applied sciences

Article Investigation on the Differential Quadrature Fabry–Pérot Interferometer with Variable Measurement Mirrors

Yi-Chieh Shih 1,* , Pi-Cheng Tung 1 , Wen-Yuh Jywe 2, Chung-Ping Chang 3, Lih-Horng Shyu 4 and Tung-Hsien Hsieh 2

1 Department of Mechanical Engineering, National Central University, Taoyuan 320, Taiwan; [email protected] 2 Smart Machinery and Intelligent Manufacturing Research Center, National Formosa University, Yunlin 632, Taiwan; [email protected] (W.-Y.J.); [email protected] (T.-H.H.) 3 Department of Mechanical and Energy Engineering, National Chiayi University, Chiayi 600, Taiwan; [email protected] 4 Department of Electro-Optical Engineering, National Formosa University, Yunlin 632, Taiwan; [email protected] * Correspondence: [email protected]



Received: 7 July 2020; Accepted: 4 September 2020; Published: 6 September 2020


Abstract: Due to the common path structure being insensitive to the environmental disturbances, relevant Fabry–Pérot interferometers have been presented for displacement measurement. However, the discontinuous signal distribution exists in the conventional Fabry–Pérot interferometer. Although a polarized Fabry–Pérot interferometer with low finesse was subsequently proposed, the signal processing is complicated, and the nonlinearity error of sub-micrometer order occurs in this signal. Therefore, a differential quadrature Fabry–Pérot interferometer has been proposed for the first time. In this measurement system, the nonlinearity error can be improved effectively, and the DC offset during the measurement procedure can be eliminated. Furthermore, the proposed system also features rapid and convenient replacing the measurement mirrors to meet the inspection requirement in various measuring ranges. In the comparison result between the commercial and self-developed Fabry–Pérot interferometer, it reveals that the maximum standard deviation is less than 0.120 µm in the whole measuring range of 600 mm. According to these results, the developed differential Fabry–Pérot interferometer is feasible for precise displacement measurement.

Keywords: differential Fabry–Pérot interferometer; homodyne interferometer; nonlinearity error; linear displacement

1. Introduction From the comparison between the resolution and measuring range of different measurement devices, high resolution and contactless measurement can be realized by the laser interferometer in various measuring ranges. Hence, the industrial application, including the calibration of the linear axis for the machine tool, the positioning of the microelectromechanical equipment, and the wafer dicing, must rely on it to ensure the quality of production and the high-precision inspection requirement [1–4]. Currently, the Michelson interferometer is the primary tool for displacement measurement in a large dynamic range. In the optical structure, the laser beam is divided into a reference beam and a measurement beam by a non-polarizing beam splitter (BS), and corresponding mirrors reflect each beam and then form the interferometric signal. The phase of this signal depends on the optical path difference between the reference beam and the measurement beam. Because the reference path is separate from

Appl. Sci. 2020, 10, 6191; doi:10.3390/app10186191 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 of 13

measurement beam by a non-polarizing beam splitter (BS), and corresponding mirrors reflect each Appl. Sci. 2020, 10, 6191 2 of 12 beam and then form the interferometric signal. The phase of this signal depends on the optical path difference between the reference beam and the measurement beam. Because the reference path is theseparate measurement from the path, measurement this kind of interferometerpath, this kind is susceptible of interferometer to relative is environmental susceptible to changes. relative Inenvironmental contrast, the Fabry–P changes.érot In interferometercontrast, the Fabry–Pé is basedrot on theinterferometer common path is based structure. on the For common this reason, path laserstructure. beams For propagate this reason, in the laser same beams optical propagate path, and in thisthe same interferometer optical path, possesses and this high interferometer resistance ofpossesses environmental high resistance disturbances of environmental that contain thermaldisturbances expansion, that contain micro-flow thermal gradient, expansion, and micro-flow the tiny vibrationgradient, [5 and–7]. the tiny vibration [5–7]. TheThe optical optical arrangement arrangement of of the the conventional conventional Fabry–P Fabry–Pérotérot interferometer, interferometer, which which contains contains a a laser laser lightlight source, source, a detector, a detector, and anand optical an optical cavity cavity composed composed of two planeof two mirrors plane (PMs) mirrors had (PMs) been proposed had been byproposed Charles Fabry by Charles and Alfred Fabry Pé rotand in Alfred 1897 (illustrated Pérot in 1897 in Figure (illustrated1a) [ 8]. in The Figure laser light1a) [8]. source The is laser incident light intosource the opticalis incident cavity into in the a nearlyoptical verticalcavity in direction, a nearly andvertical the direction, laser beam and is dividedthe laser intobeam numerous is divided transmittedinto numerous beams transmitted from the cavity beams reciprocally. from the cavity Then thereciprocally. interferometric Then signalthe interferometric can be acquired signal by the can detector.be acquired Because by the the detector. optical cavity Because of thethe conventionaloptical cavity Fabry–P of the conventionalérot interferometer Fabry–Pérot is composed interferometer of two PMsis composed with a high of reflectance, two PMswhich with resultsa high in reflectanc the discontinuouse, which signalresults distribution in the discontinuous with high finesse. signal Furthermore,distribution displacementwith high finesse. or vibration Furthermore, can only displa be realizedcement byor thevibration fringe countingcan only method,be realized and by it isthe rarelyfringe utilized counting in large-scalemethod, and dynamic it is rarely displacement utilized in measurements. large-scale dynamic displacement measurements.

