A Vibrometer Based on Magnetorheological Optical Resonators

vibration

Article A Vibrometer Based on Magnetorheological Optical Resonators

Edoardo Rubino 1,* and Tindaro Ioppolo 2

1 Mechanical and Industrial Engineering Department, University of Wisconsin Platteville, Platteville, WI 53818, USA 2 Mechanical Engineering Department, Southern Methodist University, Dallas, TX 75275, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-(608)-342-6058

Received: 23 August 2018; Accepted: 9 October 2018; Published: 17 October 2018

Abstract: This paper addresses the feasibility of an optical vibrometer that is based on the shift of the optical modes, also known as whispering gallery modes (WGMs), of a magnetorheological optical resonator. The optical resonator that is used in this study is fabricated by mixing polyvinyl chloride plastisol with magnetically polarizable particles. When a permanent magnet that is located nearby the optical resonator is moved, it induces a perturbation of the morphology of the resonator, due to the magnetostrictive effect. This change in the morphology induces a shift in the optical modes of the resonator. The shift of the optical modes can be related to the displacement of the permanent magnet. The proposed sensor concept is based on monitoring the displacement of a tiny magnet that is attached to a moving surface. The optical quality factor of the resonator used in these studies was of the order of 106. The experimental results show a sensitivity of 0.32 pm/µm and a resolution that is less than 300 nm.

Keywords: vibrometer; magnetorheological; optical resonator; composites; smart materials; whispering gallery modes

1. Introduction Several techniques have been developed so far, to detect and monitor the amplitude and the frequency of the displacement of vibrating systems. Laser vibrometers are interferometric instruments [1] that make use of the coherence properties of a laser beam [2]. A laser light is pointed toward the vibrating object, and the backscattered light is detected with a photodiode. By comparing the Doppler frequency between the source beam and the backscattered light [3], it is possible to determine the vibrating characteristics of the tested object. Laser vibrometers have been studied extensively [4], and they are employed in several fields such as structural health monitoring [5,6] and fruit textural changes [7,8]. Other studies reported that it is possible to obtain a resolution of the order of attometers with this technique [9]. Other optical methods have been developed to determine the displacement of vibrating objects. Light emission diodes (LED) were, for instance, used to measure the bending and torsional vibrations of pipes [10–12]: the pipe is located between a projector and a receiver to create a shaded area on the receiver. The vibration of the pipe can then be detected by measuring the amount of light that reaches the receiver. With this technique, a resolution of 60 nm can be obtained, and the sensor size is in the order of tens of centimeters. Optical digital techniques have been developed to detect the displacement of flexible bridges for structural analysis purposes [13]: the motion of a marked panel placed on the bridge was followed by a camera placed on a fixed point, and the recorded images were analyzed through digital image processing software. The resolution and

Vibration 2018, 1, 239–249; doi:10.3390/vibration1020017 www.mdpi.com/journal/vibration Vibration 2018, 1 240 Vibration 2018, 2, x FOR PEER REVIEW 2 of 11 sensitivitysoftware. ofThe the resolution proposed and device sensitivity are a function of the of proposed the camera device and lensesare a used,function as well of the as thecamera distance and betweenlenses used, the cameraas well andas the the distance targeted between point. the camera and the targeted point. AnotherAnother opticaloptical systemsystem thatthat isis usedused toto developdevelop vibrometersvibrometers makesmakes useuse ofof anan opticaloptical fiberfiber thatthat pointspoints the the light light to to thethe surfacesurface ofof thethe vibratingvibrating objectobject andandof of aa photodiodephotodiodethat that measuresmeasuresthe the variationvariation ofof thethe intensityintensity ofof the reflectedreflected light [14,15]. [14,15]. With With this this method, method, a asensitivity sensitivity of of0.893 0.893 V/mm V/mm in the in thefrequency frequency range range between between 75 and 75 275 and Hz 275 was Hz obtained. was obtained. Also, this Also, technique this technique was used wasfor multipoint used for multipointmeasurements measurements [16], obtaining [16], a obtaining resolution a resolutionin the order in of the 1 orderμm. Another of 1 µm. study Another reported study the reported optical thedetection optical of detection the resonant of the frequency resonant of frequency a quartz crys of atal quartz resonator crystal [17]. resonator A resolution [17]. in A the resolution order of in10 theμm order has been of 10 reachedµm has in been developing reached a in displacement developing asensor displacement based on sensora Fabry-Perot based on device a Fabry-Perot [18], or of device142 μm [18 with], or a of vibrometer 142 µm with for a vibrometercryogenic applications for cryogenic ba applicationssed on fiber based Bragg on gratings fiber Bragg [19]. gratings In addition, [19]. Inrecent addition, studies recent have studies reported have reported that the that optical- the optical-knifeknife edge edge technique technique allows allows for for displacementdisplacement √ measurementsmeasurements inin thethe rangerange betweenbetween 13 MHz andand 895 MHz, with with a a resolution resolution of of 455 455 fm/ fm/√HzHz [20]. [20 In]. Inthis this paper, paper, we we present present a a novel novel technique technique to to meas measureure the out-of-plane displacementdisplacement ofof aa vibratingvibrating objectobject oror surface.surface. TheThe sensingsensing conceptconcept isis basedbased onon thethe shiftshift ofof thethe morphology-dependentmorphology-dependent resonancesresonances (MDR),(MDR), also also known known as whisperingas whispering gallery gallery modes modes (WGM) (WGM) of a magnetorheological of a magnetorheological spherical resonator.spherical Theresonator. WGM phenomenonThe WGM phenomenon has been used has in been the pastused for in variousthe past applications, for various applications, due to the high due optical to the qualityhigh optical factor ofquality the optical factor modes. of the These optical micro-cavities modes. These have beenmicro-cavities used for the have development been used of devicesfor the fordevelopment telecommunication of devices (filtering, for telecommunication switches, multiplexing, (filtering, etc.)switches, [21–24 multiplexing,] as well as mechanical etc.) [21–24] [25 as– well32], thermalas mechanical [33–36 ],[25–32], and biological thermal [[33–36],37–42] sensing and biological applications. [37–42] sensing applications.

