value is almost 30% smaller. One part of the difference can be attributed to the tolerance in the radii (±5%) on which *k* strongly depends. Furthermore, the elasticity of the polycrystalline material is only imprecisely known, since the elastic modulus *E* depends on purity, size of the crystallites, and the porosity, which all result from the manufacturing method.

Finally, we discuss the influence of the cantilever temperature *T* on the deflection-to-force conversion factor. Young’s modulus *E*(*T*) of the alumina specimen investigated in the above mentioned study [24] decreased linearly with increasing temperatures *T*<900 °C, and a plot in that publication shows a negative slope of approximately d*E*/d*T*=−4×10^{7} Pa K ^{−1}. A temperature rise of *Δ**T*=+100 K starting from room temperature (293 K) lowers the elastic modulus *E* by 1.3% (*E*=3.05×10^{11} Pa), and the calibration constant *k* decreases consequently by 1.3%, too. When a significant warming of the cantilever is expected, this effect should be considered.

There is a simple way how a changing “spring constant” *k* could be monitored without additional temperature sensors, which obviously should not be attached to the cantilever. However, this technique works only when the damping unit is not used, because it relies on an evaluation of small shifts of the natural frequency *f*_{0}. For an ideal beam with one free end and no additional masses attached to it, the square of the natural frequency *f*_{0} is proportional to Young’s modulus *E*, i.e. \({f_{0}^{2}} \propto E\). One can easily reason that this holds for the geometrically more complicated force probe cantilever as well, since the elastic modulus *E* still enters linearly in all the forces that move the cantilever. Hence, an increase of *E* by a factor of *λ* makes the motion by the factor *λ*^{2} faster (time scales proportionally to *λ*^{−2}), i.e. frequencies of oscillations would increase by the factor *λ*^{2}. This is confirmed by the above mentioned theoretical modal analysis in the Appendix, in particlular Eq. (8), which results in a decrease from 10.6 Hz to 10.5 Hz for the first mode, when the temperature rises by *Δ**T*=+100 K from room temperature. A by 1% (*λ*=0.99) lower frequency *f*_{0} would indicate a by 2% (*λ*^{2}=0.98) smaller conversion factor *k*. A frequency drift of such an amount is detectable, see Fig. 3.

## Example of a vectorial force measurement

^{+}beam that impinges at variable angles of incidence

*α*at the probe target, see Fig. 6a [12].

The force probe is mounted on a turnable platform driven by a step motor with an angular resolution of *Δ**α*=1.8°. The cantilever coincides with the vertical rotational axis of the turnable platform, so that the target remains at the symmetry axis of the ion beam when the probe is rotated. The beam is provided by a broad-beam ion source with grid system [17, 26]. The distance from the beam source to the target is 88 cm. At this distance from the grid system of the broad-beam ion source (diameter 125 mm), inhomogeneities of the beam can be neglected (compare [11], where the target was bigger and the distance from the source was shorter).

The plasma in the ceramic source chamber, from where the ions are extracted, is generated by a 2.45 GHz electron cyclotron resonance microwave discharge at an Ar gas flow of 5 sccm. The potential of the source plasma is anchored by means of an anode at a potential of *U*_{a}=+1200 V in order to accelerate the extracted ions in the grid system, which separates the source chamber plasma at the high potential from the target chamber plasma at a low potential. The kinetic energy of the accelerated ions in the target chamber is therefore approximately *E*_{kin}=1.2 keV. A more detailed discussion of the kinetic energies in the beam can be found in [9].

The vacuum chamber is a stainless steel cylinder with an inner diameter of 65 cm and a length of 160 cm, see Fig. 6. The argon gas pressure in the target chamber is 2×10^{−2} Pa achieved by a turbo molecular pump (pumping speed 500 l s ^{−1}) in combination with a rotary vane pump (60 m^{3} h ^{−1}).

*α*=45° in this case.

Figure 7c and d demonstrate the effect of the eddy current damping. In case of the removed damping system, an overshooting and subsequent persistant oscillations occur at every switching. The amplitude of the overshooting and the oscillations is comparable to the causal signal itself, i.e. the signal-to-noise ratio is here about unity.

In case of the time series shown in Fig. 7, the probe target is a copper foil, so that sputtering and a release of momentum together with the outgoing particles (repulsion) is to be expected. Consequently, the measured force is larger than the force that a perfectly absorbing and not sputtering material experiences. Carbon has a low sputtering yield and is therefore often used for beam dumps in test chambers. Moreover, the effective sputter yields can be reduced even more by means of an appropriate topography of the surface. Carbon fiber velvet (produced by the Energy Science Laboratories, Inc.) is such a material; it consists of approximately 2.2 mm long and 7 *μ*m thin carbon fibers that arise perpendicularly from a plane base. Figure 6b shows a scanning electron microscopy image of the fibers. The geometry, which resembles a brush with extremely thin bristles, allows the ions to enter deeply into the carbon fiber velvet. Consequently, sputtered carbon from one fiber is re-deposited at other fibers with high probability, and reflected ions likely undergo more than one contact with the fibers before they leave the material.