(a) (b)

FigureFigure 1. 1.Optical Optical arrangement arrangement of of Fabry–P Fabry–Pérotérot interferometer: interferometer: ( a()a) Conventional Conventional structure, structure, and and ( b()b) quadraturequadrature phase-shifted phase-shifted fiber-optic fiber-optic structure. structure.

AccordingAccording to to the the development development of of the the optical optical encoder encoder and and the the interferometer interferometer system, system, to to improve improve thethe measurement measurement accuracy, accuracy, correlated correlated technologies, technologies including, including orthogonal orthogonal signal processingsignal processing and signal and subdivision,signal subdivision, have been have proposed. been proposed. Therefore, Therefor the quadraturee, the quadrature phase-shifted phase-shifted fiber-optic fiber-optic Fabry–P Fabry–érot interferometerPérot interferometer demonstrated demonstrated in Figure in 1Figureb had 1b been had presented been presented by Kent by A. Kent Murphy A. Murphy et al. inet 1999al. in [19999]. This[9]. structure This structure is based is onbased the on spatial the spatial phase-shifted phase-shifted method method to form to the form orthogonal the orthogonal signal, which signal, will which be obtainedwill be obtained by two detectors by two detectors (D1,D2). (D The1, D phase2). The shift phase is mainly shift is decidedmainly decided by the position by the position and angle and of angle the probeof the and probe the measured and the surface.measured For surface. this reason, For thethis measuring reason, the range measuring is limited range by the is performance limited by ofthe theperformance optical-mechanical of the optical-mechanical alignment. alignment. InIn the the recent recent study, study, polarized polarized Fabry–P Fabry–Pérotérot interferometer interferometer with with low low finesse, finesse, shown shown in in Figure Figure2, 2, hadhad been been proposed proposed by Changby Chang et al. et in al. 2013 in 2013 [10,11 [10,11].]. Theoctadic-wave The octadic-wave plate isplate placed is placed in the optical in the cavity,optical andcavity, its polarization and its polarization axis must beaxis the must same be as the that same of the as polarizing that of the beam polarizing splitter (PBS)beam to splitter introduce (PBS) the to orthogonalintroduce phasethe orthogonal shift between phase two shift interferometric between two inte signals.rferometric By reducing signals. the By reflectance reducing the of thereflectance plane mirror,of the continuousplane mirror, signal continuous distribution signal can distribution be acquired, can and be theacquired, interferometric and the interferometric signals (Is,Ip) can signals be s p 1 2 detected(I , I ) can by be the detected photodiodes by the (PD photodiodes1, PD2). The (PD orthogonal, PD ). The interferometric orthogonal interferometric signal, shown insignal, Figure shown3a, 0 canin beFigure expressed 3a, can by be Equations expressed (1) by and Equations (2) [12], (1) where and (2) A0 [12],is the where amplitude A is the of the amplitude incident of beam, the incident R and Tbeam, are the R reflectance and T are the and reflectance transmittance and oftransmittance the coated plane of the mirror, coated T’ plane is the mirror, transmittance T’ is the of transmittance the optical cavity,of the and opticalδ is thecavity, phase and di ffδ erence.is the phase In the difference. simulation, In optical the simula parameterstion, optical of R, T,parameters and T’ are of 0.25, R, T, 0.75, and andT’ are 0.86, 0.25, respectively. 0.75, and 0.86, The respectively. analysis method The analysis of the mean method normalization of the mean isnormalization adopted to evaluate is adopted the to amplitudeevaluate the and amplitude DC offset and of the DC orthogonal offset of the signal orthogonal in this study.signal in Theoretically, this study. Theoretically, if the signal amplitude if the signal is uniform change and there is no DC offset existing, the center (offset) of the orthogonal signal will Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 13 Appl. Sci. 2020, 10, 6191 3 of 12 amplitude is uniform change and there is no DC offset existing, the center (offset) of the orthogonal besignal at thewill zero be at points, the zero and points, it will becomeand it will a circle become shape a circle after shape the mean after normalization the mean normalization processing. processing. Interferometric signals (Is, Ip) can be processed by the mean normalization (Is-nom, Ip-nom) Interferometric signals (Is,Ip) can be processed by the mean normalization (Is-nom,Ip-nom) individually representedindividually inrepresented Equations in (3) Equations and (4). Then, (3) and the (4). normalized Then, the orthogonalnormalized signal orthogonal illustrated signal in illustrated Figure3b in Figure 3b can be obtained, where I_max, I_min, and μ are the maximum, minimum, and average can be obtained, where I_max,I_min, and µ are the maximum, minimum, and average intensity. intensity. 1 2 2 2 A0 T T Is = × × 0 (1) 2 2  π  1 + R T0 2 T R cos δ × − × 0 × × − 4

1 2 2 Appl. Sci. 2020, 10, x FOR PEER REVIEW 2 A0 T T 3 of 13 Ip = × × 0 (2) 2 2  π  1 + R T0 2 T R cos δ + 4 amplitude is uniform change and there is× no DC− offs× 0et × existing,× the center (offset) of the orthogonal Is µ signal will be at the zero points, andI sitnom will= become− a circle shape after the mean normalization(3) − Is_max I processing. Interferometric signals (Is, Ip) can be processed− s_min by the mean normalization (Is-nom, Ip-nom) individually represented in Equations (3) and (4). Then,Ip theµ normalized orthogonal signal illustrated Ip nom = − (4) in Figure 3b can be obtained, where I_max− , I_minI,p_max and μ Iarep_min the maximum, minimum, and average intensity.Figure 2. The optical arrangement of polarized Fabry–Pérot− interferometer. PBS, polarizing beam splitter; PD, photodiode.