2.2. SensorSensor ConceptConcept TheThe proposed proposed sensing sensing modality modality exploits exploits the optical the op modestical ofmodes spherical of dielectricspherical opticaldielectric resonators. optical Theresonators. optical resonancesThe optical inresonances a spherical in resonatora spherical can resonator be described can be using described geometric using optics geometric as long optics as theas long wavelength as the wavelength of the light of that the is light used that to excite is used the to optical excite modesthe optical is much modes smaller is much than smaller the radius than ofthe the radius resonator. of the Usingresonator. this description,Using this description, an optical resonancean optical resonance is excited whenis excited the lengthwhen the of thelength path of ofthe the path light of traveling the light on traveling the internal on the surface internal of the surf resonatorace of the is resonator a multiple is integer a multiple of the integer wavelength, of the namelywavelength, when namely 2πrn = whenlλ, where 2πrn r= islλ the, where radius r is of the the radius microsphere, of the microsphere,l is an integer, l is ann isinteger, the index n is the of refractionindex of refraction of the resonator of the resonator and λ is theand wavelength λ is the wavelength of the light. of the Figure light.1 Figureshows 1 the shows coupling the coupling of the opticalof the optical ﬁber and fiber the and optical the optical resonator. resonator.

From Laser To Photodiode

FigureFigure 1.1. SchematicSchematic ofof thethe pathpath ofof thethe lightlight fromfrom thethe ﬁberfiber totothe theresonator resonator and and back back to to the the ﬁber. fiber.

When excited, the optical resonances, also known as optical modes or Whispering Gallery When excited, the optical resonances, also known as optical modes or Whispering Gallery Modes Modes (WGMs), are seen as sharp dips in the transmission spectrum (see Figure 2). If the radius or (WGMs), are seen as sharp dips in the transmission spectrum (see Figure2). If the radius or the index the index of refraction (or both) of the resonator are perturbed by an external effect, a shift of the of refraction (or both) of the resonator are perturbed by an external effect, a shift of the optical modes optical modes (Δλ) can be written as follows: (∆λ) can be written as follows: ∆λ Δ ∆rΔ ∆Δn = = + (1)(1) λ r n TheThe variationvariation ofof thethe radiusradius andand thethe indexindex ofof refractionrefraction ofof thethe microspheremicrosphere representrepresent thethe strainstrain effecteffect andand stressstress effecteffect respectively.respectively. However, However, as as reported reported in in previous previous studies studies [ 43[43,44],44] the the stressstress effecteffect is negligible compared with the strain effect. Therefore, the relative shift of the optical modes can be expressed as Δλ/λ = Δr/r.

Vibration 2018, 1 241 Vibration 2018, 2, x FOR PEER REVIEW 3 of 11 Vibration 2018, 2, x FOR PEER REVIEW 3 of 11 is negligible compared with the strain effect. Therefore, the relative shift of the optical modes can be expressed as ∆λ/λ = ∆r/r.