*α*are performed with the copper foil and the carbon fiber velvet. Figure 8a shows the measured displacements

*d*

_{1}(

*α*) and

*d*

_{2}(

*α*) for the angles

*α*=−10°⋯+100°, while Fig. 8b displays the forces

*F*

_{1}=

*k*

*d*

_{1}and

*F*

_{2}=

*k*

*d*

_{2}transformed into a more practical coordinate system: One component,

*F*

_{normal}, points along the inward directed target normal, and the other one,

*F*

_{in-plane}, is the projection of the force vector onto the target surface. The transformations of the components are

Note the equality of the displacements, *d*_{1}=*d*_{2}, for the angle *α*=0°, and that the axes change roles when *α* changes its sign due to the symmetric setup (Fig. 8). For increasing oblique incidence angles *α*>10°, the flux onto the target becomes more and more reduced, and one could expect that the flux vanishes for extremely grazing incidence, *α*≈90°. However, due to the finite diameter of the ion source and the beam divergence, some beam particles strike the probe target even at *α*>90°. This explains the not vanishing deflections for *α*≥90°.

*F*

_{1}and

*F*

_{2}can be described by the functions

where *F*_{0} is the magnitude of the force vector at perpendicular incidence, i.e. \(F_{0} = \sqrt {{F_{1}^{2}}(0^{\circ}) + {F_{2}^{2}}(0^{\circ})}\). The Eqs. (4) account by their cos*α* terms for the reduced flux at oblique incidence and express by the second cosines the projection of the force vector onto the respective directions of the displacement sensors. These theoretical functions are plotted (as the corresponding displacements) in Fig. 8a, where the amplitude *F*_{0} was taken from the measurement with the carbon fiber velvet at perpendicular incidence (*α*=0°).

The measurements with the carbon fiber velvet roughly follow the model of pure beam absorption, the small deviations can be summarized in two observations: First, both force components vanish at *α*=90° according to the discussion above; and second, the measured force components tend to be slightly more positive than the theoretical ones. The latter can be attributed to an imperfectly inhibited sputtering that still causes additional repulsion directed at some angle into the target.

In case of the copper target, the deviations from the pure absorption case are much more pronounced. Most obviously, the forces (displacements) at perpendicular incidence (*α*=0°) are significantly, by 44%, enhanced. Moreover, the force measured by the second sensor does not vanish at *α*=45°, it rather has a magnitude of approximately 0.4 of the *F*_{2} value at *α*=0°. At this oblique angle of incidence, the second sensor is directed perpendicularly to the beam direction, so that the measured force can only be caused by particles that leave the target with momentum.

Finally, we introduce an optional component that becomes necessary for measurements in environments where it is not feasible to repeatedly switch the beam on and off for the reference measurements. For this purpose, a beam shutter can be placed directly in front of the probe target. The shutter shown in Fig. 6c consists basically of a blade iris that has a maximum aperture of 36 mm and can be closed completely. A small direct-current motor drives the blades for quick opening and closing. Measurements in the plume of a Hall thruster have recently been performed and are planned to be published soon.

## Conclusion

This article presented an instrument for the measurement of vectorial forces that are exerted on a small target exposed to a particle beam. The functional principle is based on an elastically bendable cantilever with one fixed end and the target mounted at the free end. The deflection of the cantilever is measured with an interferometric displacement sensor. Between the free and the fixed ends, two mirrors are attached that reflect the laser light from the sensor optics. By means of this arrangement, the displacement can be recorded in two dimensions, i.e. in the plane perpendicular to the cantilever beam.

Typical deflections are in the order of magnitude of a few micrometers, corresponding to some tens of micronewtons. The Fabry-Pérot type interferometer setup uses fiber optics, one for each of the two axes, which allow that only the small cavity parts have to be integrated in the probe, while the laser and the data acquisition electronics are operated outside the vacuum chamber.

The calibration was performed by putting small weights directly on the target surface. For this purpose, the cantilever and the target were oriented horizontally, so that the weights acted roughly in direction of the surface normal.