×× Is = (1) × ×××( ) ×× Ip = (2) × ×××( )

Is-nom = (3) _ _

Ip-nom = (4) _ _ Figure 2.2. The opticaloptical arrangementarrangement ofof polarizedpolarized Fabry–PFabry–Pérotérot interferometer.interferometer. PBS, polarizingpolarizing beambeam splitter;splitter; PD,PD, photodiode.photodiode.

×× Is = (1) × ×××( ) ×× Ip = (2) × ×××( )

Is-nom = (3) _ _

Ip-nom = (4) _ _

(a) (b)

Figure 3. Orthogonal signal of polarizedpolarized Fabry–PFabry–Pérotérot interferometer. ( a)) Original signal; ( b) signal with mean normalization processing.

In this structure, structure, there there are are two two critical critical issues, issues, including including the the existing existing large large nonlinearity nonlinearity error error in inthe the signal signal and and involving involving the thecomplex complex signal signal processing processing for the for elimination the elimination of the of direct the direct current current (DC) (DC)offset. o Theffset. nonlinearity The nonlinearity error is caused error is by caused the DC by offset, the DCunequal offset, alternating unequal current alternating (AC) current amplitudes, (AC) amplitudes,and quadrature and phase quadrature errors that phase occur errors in the that orthogonal occur in signal. the orthogonal In the simulation signal. results In the in simulation Figure 3, resultsthe orthogonal in Figure signal3, the after orthogonal the mean signal normalization after the mean processing normalization reveals that processing the signal reveals possesses that thethe signalDC offset possesses and unequal the DC oACffset amplitudes, and unequal which AC amplitudes, will lead to which the nonlinearity will lead to theerror. nonlinearity Therefore, error. the Therefore,measurement the measurementaccuracy of this accuracy measurement of this measurement structure will structure be affected will be abyffected the bynonlinearity the nonlinearity error (a) (b)

Figure 3. Orthogonal signal of polarized Fabry–Pérot interferometer. (a) Original signal; (b) signal with mean normalization processing.

In this structure, there are two critical issues, including the existing large nonlinearity error in the signal and involving the complex signal processing for the elimination of the direct current (DC) offset. The nonlinearity error is caused by the DC offset, unequal alternating current (AC) amplitudes, and quadrature phase errors that occur in the orthogonal signal. In the simulation results in Figure 3, the orthogonal signal after the mean normalization processing reveals that the signal possesses the DC offset and unequal AC amplitudes, which will lead to the nonlinearity error. Therefore, the measurement accuracy of this measurement structure will be affected by the nonlinearity error Appl. Sci. 2020, 10, 6191 4 of 12 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 13 errorduring during the measurement. the measurement. The Theformula formula for forthe theanalys analysisis of ofthe the nonlinearity nonlinearity error error is is revealed inin EquationEquation (5),(5), wherewhere IIxx and Iy are the interferometric signals, ψ represents the ideal phase, and mm isis aa constantconstant [[13–15].13–15]. Nonlinearity error = arctan (Iy/Ix) ψ + mπ (5) Nonlinearity error = arctan (Iy/Ix) −− ψ + mπ (5) In accordance with the measurement structure of the polarized Fabry–Pérot interferometer, In accordance with the measurement structure of the polarized Fabry–Pérot interferometer, the the analysis of the nonlinearity error is based on the simulated orthogonal signal, shown in Figure3. analysis of the nonlinearity error is based on the simulated orthogonal signal, shown in Figure 3. The The simulation result (Figure4) reveals that the nonlinearity error is ranging from 48 to 131 nm in the simulation result (Figure 4) reveals that the nonlinearity error is ranging from −−48 to 131 nm in the polarized Fabry–Pérot interferometer. polarized Fabry–Pérot interferometer.

Figure 4. Nonlinearity error of polarized Fabry–PFabry–Pérotérot interferometer.