Amplitude Amplitude

Δλ Δλ

Light Light λ λ + Δλ wavelength λ λ + Δλ wavelength Figure 2. Schematic of the transmission spectrum. FigureFigure 2. 2. SchematicSchematic of of the the transmission transmission spectrum. spectrum. To demonstrate the proposed sensing modality, we used a polymeric optical resonator that is To demonstrate the proposed sensing modality, we used a polymeric optical resonator that is dopedTo with demonstrate magnetically the proposed polarizable sensing particles. modality, When we the used resonator a polymeric is placed optical in the resonator vicinity thatof a doped with magnetically polarizable particles. When the resonator is placed in the vicinity of a ispermanent doped with magnet, magnetically the magnetic polarizable forces acting particles. on it Whenwill induce the resonator an elastic is deformation placed in the and vicinity therefore of permanent magnet, the magnetic forces acting on it will induce an elastic deformation and therefore aa permanentshift of the magnet, optical the modes. magnetic When forces the actingpermanent on it will magnet induce is anattached elastic deformationto a vibrating and object, therefore the a shift of the optical modes. When the permanent magnet is attached to a vibrating object, the aamplitude shift of the of optical the vibration modes. Whencan be the measured permanent by magnetplacing is the attached optical to resonator a vibrating in object,the proximity the amplitude of the amplitude of the vibration can be measured by placing the optical resonator in the proximity of the ofvibrating the vibration magnet can and be by measured observing by the placing shift theof the optical optical resonator modes in in the the transmission proximity of spectrum the vibrating (see vibrating magnet and by observing the shift of the optical modes in the transmission spectrum (see magnetFigure 2). and by observing the shift of the optical modes in the transmission spectrum (see Figure2). Figure 2). 3.3. AnalysisAnalysis 3. Analysis Here,Here, wewe assumedassumed thatthat thethe magnetorheologicalmagnetorheological micro-opticalmicro-optical resonatorresonator isis placedplaced nearbynearby thethe Here, we assumed that the magnetorheological micro-optical resonator is placed nearby the surfacesurface ofof aa movablemovable tinytiny permanentpermanent magnet,magnet, asas shownshown inin FigureFigure3 .3. surface of a movable tiny permanent magnet, as shown in Figure 3.

Permanent magnet Permanent magnet z B z B θ θ R R y y x x

FigureFigure 3.3.Schematic Schematic ofof thethe microspheremicrosphere subjectedsubjected toto thethe inductiveinductive magneticmagnetic ﬁeldfield BB generatedgenerated byby thethe Figure 3. Schematic of the microsphere subjected to the inductive magnetic field B generated by the permanentpermanent magnet.magnet.θ θ,, x, x, y y and and z z are are the the spatial spatial coordinates. coordinates. permanent magnet. θ, x, y and z are the spatial coordinates. Since the magnet is allowed to move in the direction parallel to the z-axis (see Figure 3), the SinceSince the the magnet magnet is is allowed allowed to tomove move in inthe the direct directionion parallel parallel to the to thez-axis z-axis (see (seeFigure Figure 3), the3), intensity of the inductive magnetic field at the microsphere location changes, inducing changes in intensitythe intensity of the of inductive the inductive magnetic magnetic field ﬁeld at the at themicrosphere microsphere location location changes, changes, inducing inducing changes changes in the magnetic force acting on it. These forces, acting on the optical resonator, induce a shift in the thein themagnetic magnetic force force acting acting on onit. it.These These forces, forces, acting acting on on the the optical optical resonator, resonator, induce induce a ashift shift in in the the optical resonances as in [43]: opticaloptical resonances resonances as as in in [43]: [43]: ∆ 7−4 − 3 ∆ = − 7−4 −1 − 32 (2) ∆λ 47 −754ν 2 b1−b2 3B 2 = =− − (µr−−11) + (2)(2) 475+ 22 λ 4G(7 5ν) µ0 µ0(µr + 2) where G, ν, and μr are respectively, the shear modulus, the Poisson ratio, and the relative magnetic where G, ν, and μr are respectively, the shear modulus, the Poisson ratio, and the relative magnetic permeability of the microsphere. Coefficients b1 and b2 are defined as =⁄ and =⁄ permeability of the microsphere. Coefficients b1 and b2 are defined as =⁄ and =⁄

Vibration 2018, 1 242

where G, ν, and µr are respectively, the shear modulus, the Poisson ratio, and the relative magnetic Vibrationpermeability 2018, 2, x FOR of the PEER microsphere. REVIEW Coefficients b1 and b2 are defined as b1 = ∂µ∂eii and b2 = ∂µ∂4 of 11ekk , where eii and ekk are the normal components of the strain. The magnetic permeability of the surrounding fluid, where (air) is eii close and toekk thatare ofthe the normal vacuum, components and it isindicated of the strain. with Theµ0. Bmagneticis the intensity permeability of the externalof the surroundinginductive magnetic fluid (air) field. is close to that of the vacuum, and it is indicated with μ0. B is the intensity of the externalThe above inductive relationship magnetic shows field. that the WGM shift is a quadratic function of the applied external inductiveThe above magnetic relationship field, and shows it is that also the a function WGM shift of the is a elastic quadratic and magneticfunction of properties the applied of the external optical inductiveresonator. magnetic If the spatial field, distributionand it is also of a thefunction inductive of the magnetic elastic and field magnetic is known, properties the induced of the WGM optical shift resonator.can be related If the to spatial the position distribution of the magnet.of the inductive Thus, Equation magnetic (2) field can be is writtenknown, as: the induced WGM shift can be related to the position of the magnet. Thus, Equation (2) can be written as: ∆λ = k B(z)2 (3) λ = (3) [ ( − )] n o where == −− 3 7 4ν −1( − )2 + b1 −(seeb2 Equation (2)). If we assume that the where k 2 µr 1 µ (see Equation (2)). If we assume that the [4G(7+5ν)µ0(µr+2) ] 0 displacementdisplacement of of the the magnet magnet is isrelatively relatively small, small, we we can can linearize linearize Equation Equation (3) (3) as: as:

Δ =∆ 2 λ ∆0 (4) = 2 k B(Z )B (Z )∆z (4) λ 0 0 Here, Z0 is the initial distance between the surface of the magnet and the center of the sphere, B’ Here, Z is the initial distance between the surface of the magnet and the center of the sphere, B’ is is its derivative0 and Δz is the displacement of the magnet. From the above equation, we can calculate z theits displacement derivative and as∆ ais function the displacement of the WGM of the shift. magnet. Therefore, From the if abovea magnet equation, is placed we can on calculatea moving the surface,displacement the displacement as a function of that of the surface WGM can shift. be measured Therefore, using if a magnet the presented is placed approach. on a moving surface, the displacement of that surface can be measured using the presented approach. 4. Results 4. Results A schematic of the optoelectronic setup used in this experiment is reported in Figure 4. A schematic of the optoelectronic setup used in this experiment is reported in Figure4.

Optical Signal Function Tunable Resonator Photodiode Generator PC/Data Laser Acquisition

Beam Laser Splitter Controller

Reference Photodiode

FigureFigure 4. 4.SchematicSchematic of ofthe the optoelectronic optoelectronic setup. setup.

ThisThis is isanalogous analogous to to that that one one used used in in our our previous previous studies studies [25,45]. [25,45 ].Briefly, Briefly, a asingle single mode mode optical optical fiberfiber is iscoupled coupled to to a atunable tunable distributed distributed feedbac feedbackk (DFB) (DFB) laser laser diode diode that that has has a anominal nominal central central wavelengthwavelength of of1.312 1.312 μmµ mand and 10 10mW mW maximum maximum power. power. A section A section of the of optical the optical fiber fiberis tapered is tapered (by heating(by heating and stretching and stretching the fiber) the to fiber) couple to evanescent couple evanescentlyly the laser the light laser from light the fromtapered the fiber tapered into the fiber microsphere.into the microsphere. The other Theend otherof the end optical of the fibe opticalr is fiberbrought is broughtto a photodiode to a photodiode to monitor to monitor the transmissionthe transmission spectrum. spectrum. The output The output from the from photod the photodiodesiodes is analyzed is analyzed with in-house with in-house software software that calculatesthat calculates the WGM the WGMshift. The shift. microsphere The microsphere was fabricated was fabricated using polyvinyl using polyvinyl chloride-plastisol chloride-plastisol (PVC) that(PVC) was thatmixed was with mixed magnetically with magnetically polarizable polarizable particles. particles.The mixture The of mixture polymer–magnetic of polymer–magnetic particles wasparticles placed was in placedan oven in at an a oven temperature at a temperature of 230 °C of 230for ◦20C formin. 20 The min. tip The of tipan of optical an optical fiber fiber with with a diametera diameter of 125 of 125μmµ wasm was immersed immersed in the in thepolymer polymer mixture, mixture, and and due dueto surface to surface tension tension and andgravity, gravity, a spherical droplet was formed at the tip of the fiber. The solid sphere was formed by cooling it at room temperature. Once the solid sphere was formed, it was coated with a thin layer of pure polymer (PVC) that served as an optical wave-guide for the propagation of the optical modes. A photograph of a resonator used in this study is reported in Figure 5. It has a radius of ~600 μm and a

Vibration 2018, 1 243 a spherical droplet was formed at the tip of the ﬁber. The solid sphere was formed by cooling it at roomVibration temperature. 2018, 2, x FOR OncePEER REVIEW the solid sphere was formed, it was coated with a thin layer of pure polymer5 of 11 (PVC) that served as an optical wave-guide for the propagation of the optical modes. A photograph of avolume resonator fraction used in(the this ratio study between is reported the in volume Figure5 .of It hasparticles a radius and of ~600the volumeµm and of a volumethe composite fraction (theelastomer)Vibration ratio 2018 between of, 2 ,0.32. x FOR the PEER volume REVIEW of particles and the volume of the composite elastomer) of 0.32. 5 of 11 volume fraction (the ratio between the volume of particles and the volume of the composite elastomer) of 0.32.

Figure 5. AA picture picture of the magnetorheological spherical resonator.

