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sensors Article An Optical Fiber Lateral Displacement Measurement Method and Experiments Based on Reflective Grating Panel Yuhe Li *, Kaisen Guan, Zhaohui Hu and Yanxiang Chen Department of Precision Instruments, Tsinghua University, State Key Laboratory of Precision Measurement Technology and Instruments, Beijing 100084, China; [email protected] (K.G.); [email protected] (Z.H.); [email protected] (Y.C.) * Correspondence: [email protected]; Tel.: +86-10-6279-4630 Academic Editor: Jose Miguel Lopez-Higuera Received: 10 March 2016; Accepted: 30 May 2016; Published: 2 June 2016 Abstract: An optical fiber sensing method based on a reflective grating panel is demonstrated for lateral displacement measurement. The reflective panel is a homemade grating with a periodic variation of its refractive index, which is used to modulate the reflected light intensity. The system structure and operation principle are illustrated in detail. The intensity calculation and simulation of the optical path are carried out to theoretically analyze the measurement performance. A distinctive fiber optic grating ruler with a special fiber optic measuring probe and reflective grating panel is set up. Experiments with different grating pitches are conducted, and long-distance measurements are executed to accomplish the functions of counting optical signals, subdivision, and discerning direction. Experimental results show that the proposed measurement method can be used to detect lateral displacement, especially for applications in working environments with high temperatures. Keywords: fiber optic sensing; lateral displacement measurement; reflective grating panel 1. Introduction Displacement sensors and measurement techniques have been widely applied in many fields, including advanced fabrication machines, health monitoring devices, and medicine [1–3]. Capacitive sensors could provide excellent linearity, resolution, and bandwidth in short-range applications [4]. A sensing method with scanning probe microscopy (SPM) has become an indispensible tool for topographical measurement with nano- and sub-nanometer resolution [5], but usually the single scanning distance is of the order of 150 ˆ 150 µm, and the scanning speed is also a limitation. Eddy current sensors have a large range with linearity error of less than ˘3% [6,7], but they are not widely used because of the temperature sensitivity. Both linear variable displacement transformers [8] and linear encoders can supply nanoscale resolution with a large measurement range [9,10], however, they are unreliable in electromagnetic interference environments. Fiber optic sensors (FOS) have attracted much attention of researchers over the past few decades due to some innovative characteristics, such as high bandwidth, low loss, and can work under harsh environmental conditions compared to traditional sensors [11]. Fiber optic displacement sensors (FODS) are playing a more important role in the area of displacement measurement. Fiber Bragg grating-based sensors have become most widely known in the market due to their multiple functions [12–17]. Including interferometric sensors and intensity modulated sensors, many kinds of FODS have been presented in recent years [18–29]. The interferometric sensors could realize high sensitivity and accuracy, but they are usually complicated and of higher cost to set up wavelength shift interrogation systems [19]. The intensity-modulated Sensors 2016, 16, 808; doi:10.3390/s16060808 www.mdpi.com/journal/sensors Sensors 2016, 16, 808 Sensors 2016, 16, 808 2 of 15 2 of 15 intensity-modulated sensors, contrast, are [20–27]. simple and low cost [20–27]. transmissive fiber design lens sensors, by contrast, are simplebyand low cost A transmissive fiberAlens double-fiber double-fiber proposed to achieve displacement measurement [20]. A seond transmissive is proposed to design achieveis displacement measurement [20]. A seond transmissive one could realize a one could realize accuracy of of 6 µm over range of 3000 µm However, measurement with aanmeasurement accuracy of with 6 µman over a range 3000 µma[21]. However, in [21]. the transmissive in the transmissive fiber sensor, the emitting fibers and receiving fibers are, respectively, arranged fiber sensor, the emitting fibers and receiving fibers are, respectively, arranged on two sides of the on two sides of the measuring which occupies tooReflective much room [22]. Reflective fiber measuring probe, which usuallyprobe, occupies toousually much room [22]. fiber optic displacement optic displacement sensors (RFODS) simplify their the structure the of fibers one [23,24]. side sensors (RFODS) could simplify their could structure by fixing fibersby onfixing one side the on probe of the probe [23,24]. In addition, by adopting different reflective targets, RFODS show many In addition, by adopting different reflective targets, RFODS show many advanced characteristics advanced characteristics and could measure different measurands [28–32]. Although RFODS have and could measure different measurands [28–32]. Although RFODS have high accuracy and simple high accuracy and simple structure, small measurement ranges, to some degree, limit their practical structure, small measurement ranges, to some degree, limit their practical applications [33]. applications [33]. In this study, we propose a novel type of optical fiber measurement method based on a reflective In this study, we propose a