ForFor subsequent signal signal subdivision, subdivision, significant significant in interpolationterpolation error error will will occur. occur. Furthermore, Furthermore, the theconventional conventional correction correction method method of nonlinearity of nonlinearity error erroris also is difficult also diffi tocult implement to implement in this insignal this signal[16,17].[16 Subsequently,,17]. Subsequently, a look-up a look-up table method table method was pr wasovided provided to carry to carryout the out signal the signal subdivision subdivision with withthe resolution the resolution of 0.1 of nm 0.1 in nm the in polarized the polarized Fabry–Péro Fabry–Pt interferometerérot interferometer [18]. [Still,18]. Still,this method this method can only can onlybe processed be processed for a for single a single Lissajous Lissajous signal signal under under specific specific measurement measurement conditions, conditions, e.g., e.g., measuring speedspeed and range, etc. etc. If If the the measurement measurement signal signal change changes,s, this this method method will will not not be beable able to tocarry carry out. out. In Inthe the correlated correlated study, study, the the Lissajous Lissajous signal signal of of the the st start-pointart-point and and the the endpoint endpoint in in the whole measuring rangerange of 500 mm mm is is different, different, which which means means that that the the signal signal phase phase and and amplitude amplitude are are changing. changing [11]. [11 If]. Ifthis this signal signal processing processing method method is is utilized utilized in in th thisis situation, situation, the inherent measurementmeasurement performanceperformance withwith thethe resolutionresolution ofof 0.10.1 nmnm willwill notnot bebe achievedachieved inin thethe actualactual experiment.experiment. Furthermore, toto avoidavoid thethe signalsignal driftdrift during during the the measurement measurement procedure, procedure, the DC the off setDC compensation offset compensation module is module also introduced is also inintroduced the polarized in the Fabry–P polarizedérot interferometer. Fabry–Pérot interferometer. The conversion The of analog conversion to digital of andanalog digital to todigital analog and is performeddigital to analog in real-time is performed to acquire in real-time the signal to center acquire and the then signal is to center eliminate and then the DC is to off eliminateset. Since the signal DC processingoffset. Since involves signal processing more complex involves feedback more complex control, feedback if the noise control, occurs if the during noise the occurs measurement, during the themeasurement, measurement the accuracy measurement will be affaccuracyected severely. will be Therefore, affected thisseverely. kind of Therefore, interferometer this cannot kind beof performedinterferometer in high cannot precision be performed and high-speed in high precision industrial and applications. high-speed industrial applications. ByBy reviewing the development of the Fabry–PFabry–Pérotérot interferometer employedemployed forfor thethe displacement measurement,measurement, thethe majormajor problemsproblems inin criticalcritical structuresstructures areare summarizedsummarized inin TableTable1 1.. Therefore, Therefore, how how to to provideprovide aa measurementmeasurement systemsystem basedbased onon thethe Fabry–PFabry–Pérotérot interferometerinterferometer withwith highhigh measurementmeasurement performanceperformance forfor employingemploying inin thethe largelarge rangerange isis anan essentialessential issue.issue. FromFrom thethe above-mentionedabove-mentioned description,description, aa didifferentialfferential quadraturequadrature Fabry–PFabry–Pérotérot interferometerinterferometer withwith variablevariable measurement mirrorsmirrors employingemploying inin di differentfferent measuring measuring ranges ranges is proposed is proposed in this in study. this Bystudy. the integration By the integration of the differential of the differential optical structure and the Fabry–Pérot interferometric technique, the nonlinearity error Appl. Sci. 2020, 10, 6191 5 of 12 optical structure and the Fabry–Pérot interferometric technique, the nonlinearity error can be improved effectively, and the DC offset can also be eliminated during the measurement process to realize high precision measurement performance conveniently and flexibly.

Table 1. Comparison between the previous Fabry–Pérot interferometers.

Item Signal Signal Processing Problems Fabry–Pérot Interferometer Distribution Method Not suitable for Discontinuous dynamic Conventional structure Reference 8 Fringe counting (high finesse) measurement in large range 1. With large Quadrature-based nonlinearity error Continuous (low Polarized structure Reference 11 sensing with 2 2. Involved with finesse) photodiodes the complex signal processing

2. Proposed Differential Quadrature Fabry–Pérot Interferometer Based on the pending patent (application number: 108142443), the optical structure of the proposed differential quadrature Fabry–Pérot interferometer is extended and realized experimentally. It enables the arrangement with two measurement types simultaneously for determining the linear displacement. This structure contains a laser source, an optical cavity, and an optical element and signal sensing module (Figure5a). The laser beam is incident to the optical cavity module from the laser source module and the optical element and signal sensing module. Then the laser beam is divided into multiple beams and output from the cavity, and PDs placed in the optical element and signal sensing module will inspect the interferometric signals. The laser source module is composed of the high stabilized He-Ne laser and the optical isolator. The purpose of placing the optical isolator is to avoid the laser beam to reflect back from other optical components. The optical cavity module contains two measurement types depending on the selection of different measurement mirrors (Figure5b). One is the PM type whose optical cavity is composed of two PMs, and the other is the corner cube retro-reflector (CCR) type whose optical cavity consists of a PM and a CCR. In the CCR type, due to its folded measurement structure, the optical resolution can be improved by a factor of two compared to the type of the PM [19]. In addition, owing to CCR bearing large tilt angles of the measurement mirror, this type is more suitable for the displacement measurement within the large range. Due to the multi-beam interferometric measurement structure, the incident beam travels forward and backward in the cavity and is divided into numerous reflected beams (order number of beam: 1st, 2nd, 3rd ... Nth). The Nth beam will pass through the quarter-wave plate twice more than the N-1th beam, so the polarization direction of each laser beam will be orthogonally converted. Then they will be transmitted back to the optical element and signal sensing module. In the optical element and signal sensing module, it contains a quarter-wave plate, two BS, two PBS, and four PDs. According to the proposed optical structure, the interferometric signal can be acquired by PDs for further displacement measurement. The linearly polarized laser beam with a polarization direction angle of –π/4 with respect to the y-axis will be acquired after passing through the optical isolator, and it is converted to the right-handed circularly polarized beam by the first quarter-wave plate, whose fast axis is along the y-axis. Then the laser beam will be incident to the BS1 and the optical cavity. In this cavity, the beam is transmitted to the second quarter-wave plate whose fast axis is at a zero angle with respect to the y-axis multiple times. Numerous reflected laser beams output by the optical cavity are sequentially converted into the orthogonally polarized beams (Figure5c). The backward laser beam is split into two beams by BS2, and then each beam is transmitted to the PBS1 and the PBS2, respectively. The optical axis of the PBS2 is rotated around the incident beam by π/4 relative to that of PBS1. In the optical arrangement, two pairs of the interferometric signal with a phase difference of π can be obtained. And by subtracting each pair signal, the orthogonal signal for displacement measurement will be determined. Appl.Appl. Sci. Sci.2020 2020, 10, ,10 6191, x FOR PEER REVIEW 6 of6 of 12 13

(a)

(b)1 (b)2

(c)

FigureFigure 5. 5.Proposed Proposed di differentialfferential quadrature quadrature Fabry–PFabry–Pérotérot interferometer: ( (aa)) Structure Structure composition; composition; (b) (b)transmission transmission of of laser laser beams beams in intwo two measurement measurement types types (b)1 (b plane)1 plane mirrors mirrors (PM) (PM) type, type, (b)2 ( bcube)2 cube retro- retro-reflectorreflector (CCR) (CCR) type, type, (c) (opticalc) optical arrangement arrangement and and the thepolarizat polarizationion direction direction of oflaser laser beams. beams. PD, PD,photodiode, photodiode, PBS, PBS, polarizing polarizing beam beam splitter; splitter; BS, BS, non-polarizing non-polarizing beam beam splitter. splitter.