An example of a force vector measurement was presented, where the energetic particles of a broad-beam ion source impacted at variable angles at the probe target. Two different target materials were used, namely copper and a microstructured carbon fiber material with low volume filling factor (carbon fiber velvet). The forces measured with the carbon material showed an angle-of-incidence dependence expected from a perfect particle absorber without sputtering. In case of the copper target, the measured forces were not only enhanced in comparison with the carbon fiber material, but also the angle dependence changed. In particular, at an angle of 45°, the force components perpendicular to the beam direction differed qualitatively from each other: While this component practically vanished in case of the carbon fiber material, almost 40% of the value for perpendicular incidence was measured with the copper target. This was clear evidence for the repulsion caused by sputtered target atoms and re-emitted argon beam particles.

The force probe is therefore a promising diagnostic for the study of sputtering. When equipped with target materials like the carbon fiber velvet, which are characterized by very low effective sputter yields, the measured forces equal the momentum flux of the absorbed beam without any modification by the momenta of released particles. A force probe with such materials is thus ideally suited for thruster development and testing in the field of electric spacecraft propulsion.

## Appendix

The cantilever with the mirror and the target units as a vibrating system can be treated by means of the Euler-Bernoulli beam theory, e.g. [25], which is demonstrated in this Appendix.

*x*, while the small deflections

*y*(

*x*,

*t*) at the time

*t*are perpendicular to the beam axis. The mirror and the target are represented by two point masses

*m*

_{1}and

*m*

_{2}at the positions

*x*

_{1}and

*x*

_{2}. At the positions

*x*between the fixed end and the first point mass and between the two masses, i.e. 0≤

*x*≤

*x*

_{1}and

*x*

_{1}≤

*x*≤

*x*

_{2}, the Euler-Bernoulli beam equation

describes the shape of the beam, where *E**I* is the flexural rigidity for the bending direction, and *m* is the mass per length unit of the beam.

of the modes *n* with the corresponding eigenfrequencies *ω*_{ n }, where the individual modes are separated into a spatial amplitude function *w*_{ n }(*x*) and the oscillation in time with an arbitrary phase constant *φ*_{ n }.

*y*(

*x*)=

*y*

_{1}(

*x*) for 0≤

*x*≤

*x*

_{1}and

*y*(

*x*)=

*y*

_{2}(

*x*) for

*x*

_{1}<

*x*≤

*x*

_{2}; the parts are

*x*=0 means

*x*=

*x*

_{1}is expressed by

*EI*), which is continuous for all

*x*on the beam. In particular, the bending moment vanishes at the free end,

*M*

_{1}and

*M*

_{2},

The mass *M*_{1} is moved by the shear forces *E**I**y*^{‴} from both sides, whereas *M*_{2} feels a force only from one side. Note, that the shear force along the beam has a discontinuity at *x*=*x*_{1} due to the force *M*_{1} exerts at that point. The equation of motion for a line element *dx* of the beam is already expressed by the Euler-Bernoulli beam equation Eq. (5).

*a*

_{1},…

*a*

_{8}for a specific

*κ*. Therefore, it can be written with the ansatz for

*w*

_{ n }(

*x*) from Eqs. (7) and its three derivatives as a matrix equation

with, for the sake of printability, the *κ* dependent functions *c*_{1}(*κ*)= sin(*x*_{1}*κ*), *c*_{2}(*κ*)= cos(*x*_{1}*κ*), *c*_{3}(*κ*)= sinh(*x*_{1}*κ*), *c*_{4}(*κ*)= cosh(*x*_{1}*κ*), *c*_{5}(*κ*)= sin((*x*_{2}−*x*_{1})*κ*), *c*_{6}(*κ*)= cos((*x*_{2}−*x*_{1})*κ*), *c*_{7}(*κ*)= sinh((*x*_{2}−*x*_{1})*κ*), and *c*_{8}(*κ*)= cosh((*x*_{2}−*x*_{1})*κ*), and the constants *μ*_{1}=*M*_{1}/*m* and *μ*_{2}=*M*_{2}/*m*.

In general, Eq. (14) has only the trivial solution *a*_{ i }=0 for all *i*=1,…,8, which expresses that the system cannot oscillate at all frequencies *κ*. The eigenmodes can be found by numerically searching for the parameters *κ* that satisfy det(**M**(*κ*))=0. The resulting *κ*_{ n } yield with Eq. (8) the discrete spectrum of eigenfrequencies *ω*_{ n }. If desired, the system of eight linear equations expressed by Eq. (14) can be solved, too, in order to obtain the shape functions *y*_{ n }(*x*) for a specific mode *n*, by arbitrarily choosing one coefficient, e.g. *a*_{n1}=1.

## Notes

### Funding

This work was financially supported by the German Aerospace Center (DLR), Project No. 50 RS 1301.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

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