novel type of optical fiber measurement method based on a grating panel for lateral displacement. The signal subdivision model with the tangent and cotangent reflective grating panel for lateral displacement. The signal subdivision model with the tangent and functions is given, and error analysis and processing are studied. Based on the relationship between cotangent functions is given, and error analysis and processing are studied. Based on the displacement received optical power phaseoptical difference is adopted in order to realize discerning relationshipand between displacement and the received power the phase difference is adopted in direction displacement and the center distance ofand twothe receiving fibers isofderived order tofor realize discerningmeasurement, direction for displacement measurement, center distance two theoretically to ensure the orthogonal between two receiving A distinctive fiber optic receiving fibers is derived theoreticallyphase to ensure the orthogonal phasefibers. between two receiving fibers. grating ruler is set upoptic to measure includes a special fiber optic measuring probe A distinctive fiber grating dispalcement, ruler is set up which to measure dispalcement, which includes a special and reflective grating panel. fiber optic measuring probe and reflective grating panel. In In this paper, methodfor forlateral lateral displacement is presented, this paper,an anoptical opticalfiber fiber measurement measurement method displacement is presented, and and a a reflective Themeasurement measurementmethod methodand andsystem systemforfordisplacement displacement reflective grating grating panel panel is fabricated. fabricated. The measurement is detailed. The simulation calculationofofthe the opticalpath path and resolutionare are carred measurement is detailed. The simulation calculation optical and resolution carred out, out, and signal analysis and processing follows. Finally, some experiments including the and signal analysis and processing follows. Finally, some experiments including the application test in application test inworking a high-temperature working environment, arethe presented to measurement verify the proposed a high-temperature environment, are presented to verify proposed method measurement method andruler. the fiber optic grating ruler. and the fiber optic grating MeasurementMethod Methodand and System System for 2. 2.Measurement for Lateral LateralDisplacement Displacement Working Principle 2.1.2.1. Working Principle The displacement measurement system is composed a light source, a sensor a The displacement measurement system is composed of a of light source, a sensor probe, probe, a reflective reflective grating panel (RGP), photodetectors and anpower opticalmeter, poweras meter, as shown in Figure 1. grating panel (RGP), photodetectors (PD), and(PD), an optical shown in Figure 1. Figure1.1.Displacement Displacement measurement Figure measurementsystem. system. The RGP, which comprises highly-reflective areas and low reflective areas, is attached to the The RGP, which comprises highly-reflective areas and low reflective areas, is attached to the measurand. An emitting fiber and two receiving fibers make up the fiber bundle. The working measurand. An emitting fiber and two receiving fibers make up the fiber bundle. The working process process of the measurement system can be summarized as follows. The RGP moves with the of the measurement system can be summarized as follows. The RGP moves with the measurand and measurand and the emitted light from the fiber is modulated by the moving RGP. When the light theirradiates emitted light from the fiber is modulated by the moving RGP. When the light irradiates on the highly on the highly reflective area, the reflected light intensity is high, and vice versa. Through reflective area, the reflected light intensity is high, and Through two receiving two receiving fibers, the modulated light transmits into vice PDs,versa. whose signals enter the opticalfibers, powerthe modulated light transmits into PDs, whose signals enter the optical power meter. The reflected signals Sensors 2016, 16, 808 3 of 15 vary like sinusoidal functions, and they would vary one period when the RGP moves one grating pitch [33]. Therefore, the displacement measurement can be achieved by counting the sinusoidal signals. In the measurement system, the two receiving fibers are arranged at a specific location, so that the PDs could obtain two light signals S1 and S2 with a phase difference of 90˝ , which can be expressed as: ˆ ˙ 2π ¨ x ptq S1 “ sin (1) p ˆ ˙ 2π ¨ x ptq S2 “ cos (2) p where x(t) is the measurand displacement, and p is the grating pitch. The displacement x(t) could be inferred as follows: ˆ ˙ p S1 x ptq “ arctan (3) 2π S2 The signal processing methods adopted in the measurement system are similar to that used for traditional grating signals [34], such as counting, subdivision, and discerning direction. 