TheThe Jones linearly matrix polarized of the incidentlaser beam beam with and a polarization the relevant direction optical elementsangle of – inπ/4 this with structure respectis to as the Equationy-axis will (6) tobe Equation acquired (11). after passing through the optical isolator, and it is converted to the right- ! handed circularly polarized beam by the first quarter-wavecos Φ plate, whose fast axis is along the y-axis. E(Φ) = (6) Then the laser beam will be incident to the BS1 sinandΦ the optical cavity. In this cavity, the beam is transmitted to the second quarter-wave plate whose fast axis is at a zero angle with respect to the y- i δ i 4πnd axis multiple times. Numerous reflectedPF laser= e · beam= e ·s λoutput by the optical cavity are sequentially(7) converted into the orthogonally polarized beams (Figure 5c). The backward laser beam is split into two beams by BS2, and then each beam is transmitted to the PBS1 and the PBS2, respectively. The Appl. Sci. 2020, 10, 6191 7 of 12

! 1 1 0 BS = (8) √2· 0 1 ! " i ( π ) # ! cos(θ) sin(θ) e− · 4 0 cos(θ) sin(θ) QWP(θ) = − i ( π ) (9) sin(θ) cos(θ) · 0 e 4 · sin(θ) cos(θ) · −  2   sin(α) sin(α) cos(α)  PBSo(α) =  − · 2  (10)  sin(α) cos(α) cos(α)  − ·  2   cos(α) sin(α) cos(α)  PBSe(α) =  · 2  (11)  sin(α) cos(α) sin(α)  · Where E (Φ) is the electric field of the linearly polarized laser beam after traveling the optical isolator, the angle of the polarization direction with respect to the y-axis is Φ. PF is defined as the phase i δ factor (e · ) accounts for the optical phase difference (δ) caused by the changing of the cavity distance (d). δ equals 4πnd/λ, where λ is the laser wavelength, and n is the refractive index. For the type of the plane mirror and the corner cube retro-reflector, the optical phase difference equals to 4πnd/λ (δ) and 8πnd/λ (2δ), respectively. BS is the matrix of the non-polarizing beam splitter, and QWP (θ) represents a matrix that corresponds to a quarter-wave plate whose fast axis is at an angle θ with respect to the y-axis. PBSo (α) and PBSe (α) represent two separated beams rotated by an angle α relative to the x-axis. The sum of the electric field can be express as Equation (12), where R and T are the reflectance and transmittance of the coated plane mirror, respectively, Q is the transmittance of the optical cavity, and N is the order number of the backward reflected beam.   X  3 3  2 ∞  2N 3 N 1 N 1 2 N 2 Esum = BS √R + BS √T √R − Q − PF − QWP(θ ) · −  QWP( θ ) E(Φ) (12)  · · · · · · 2 · 1 · N=2

For PD1 to PD4, the corresponding electric fields of each incident beam are shown from Equation (13) to Equation (16). E = PBSo(α ) Esum (13) 1 1 · E = PBSe(α ) Esum (14) 2 1 · E = PBSo(α ) Esum (15) 3 2 · E = PBSe(α ) Esum (16) 4 2 · The light intensity can be expressed as Equation. (17) and two orthogonal signals can be determined by Equation (18) and Equation (19).

I = E E∗ (17) ·

Ix = I I (18) 1 − 2 Iy = I I (19) 3 − 4 For the proposed differential measurement structure, the mean normalization processing is similar to the analysis method in the polarized Fabry–Pérot interferometer, shown in Section1. By the mean normalization (Ix-nom,Iy-nom) for each interferometric signal (Ix,Iy) indicated in Equations (20) and (21), the orthogonal signal with normalization processing can be acquired. In order to illustrate this method, an assumed ellipse is utilized to perform the mean normalization. Its center is in the coordinate (0, 0), the length of the semi-major axis, and the semi-minor axis equal 2 and 1, respectively. The original graph, shown in Figure6a, can be converted into a circle (Figure6b) by this normalization. In summary, the reflectance of the coated plane mirror and transmittance of the optical cavity are 0.25 and 0.86. Appl.Appl. Sci. Sci. 2020 2020, ,10 10, ,x x FOR FOR PEER PEER REVIEW REVIEW 88 of of 13 13