An experiment was carried out to characterize the response of the elastic magnetorheological An experiment was carried out to characterize the response of the elastic magnetorheological sphere to the appliedFigure external 5. A magnetic picture of thefield. magnetorheological The microsphere spherical was placed resonator. in a uniform inductive sphere to the applied external magnetic field. The microsphere was placed in a uniform inductive magnetic field that was generated by a magnetic coil. The intensity of the field was changed by magneticAn experiment field that was was generated carried out by to a characterize magnetic coil. the Theresponse intensity of the of elastic the field magnetorheological was changed by changing the current flowing into the coil, and it was measured using a Gauss meter. In addition, the changingsphere to thethe currentapplied flowing external into magnetic the coil, field. and The it was microsphere measured was using placed a Gauss in a meter. uniform In addition,inductive temperature was kept constant during the measurements. Figure 6 shows the relationship between themagnetic temperature field wasthat keptwas constantgenerated during by a themagnetic measurements. coil. The Figure intensity6 shows of the the field relationship was changed between by the applied external inductive magnetic field and the induced WGM shift. As shown in the figure, thechanging applied the external current inductive flowing into magnetic the coil, field and and it was the measured induced WGMusing a shift. Gauss As meter. shown In in addition, the figure, the there is a quadratic dependency between the induced WGM shift and the applied inductive theretemperature is a quadratic was kept dependency constant betweenduring the the measur inducedements. WGM Figure shift and 6 shows the applied the relationship inductive magnetic between magnetic field. In addition, we carried out a series of experiments where the optical resonator was field.the applied In addition, external we carriedinductive out magnetic a series of field experiments and the induced where the WGM optical shift. resonator As shown was in placed the figure, near placed near a disk-shaped permanent magnet (K&J Magnetics D8H2-Plumsteadville, PA, USA), athere disk-shaped is a quadratic permanent dependency magnet (K&J between Magnetics the D8H2-Plumsteadville, induced WGM shift PA, and USA), the axiallyapplied magnetized inductive axially magnetized with a thickness of 5.08 mm and a diameter of 12.7 mm, as shown in Figure 7. withmagnetic a thickness field. In of 5.08addition, mm andwe acarried diameter out ofa 12.7series mm, of experiments as shown in Figurewhere 7the. optical resonator was placedInitially, near thea disk-shaped permanent permanent magnet0 was 10203040 magnet placed on(K&J a translationMagnetics stage,D8H2-Plumsteadville, while the microsphere PA, USA), was keptaxially steady magnetized at a distance withZ a0 thicknessof0 2 mm fromof 5.08 the mm surface and a of diameter the permanent of 12.7 magnetmm, as shown (see Figure in Figure7). 7. -2 -4 0 10203040 0 -6 -2 -8 -4 -10

WGM Shift, pm Shift, WGM -6 -12 -8 Δλ = −9.3610 Δ -14 -10 -16

WGM Shift, pm Shift, WGM -12 -18 Δλ = −9.3610Δ -14 Magnetic Field, mT -16 Figure 6. Relationship between the applied external inductive magnetic field and the whispering -18 gallery mode (WGM) shift. Magnetic Field, mT

FigureInitially,Figure 6.6. the RelationshipRelationship permanent betweenbetween magnet thethe was appliedapplied placed externalexternal on a inductive inductivetranslation magneticmagnetic stage, ﬁeldwhilefield andand the thethe microspherewhispering whispering was gallery mode (WGM) shift. kept gallerysteady modeat a distance (WGM) shift.Z0 of 2 mm from the surface of the permanent magnet (see Figure 7).

Initially, the permanent magnet was placed on a translation stage, while the microsphere was kept steady at a distance Z0 of 2 mm from the surface of the permanent magnet (see Figure 7).

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From Laser OpticalOptical Fiber Fiber From Laser

DisplacementDisplacement AccelerometerAccelerometer OpticalOptical ResonatorResonator TranslationTranslation StageStage SphereSphere Holder Holder

PermanentPermanent Magnet Magnet ToTo Photodiode Photodiode

FigureFigure 7. 7.Schematic Schematic of of the the experimental experimental setup. setup.

TheThe magnet magnet was was then then moved moved in in steps steps with with an an amplitude amplitude of of 25.4 25.4 μµ m,μm, and and the the WGM WGM shift shift was was recorded.recorded. Figure Figure Figure 88 shows8shows shows thethe the responseresponse response ofof of thethe the opticaloptica optical resonatorresonatorl resonator toto to thethe the stepwisestepwise stepwise displacementdisplacement displacement ofof of thethe the permanentpermanent magnet. magnet. As As As shown shown shown in in in Figure Figure 88,, 8, thethe the WGMWG WGMM shiftshift shift followedfollowed followed thethe the positionposition position ofof of thethe the permanentpermanent permanent magnetmagnet relative relative to to the the microsphere microsphere very very well. well.

4040 140140 WGMWGM Shift Shift 35 m 35 120120 m μ μ 30 MagnetMagnet Displacement Displacement 30 100100 2525 8080 2020 6060 1515 WGM SHift, pm SHift, WGM WGM SHift, pm SHift, WGM 40 1010 40

20 Magnet Displacement, 5 5 20 Magnet Displacement, 0 0 0 0 05100510 Time,Time, s s

FigureFigure 8. 8.WGM WGM shift shift and and displacement displacement of of the the perm permanent permanentanent magnet magnet as as a functiona function of of time. time.

As the magnet is moved far away from the optical resonator, the WGM experienced a red shift, As the magnet is moved far away from the opti opticalcal resonator, resonator, the WGM experienced a a red red shift, shift, since the intensity of the inductive magnetic field decreased and the microsphere tended to reach its since the intensity of the inductive magnetic field ﬁeld decreased and the microsphere tended to reach its original size (ΔR > 0). original size (Δ∆R > 0). 0). Figure 9 shows the relationship between the displacement of the permanent magnet and the Figure 99 showsshows thethe relationshiprelationship betweenbetween thethe displacementdisplacement ofof thethe permanentpermanent magnetmagnet andand thethe induced WGM shift together with the analytical results obtained from Equation (4) using the values induced WGMWGM shiftshift togethertogether withwith the the analytical analytical results results obtained obtained from from Equation Equation (4) (4) using using the the values values of ofof B (BZ(0Z) 0and) and B ’(BZ’(0Z) 0and) and k, kdiscussed, discussed previously. previously. B(Z0) and B’(Z0) and k, discussed previously.