ForFor thethe proposedproposed differentialdifferential measurementmeasurement strustructure,cture, thethe meanmean normalizationnormalization processingprocessing isis similarsimilar toto thethe analysisanalysis methodmethod inin thethe polarizedpolarized Fabry–PérotFabry–Pérot interferometer,interferometer, shownshown inin SectionSection 1.1. ByBy thethe meanmean normalizationnormalization (I(Ix-nomx-nom,, IIy-nomy-nom)) forfor eacheach interferometricinterferometric signalsignal (I(Ixx,, IIyy)) indicatedindicated inin EquationsEquations (20)(20) andand (21),(21), thethe orthogonalorthogonal signalsignal withwith normalizationnormalization processingprocessing cancan bebe acquired.acquired. InIn orderorder toto illustrateillustrate thisAppl.this method, Sci.method,2020, 10 anan, 6191 assumedassumed ellipseellipse isis utilizedutilized toto performperform thethe meanmean normalization.normalization. ItsIts centercenter isis inin8 ofthethe 12 coordinatecoordinate (0, (0, 0), 0), the the length length of of the the semi-major semi-major axis, axis, and and the the semi-minor semi-minor axis axis equal equal 2 2 and and 1, 1, respectively. respectively. TheThe originaloriginal graph,graph, shownshown inin FigureFigure 6a,6a, cancan bebe convertedconverted intointo aa circlecircle (Figure(Figure 6b)6b) byby thisthis And the angle of Φ, θ, α1 and α2 are π/4, 0, 0, and π/4, respectively. The Lissajous signal of the proposed normalization.normalization. In In summary, summary, the the reflectance reflectance of of the the co coatedated plane plane mirror mirror and and transmittance transmittance of of the the optical optical differential Fabry–Pérot interferometer is demonstrated in Figure7a,b. cavitycavity areare 0.250.25 andand 0.86.0.86. AndAnd thethe angleangle ofof ΦΦ,, θθ,, αα11 andand αα22 areare ππ/4,/4, 0,0, 0,0, andand ππ/4,/4, respectively.respectively. TheThe LissajousLissajous signalsignal ofof thethe proposedproposed differentialdifferential Fabry–PérotFabry–PérotIx µ interferometerinterferometer isis demonstrateddemonstrated inin FigureFigure Ix nom = − (20) 7a,b.7a,b. − Ix_max I − x_min IIx-nomx-nom == (20)(20) __Iy µ__ Iy nom = − (21) − Iy_max Iy_min IIy-nomy-nom == − (21)(21) __ __

((aa)) ((bb))

FigureFigure 6. 6. AssumedAssumed ellipse. ellipse. (( aa)) OriginalOriginal graph; graph; ( (bb)) Graph Graph with with mean mean normalization normalization processing. processing.processing.

((aa)) ((bb))

FigureFigureFigure 7.7.7. Orthogonal OrthogonalOrthogonal signal signalsignal of proposedofof proposedproposed diff erential differentiadifferentia quadraturell quadraturequadrature Fabry–P Fabry–PérotFabry–Pérotérot interferometer. interferometer.interferometer. (a) Original ((aa)) Originalsignal;Original (b signal;signal;) Signal ((bb with)) SignalSignal mean withwith normalization meanmean normalizationnormalization processing. processing.processing.

AccordingAccordingAccording tototo the thethe proposed proposedproposed diff erentialdifferentialdifferential Fabry–P Fabry–Fabry–érotPérot interferometer,Pérot interferometer,interferometer, the simulated thethe simulatedsimulated results illustrated resultsresults illustratedinillustrated Figure7 reveal inin FigureFigure that 77 it revealreveal is an ellipse thatthat itit orthogonal isis anan ellipseellipse signal orthogonalorthogonal without signalsignal the DC withoutwithout o ffset. thethe It means DCDC offset.offset. that by ItIt meanssubtractingmeans thatthat bytwoby subtractingsubtracting pairs of the twotwo signal pairspairs with ofof the athe phase sisignalgnal di withffwitherence aa phasephase of π, difference thedifference DC off setofof ππ can,, thethe be DCDC eliminated. offsetoffset cancan Then, bebe eliminated.eliminated. the equal Then,ACThen, amplitudes thethe equalequal canACAC be amplitudesamplitudes realized by caca thenn bebe hardware realizedrealized circuit,byby thethe hardware andhardware that minimized circuit,circuit, andand the thatthat nonlinearity minimizedminimized error. thethe nonlinearityTherefore,nonlinearity by error.error. the integration Therefore,Therefore, of byby the thethe di integrationffintegrationerential optical ofof thethe structure, differentialdifferential and opticaloptical common structure,structure, path interferometric andand commoncommon pathtechnique,path interferometricinterferometric the DC off set technique,technique, can be eliminated, thethe DCDC offsetoffset and thecancan nonlinearity bebe eliminated,eliminated, error andand can thethe also nonlinearitynonlinearity be improved. errorerror The can analysiscan alsoalso beofbe theimproved.improved. nonlinearity TheThe erroranalysisanalysis is similar ofof thethe to nonlinearitynonlinearity the polarized errorerror Fabry–P isis similarsimilarérot interferometer. toto thethe polarizedpolarized Compared Fabry–PérotFabry–Pérot to the interferometer.polarizedinterferometer. structure, ComparedCompared the result toto thethe reveals polarizedpolarized thatthe structure,structure, nonlinearity thethe resultresult error revealsreveals is significantly thatthat thethe nonlinearity reducednonlinearity to less errorerror than isis significantlyonesignificantly nanometer, reducedreduced which toto equals lessless thanthan to two oneone magnitude nanometer,nanometer, orders whichwhich (Figure equalsequals8 to).to twotwo magnitudemagnitude ordersorders (Figure(Figure 8).8). In summary, by the proposed differential measurement structure, the DC offset occurs during the measurement can be eliminated without a complex signal processing module. The nonlinearity error can also be improved significantly to realize the precise industrial and scientific applications. Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13

Appl. Sci. 2020, 10, 6191 9 of 12

Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 13

Figure 8. Comparison of nonlinearity error between polarized and proposed differential Fabry–Pérot interferometer.

In summary, by the proposed differential measurement structure, the DC offset occurs during Figure 8. Comparison of nonlinearity error between polarized and proposed differential Fabry–Pérot the measurementFigure 8. Comparison can be eliminated of nonlinearity without errora comp betweenlex signal polarized processing and module. proposed The diff nonlinearityerential interferometer. errorFabry–P can alsoérot be interferometer. improved significantly to realize the precise industrial and scientific applications. 3. ExperimentalIn summary, Results by the proposed differential measurement structure, the DC offset occurs during 3. Experimental Results the measurement can be eliminated without a complex signal processing module. The nonlinearity To verify the measurement performance of the proposed differential quadrature Fabry–Pérot error can also be improved significantly to realize the precise industrial and scientific applications. interferometer,To verify the a commercial measurement interferometer performance with of the the proposed resolution differential of 1 nm is employedquadrature as Fabry–Pérot a reference standardinterferometer, to carry a outcommercial the comparison interferometer experiment with simultaneously. the resolution of After 1 nm the is experiment, employed as the a deviationreference 3. Experimental Results canstandard be obtained to carry by out the the diff comparisonerence between experiment the measured simultaneously. values of two After interferometers. the experiment, For the determining deviation thecan measurementTobe verifyobtained the repeatability,bymeasurement the difference theperformance experimental between ofthe the standard measured proposed deviation valuesdifferential isof calculated two quadrature interferometers. by 10 Fabry–Pérot deviations For interferometer,(10determining repeated the cycles) a measurementcommercial at each position,interferometer repeatability, and thenwiththe expe thethe rimental linearityresolution standard can of 1 be nm confirmed deviation is employed is through calculated as a reference dividing by 10 standardthreedeviations times to (10carry the repeated maximum out the comparisoncycles) standard at each deviation experiment position, by simultaneously. theand whole then measuringthe linearity After the range can experiment, [be20 ].confirmed The experimentalthe deviation through canarrangementdividing be obtained three is demonstratedtimes by the maximumdifference in Figure standardbetween9. In this experiment,thedeviation measured by the the values measurement whole of measuring two structure interferometers. range of PM [20]. type TheFor in determiningtheexperimental small measuring the arrangement measurement range andis repeatability,demonstrated that of CCR the type in expe inFigure therimental large9. In rangestandard this haveexperiment, deviation been utilized. isthe calculated measurement Results by can 10 structure of PM type in the small measuring range and that of CCR type in the large range have been deviationsbe gained in(10 both repeated structures. cycles) The at Fabry–P each position,érot interferometer and then the and linearity the commercial can be confirmed interferometer through are utilized. Results can be gained in both structures. The Fabry–Pérot interferometer and the commercial dividinginstalled onthree the times left and the right maximum sides of standard the linear deviation stage, and by the the corresponding whole measuring measurement range [20]. mirror The is interferometer are installed on the left and right sides of the linear stage, and the corresponding experimentalfixed on it to carryarrangement out the measurement is demonstrated procedure. in Figure 9. In this experiment, the measurement structuremeasurement of PM mirror type in is thefixed small on it measuring to carry out range the measurementand that of CCR procedure. type in the large range have been utilized. Results can be gained in both structures. The Fabry–Pérot interferometer and the commercial interferometer are installed on the left and right sides of the linear stage, and the corresponding measurement mirror is fixed on it to carry out the measurement procedure.

Figure 9.9. TheThe optical arrangement of the comparison experiment between the commercial andand proposedproposed interferometer.interferometer.

InIn thethe comparisoncomparison experimentexperiment between the Fabry–PFabry–Pérotérot interferometer of the PM type with the Figure 9. The optical arrangement of the comparison experiment between the commercial and resolution of λ//88 andand commercialcommercial interferometer,interferometer, thethe forwardforward displacementdisplacement is regulated from 0 mmmm proposed interferometer. toto 150150 mmmm withwith aa stepstep intervalinterval ofof 1515 mm,mm, andand eacheach cyclecycle isis repeatedrepeated 1010 times.times. AccordingAccording toto thethe In the comparison experiment between the Fabry–Pérot interferometer of the PM type with the resolution of λ/8 and commercial interferometer, the forward displacement is regulated from 0 mm to 150 mm with a step interval of 15 mm, and each cycle is repeated 10 times. According to the Appl. Sci. 2020,, 10,, x 6191 FOR PEER REVIEW 10 of 13 12 Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 13 measurement result (Figure 10), the maximum deviation between the two interferometers is 0.219 measurement result (Figure 10), the maximum deviation between the two interferometers is 0.219 μmeasurementm, the maximum result standard (Figure 10 deviation), the maximum is 0.076 deviation μm, and betweenthe linearity the two is 1.52 interferometers × 10−6 F.S. in is the 0.219 wholeµm, μm, the maximum standard deviation is 0.076 μm, and the linearity is 1.526 × 10−6 F.S. in the whole range.the maximum standard deviation is 0.076 µm, and the linearity is 1.52 10− F.S. in the whole range. range. ×