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50 50 ExperimentExperiment 40 40 E=25kPaE=25kPa E=30kPaE=30kPa 30 30 E=35kPaE=35kPa

20 20 WGM Shift, pm Shift, WGM WGM Shift, pm Shift, WGM 10 10

0 0 00 20406080100120140 20406080100120140 Magnet Displacement, μm Magnet Displacement, μm

FigureFigureFigure 9. Relationship Relationship9. Relationship between between between the the theWGM WGM WGM shift shift and and the thedisplacement displacement of the of themagnet. magnet.

As Asshown, shown, there there was was a nearly a nearly linear linear relationship relationship between between the the displacement displacement of theof the magnet magnet and and As shown, there was a nearly linear relationship between the displacement of the magnet and the thethe WGM WGM shift. shift. In theIn the same same figure, figure, the the values values of theof the WGM WGM shift shift (calculated (calculated using using Equation Equation (2)) (2)) for for WGM shift. In the same ﬁgure, the values of the WGM shift (calculated using Equation (2)) for three threethree different different values values of Young’sof Young’s modulus modulus of theof the polymer polymer (Young’s (Young’s modulus modulus of theof the polymeric polymeric matrix matrix different values of Young’s modulus of the polymer (Young’s modulus of the polymeric matrix varies variesvaries ranges ranges from from 25 kPa25 kPa to 35to kPa35 kPa [28]) [28]) were were also also reported. reported. ranges from 25 kPa to 35 kPa [28]) were also reported. MeasurementsMeasurements of theof the sensitivity sensitivity were were repeated repeated for for different different values values of theof the initial initial distance, distance, Z0 , Z0, Measurements of the sensitivity were repeated for different values of the initial distance, betweenbetween the the resonator resonator and and the the surface surface of theof the perman permanentent magnet. magnet. Figure Figure 10 shows10 shows the the results results of theseof these Z0, between the resonator and the surface of the permanent magnet. Figure 10 shows the results of measurements.measurements. As Asexpected, expected, the the sensitivity sensitivity decrease decreases withs with an anincrease increase of theof the distance distance between between the the these measurements. As expected, the sensitivity decreases with an increase of the distance between permanentpermanent magnet magnet and and the the optical optical resonator, resonator, sinc since thee the strength strength of ofthe the inductive inductive magnetic magnetic field field the permanent magnet and the optical resonator, since the strength of the inductive magnetic ﬁeld decaysdecays with with increasing increasing distance. distance. decays with increasing distance.

0.350.35

0.3 0.3

0.250.25

0.2 0.2

0.150.15

0.1 0.1 Sensitivity pm/µm Sensitivity Sensitivity pm/µm Sensitivity 0.050.05

0 0 1.51.5 2.5 2.5 3.5 3.5 4.5 4.5 5.5 5.5

Distance,Distance, Z0, mm Z0, mm

FigureFigure 10. 10.SensitivitySensitivity Sensitivity as as aas a function functiona function of of the of theinitial initial di distancestance distance between between the thepermanent permanent magnet magnet and and the the opticaloptical resonator. resonator.

InIn additionIn addition to to these to these studies, studies, we we carried carried out out ex experiments perimentsexperiments to study to study the the dynamic dynamic response response of the of the opticaloptical resonator.resonator. resonator. For For theseFor these these experiments, experiments, experiments, the permanentth eth permanente permanent magnet magnet wasmagnet mounted was was mounted ontomounted a plate onto onto connected a platea plate connectedto aconnected shaker to that toa wasshakera shaker driven that that harmonicallywas was driven driven harmonically by harmonically a function generator.by bya functiona function An accelerometergenerator. generator. An mountedAn accelerometer accelerometer on the mountedmovablemounted plateon onthe wasthe movable usedmovable to plate calculate plate was was theused used amplitude to tocalcul calculate ofate thethe displacementthe amplitude amplitude of of ofthe the the displacement permanent displacement magnet. of ofthe the permanentFigurepermanent 11 shows magnet. magnet. the Figure WGM Figure 11 shift 11shows shows and the the the WGM magnet WGM shift displacementshift and and the the magnet magnet as a displacement function displacement of time as aas for function a anfunction input of of timefrequencytime for for an ofaninput 30 input Hz. frequency frequency As shown of 30of in Hz.30 ﬁgure, Hz. As Asshown the shown WGM in figure,in shift figure, followsthe the WGM WGM the shift displacement shift follows follows the ofthe displacement the displacement permanent of of themagnetthe permanent permanent very well. magnet magnet very very well. well.

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2020 100100

15 m 15 m μ μ 1010 5050 55 00 00 -5-5 WGM Shift, pm WGM Shift, WGM Shift, pm WGM Shift, -10-10 -50-50

-15 Magnet Displacement,

-15 Magnet Displacement, -20-20 -100-100 0.000.00 0.05 0.05 0.10 0.10 0.15 0.15 0.20 0.20 WGMWGM Shift Shift MagnetMagnet Displacement Displacement Time,Time, s s

FigureFigure 11.11. WGMWGM shift shift inducedinduced byby aa harmonicharmonic displacementdisplacement ofof thethe magnet.magnet.

FigureFigure 12 1212 shows showsshows the thethe WGM WGMWGM shift shiftshift as a asas function aa functionfunction of the ofof magnet thethe magnetmagnet displacement displacementdisplacement using the usingusing data the reportedthe datadata reportedinreported Figure in in11 FigureFigure. As expected,11.11. AsAs expected,expected, the sensitivity thethe sensitivitysensitivity dλ/d ddzλλwas/d/dzz waswas the thethe same samesame as as theas thethe one oneone reported reportedreported in inin Figure FigureFigure9 99 (static(static measurements).measurements).measurements). In InIn addition, addition,addition, Figure FigureFigure 11 1111 shows showsshows that thatthat the thethe hysteresis hysteresishysteresis of ofof the thth systemee systemsystem was waswas negligible negligiblenegligible in inthein thethe range rangerange of theofof thethe measured measuredmeasured displacement. displacement.displacement.

2020 1515 1010 55 00 -5-5 WGM Shift, pm WGM Shift, WGM Shift, pm WGM Shift, -10-10 -15-15 -20-20 -100-100 -50 -50 0 0 50 50 100 100 MagnetMagnet Displacement, Displacement, μ μmm

FigureFigure 12.12. RelationshipRelationship betweenbetween thethe inducedinduced WGWGMWGMM shiftshift andand thethe magnetmagnet displacement.displacement.

FigureFigure 13 1313 shows showsshows the thethe sensitivity sensitivitysensitivity as asas a a functiona functionfunction of ofof the thethe input inputinput frequency frequencyfrequency for forfor four fourfour different differentdifferent values valuesvalues of the ofof Vibrationthe distance 2018, 2 (, Zx 0FOR) between PEER REVIEW the sphere and the surface of the permanent magnet. 9 of 11 thedistance distance (Z0 )(Z between0) between the the sphere sphere and and the the surface surface of the of the permanent permanent magnet. magnet. AsAs shownshown inin FigureFigure 13,13, thethe sensitivitysensitivity asas aa functionfunction ofof thethe inputinput frequencyfrequency waswas nearlynearly constantconstant inin thethe investigatedinvestigated frequencyfrequency0.5 range.range. Again,Again, thethe sensitivitysensitivity ofof thethe opticaloptical resonatorresonator decreaseddecreased withwith anan increaseincrease inin thethe distancedistance ZZ00 betweenbetween thethe microspheremicrosphere andand thethe surfacesurface ofof thethe permanentpermanent magnet.magnet. TheThe 0.4 2 mm 3 mm 4 mm 5 mm maximummaximum frequencyfrequency rangerange investigatedinvestigated inin thisthis studystudy waswas limitedlimited byby thethe hardwarehardware usedused inin thethe experiments.experiments. However,However, thethe0.3 bandwidthbandwidth ofof thethe prproposedoposed sensingsensing techniquetechnique isis limitedlimited byby thethe mechanical resonance of the sphere. This limit can be tuned by choosing the sphere’s material and mechanical resonance of the0.2 sphere. This limit can be tuned by choosing the sphere’s material and size.size.

InIn particular,particular, aa smallersmaller pm/µm Sensitivity, 0.1 spheresphere wouldwould leadlead toto aa higherhigher bandwidthbandwidth (smaller(smaller sphere’ssphere’s mass),mass), duedue toto thethe factfact thatthat thethe mechanicalmechanical resonanceresonance occursoccurs laterlater inin thethe frequencyfrequency spectrum.spectrum. InIn addition,addition, aa 0 spheresphere fabricatedfabricated withwith aa stifferstiffer050100150 mamaterialterial (higher(higher springspring constant)constant) wowoulduld leadlead toto aa higherhigher bandwidth.bandwidth. TheseThese factorsfactors areare typicaltypical forfor sensorssensors thatthat relyrely ononFrequency, mechanicalmechanical Hz deformation.deformation. AA softersofter materialmaterial wouldwould reducereduce thethe bandwidthbandwidth butbut increaseincrease thethe sensitivity.sensitivity. Thus,Thus, sensitivitysensitivity andand bandwidthbandwidth presentpresent aa tradeoff.tradeoff. Figure 13. Frequency response of the resonator in the range between 00 andand 150150 Hz.Hz.