Figure 10. Comparison result between the commercial interferometer and proposed Fabry–Pérot Figure 10. Comparison result between the commercial interferometer and proposed Fabry–Pérot interferometerFigure 10. Comparison of PM type. result between the commercial interferometer and proposed Fabry–Pérot interferometerinterferometer of of PM PM type. type. The comparison experiment between the Fabry–Pérot interferometer of the CCR type with the TheThe comparison comparison experiment experiment between between the the Fabry–Pérot Fabry–Pérot interferometer interferometer of the of theCCR CCR type type with with the resolution of λ/16 and commercial interferometer is similar to the above-mentioned procedure. The resolutionthe resolution of λ/16 of λand/16 andcommercial commercial interferometer interferometer is simi islar similar to the to above-mentioned the above-mentioned procedure. procedure. The measuring range is extended to 600 mm, which is conducted with a step interval of 60 mm. The measuringThe measuring range range is extended is extended to 600 to mm, 600 mm,which which is conducted is conducted with witha step a interval step interval of 60 ofmm. 60 mm.The measurement results, shown in Figure 11, demonstrates that the maximum deviation, the maximum measurementThe measurement results, results, shown shown in Figure in Figure 11, 11demons, demonstratestrates that that the the maximum maximum deviation, deviation, the the maximum maximum standard deviation, and the linearity are 0.427 μm, 0.120 μm, and 6 × 10−7 F.S., respectively. From standardstandard deviation, andand thethe linearity linearity are are 0.427 0.427µm, μm, 0.120 0.120µm, μ andm, and 6 106 ×7 10F.S.,−7 F.S., respectively. respectively. From From these these comparison results between two measurement types demonstrated× − in Table 2, the linearity of thesecomparison comparison results results between between two measurementtwo measurement types ty demonstratedpes demonstrated in Table in Table2, the 2, linearity the linearity of CCR of CCR type is better than the PM type. Because of CCR bearing large tilt angles of the measurement CCRtype type is better is better than thethan PM the type. PM type. Because Because of CCR of bearingCCR bearing large tiltlarge angles tilt angles of the measurementof the measurement mirror, mirror, the linearity of CCR type would be better than that of the PM type within the same measuring mirror,the linearity the linearity of CCR of type CCR would type would be better be thanbetter that than of that the PMof the type PM within type within the same the measuringsame measuring range. range. range.

Figure 11. The comparison result between the commercial interferometer and proposed Fabry–Pérot FigureFigure 11. 11. TheThe comparison comparison result result between between the the commercia commerciall interferometer interferometer and and proposed proposed Fabry–Pérot Fabry–Pérot interferometer of CCR type. interferometerinterferometer of of CCR CCR type. type. In order to evaluate the improvement of the measurement performance in the large measuring

range, comparison results between the published polarized Fabry–Pérot interferometer in reference 11 and the proposed differential Fabry–Pérot interferometer of CCR type are illustrated in Table3. It reveals that the linearity in the published structure within the range of 500 mm is 8.76 10 7 F.S., × − Appl. Sci. 2020, 10, 6191 11 of 12 and the linearity in the proposed structure within the extended range of 600 mm is 6 10 7 F.S. × − In summary, the maximum deviation and linearity can be improved by 48% and 32%, respectively by the proposed measurement system. This measurement performance can be utilized for the linear displacement measurement to meet the high-precision industrial demand.

Table 2. The comparison result of the proposed measurement structure between the PM type and the CCR type.

Type PM CCR Measurement Performance Range mm 150 160 standard deviation (σ) 0.076 0.120 Maximum µm standard deviation (3σ) 0.228 0.360 σ Linearity ( 3 ) F.S. 1.52 10 6 6 10 7 Range × − × −

Table 3. The comparison result between the published and the proposed measurement structure.

Item Proposed Reference 11 Improvement (%) Measurement Performance Structure Range mm 500 600 deviation 0.824 0.427 48 Maximum standard deviation (σ)µm 0.146 0.120 standard deviation (3σ) 0.438 0.360 σ Linearity ( 3 ) F.S. 8.76 10 7 6 10 7 32 Range × − × −

4. Conclusions In this study, the differential quadrature Fabry–Pérot interferometer with variable measurement mirrors for employing in different measuring ranges has been proposed. By the integration of the differential optical structure and common path interferometric technique, the nonlinearity error can be effectively reduced, which equals two magnitude orders, and the DC offset can also be eliminated during the measurement procedure. The measurement performance of the developed Fabry–Pérot interferometer has been verified. From the experimental result, the measurement repeatability is less than 0.120 µm, and the linearity is 6 10 7 F.S. within the whole range of 600 mm. By the comparison × − of the measurement performance between the published polarized Fabry–Pérot interferometer and the proposed differential Fabry–Pérot interferometer, the maximum deviation and linearity can be improved by 48% and 32%, respectively. The capability of the proposed system is conducive to industrial applications to realize the high precision displacement measurement.

Author Contributions: Conceptualization, Y.-C.S., P.-C.T., W.-Y.J., C.-P.C., L.-H.S. and T.-H.H.; Investigation, Y.-C.S., L.-H.S., P.-C.T., W.-Y.J., C.-P.C., T.-H.H.; Writing—original draft, Y.-C.S. and P.-C.T.; Writing—review and editing, P.-C.T., Y.-C.S., W.-Y.J., C.-P.C., L.-H.S. and T.-H.H. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by the Ministry of Science and Technology under the Grant number MOST 108-2221-E-224-044, and also by the Ministry of Economic Affairs under the Grant number108-EC-17-A-05-S5-001. Acknowledgments: For the technical suggestions and the experimental analyses, authors would like to be grateful to Prof. Yung-Cheng Wang and Mr. Teng-Chi Wu in National Yunlin University of Science and Technology. We acknowledge the financial support by the Ministry of Science and Technology under the Grant number MOST 108-2221-E-224-044, and also by the Ministry of Economic Affairs under the Grant number108-EC-17-A-05-S5-001. Conflicts of Interest: The authors declare no conflict of interest.

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