5. Conclusions As shown in Figure 13, the sensitivity as a function of the input frequency was nearly constant in theThe investigated feasibility frequencyof a vibrometer range. based Again, on thethe sensitivityshift of the of optical the optical modes resonator of a magnetorheological decreased with anoptical increase resonator in the was distance demonstratedZ0 between analytically the microsphere and experimentally. and the surface For of small the permanent displacement, magnet. the Theinduced maximum WGM frequencyshift is nearly range a investigated linear function in this of studythe amplitude was limited of bythe the displacement. hardware used If a in tiny the permanent magnet is placed on a vibrating structure, the proposed approach can indeed be used to design a photonic vibrometer. The optical resonator showed a sensitivity up to 0.36 pm/μm and a resolution of 278 nm. The study shows that the sensitivity of the resonator can be increased by using a softer polymer, or by adding particles with higher magnetic permeability. Moreover, as showed by the experiments, the sensitivity can be further increased by reducing the initial distance between the surface of the permanent magnet and the optical resonator. Dynamic measurements were also carried out up to a frequency of 150 Hz. In this frequency range, the frequency response of the optical resonator was constant. In addition, the presented vibrometer does not require a reflective surface and/or focusing lens, and it can be operated at low frequencies without a change in sensitivity.

Funding: This research was funded by the National Science Foundation through grant CBET-1133876.

Acknowledgments: All sources of funding of the study should be disclosed. Please clearly indicate grants that you have received in support of your research work. Clearly state if you received funds for covering the costs to publish in open access.

Conflicts of Interest: The authors declare no conflict of interest.

References

1. Donati, S.; Norgia, M.; Giuliani, G. Self-mixing differential vibrometer based on electronic channel subtraction. Appl. Opt. 2006, 45, 7264, doi:10.1364/AO.45.007264. 2. Giuliani, G.; Bozzi-Pietra, S.; Donati, S. Self-mixing laser diode vibrometer. Meas. Sci. Technol. 2003, 14, 24– 32, doi:10.1088/0957-0233/14/1/304. 3. Stanbridge, A.B.; Ewins, D.J. Measurement of translational and angular vibration using a scanning laser Doppler vibrometer. In Proceedings of the First International Conference on Vibration Measurements by Laser Techniques: Advances and Applications, Ancona, Italy, 3–5 October 1994; Volume 2358, pp. 37–47, doi:10.1117/12.185354. 4. Rothberg, S.J.; Allen, M.S.; Castellini, P.; Di Maio, D.; Dirckx, J.J.J.; Ewins, D.J.; Halkon, B.J.; Muyshondt, P.; Paone, N.; Ryan, T.; et al. An international review of laser Doppler vibrometry: Making light work of vibration measurement. Opt. Lasers Eng. 2017, 99, 11–22, doi:10.1016/j.optlaseng.2016.10.023. 5. Siringoringo, D.M.; Fujino, Y. Experimental study of laser Doppler vibrometer and ambient vibration for vibration-based damage detection. Eng. Struct. 2006, 28, 1803–1815, doi:10.1016/j.engstruct.2006.03.006. 6. Siringoringo, D.M.; Fujino, Y. Noncontact operational modal analysis of structural members by laser doppler vibrometer. Comput. Civ. Infrastruct. Eng. 2009, 24, 249–265, doi:10.1111/j.1467-8667.2008.00585.x.

Vibration 2018, 1 247 experiments. However, the bandwidth of the proposed sensing technique is limited by the mechanical resonance of the sphere. This limit can be tuned by choosing the sphere’s material and size. In particular, a smaller sphere would lead to a higher bandwidth (smaller sphere’s mass), due to the fact that the mechanical resonance occurs later in the frequency spectrum. In addition, a sphere fabricated with a stiffer material (higher spring constant) would lead to a higher bandwidth. These factors are typical for sensors that rely on mechanical deformation. A softer material would reduce the bandwidth but increase the sensitivity. Thus, sensitivity and bandwidth present a tradeoff.

5. Conclusions The feasibility of a vibrometer based on the shift of the optical modes of a magnetorheological optical resonator was demonstrated analytically and experimentally. For small displacement, the induced WGM shift is nearly a linear function of the amplitude of the displacement. If a tiny permanent magnet is placed on a vibrating structure, the proposed approach can indeed be used to design a photonic vibrometer. The optical resonator showed a sensitivity up to 0.36 pm/µm and a resolution of 278 nm. The study shows that the sensitivity of the resonator can be increased by using a softer polymer, or by adding particles with higher magnetic permeability. Moreover, as showed by the experiments, the sensitivity can be further increased by reducing the initial distance between the surface of the permanent magnet and the optical resonator. Dynamic measurements were also carried out up to a frequency of 150 Hz. In this frequency range, the frequency response of the optical resonator was constant. In addition, the presented vibrometer does not require a reﬂective surface and/or focusing lens, and it can be operated at low frequencies without a change in sensitivity.

Author Contributions: E.R. and T.I. conceived the experiments; E.R. performed the measurements at Southern Methodist University during his Ph.D.; E.R. and T.I. analyzed the data; E.R. and T.I. wrote the paper. Funding: This research was funded by the National Science Foundation through grant CBET-1133876. Acknowledgments: All sources of funding of the study should be disclosed. Please clearly indicate grants that you have received in support of your research work. Clearly state if you received funds for covering the costs to publish in open access. Conﬂicts of Interest: The authors declare no conﬂict of interest.

References

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