2.2. Preliminary Theoretical Derivation The fiber-grating distance ZFG between the fiber end and the RGP has influence on the measurement performance, as shown in Figure 2. The decreased ZFG will result in a smaller spot size of fiber, which is related with the measurement resolution to some degree. When ZFG gets larger than 0, the optical power of the receiving fiber will become higher, and then decrease gradually [23], which would have significant influence on the measuring sensitivity. Therefore, it is necessary to make an analysis about the relationship between ZFG and the optical field intensity at the fiber end. On the basis of the intensity distribution of the output optical field formed by the source fiber end [35], the optical field intensity distribution function at the multimode fiber end is calculated by the following formula: $ ’ & I pr, zq “ r2 , / . I0 ” ı2 exp ´ ” ı2 ’ % a2 1 ` 0.146 pz{a0 q1.5 tanθ0 / a20 1 ` 0.146 pz{a0 q1.5 tanθ0 0 (4) where I0 is the intensity of the light source, a0 is the radius of the emitting fiber, θ 0 is the fiber maximum exit angle, which is equivalent to the acceptance angle and can be determined by the numerical aperture (NA), θ 0 = arctan(NA), z = 2ZFG is the axial distance between the receiving fiber end and its mirroring, r is the radial distance from the fiber center, and I(r,z) is the light field intensity. The parameters of the emitting fiber are: a0 = 52.5 µm, tanθ 0 = 0.22; then, by solving Equation (4), the following equation can be obtained: # + r2 I0 (5) I pr, zq “ ` ˘2 exp ´ ` ˘2 52.5 ` 0.00445 ˆ z1.5 52.5 ` 0.00445 ˆ z1.5 As shown in Figure 2, the emitting and receiving fibers are placed closely and parallelly, and the fiber ends are aligned as much as possible. The gray ring Lin Lout is the emitted optical field. It can be known that the optical power of the receiving fiber is proportional to the light flux BZ in the area between the rings Lin and Lout . It can be expressed as: Bz “ ş a0 `δ`2a a 0 `δ I pr, zq dr “ ş1137.5 137.5 ? I pr, zq dr “ I0 π 105`0.0089ˆz1.5 ´ ¨ erf r 52.5`0.00445ˆz1.5 ˇ ¯ ˇ 1137.5 ˇ ˇ ˇ 137.5 (6) where a = 500 µm is the radius of the receiving fibers, and δ = 85 µm is the interval between the emitting fiber and receiving fiber. It can be known from the numerical solution obtained by MATLAB software (MathWorks Inc., Natick, MA, USA) that BZ reaches a maximum value at the point of Z = 1292 µm. Sensors 2016, 16, 808 Sensors 2016, 16, 808 4 of 15 4 of 15 where a = 500 µm is the radius of the receiving fibers, and δ = 85 µm is the interval between the where a = 16, 500808µm is the radius of the receiving fibers, and δ = 85 µm is the interval between the Sensors 2016, 4 of 15 emitting fiber and receiving fiber. It can be known from the numerical solution obtained by emitting fiber and receiving fiber. It can be known from the numerical solution obtained by MATLAB software (MathWorks Inc., Natick, MA, USA) that BZ reaches a maximum value at the MATLAB software (MathWorks Inc., Natick, MA, USA) that BZ reaches a maximum value at the point of Z =when 1292 µm.fiber-grating Therefore, when the fiber-grating distance ZFG = 646 µm, the optical power of Therefore, distance the optical of the receiving fiber will FG = 646 µm,distance point of Z = 1292 the µm. Therefore, when the Zfiber-grating ZFGpower = 646 µm, the optical power of the receiving fiber will reach its maximum value. reach its maximum value. the receiving fiber will reach its maximum value. (a) (a) (b) (b) Figure 2. Schematic diagram of measurement method. (a) Layout of measurement units; and (b) measurement method. (a) Layout of measurement units; units; and (b)and spatial Figure 2. 2. Schematic Schematicdiagram diagramofof measurement method. (a) Layout of measurement (b) spatial relations and dimension parameters. relations and dimension parameters. spatial relations and dimension parameters. 3. Simulation Calculation 3. Simulation SimulationCalculation Calculation 3.1. Optical Modeling 3.1. Optical Modeling Modeling 3.1. Optical Simulation isisconducted conducted by using the optical design software, ZEMAX software (Radiant Simulation using the the optical design software, ZEMAX software (Radiant(Radiant Zemax, Simulation is conductedbyby using optical design software, ZEMAX software Zemax, LLC, Redmond, WA, USA). As shown in Figure 3, the simulation model consists of one LLC, Redmond, WA, USA). shown Figurein 3, Figure the simulation model consists one LED Zemax, LLC, Redmond, WA,AsUSA). Asinshown 3, the simulation modelofconsists oflight one LED light source, one emitting fiber, two receiving fibers and a RGP. The emitting fiber core source, one emitting fiber, two receiving fibers and a RGP. The emitting fiber core diameter is 105 µm, LED light source, one emitting fiber, two receiving fibers and a RGP. The emitting fiber core diameter is 105 µm, and its numerical aperture, NA 1 = 0.236. The receiving fiber core diameter 2a = and its numerical NA1 = 0.236. The NA receiving fiber diameter = 1000 µm, 2a and diameter is 105 µm,aperture, and its numerical aperture, 1 = 0.236. Thecore receiving fiber2acore diameter = 1000 µm, and NA 2 = 0.467. The RGP grating pitch p is 300 µm. For simplicity, the reflectivity of high NA = 0.467. The RGP grating pitch p is 300 µm. For simplicity, the reflectivity of high and low 2 1000 µm, and NA2 = 0.467. The RGP grating pitch p is 300 µm. For simplicity, the reflectivity of high and low reflective areas were set to 1 and 0, respectively, and the highly-reflective area width p1 is reflective areas were set to 1 and and the highly-reflective area width parea 1 is equal and low reflective areas were set0,torespectively, 1 and 0, respectively, and the highly-reflective widthtopthe 1 is equal to the low one p 2. low one 2 . low one p2. equal to pthe (a) (a) (b) (b) Figure 3. Simulation model of fibers and the RGP. (a) Overall view; and (b) partial enlarged view. Figure enlarged view. view. Figure 3. 3. Simulation Simulationmodel modelof offibers fibers and and the the RGP. RGP. (a) (a) Overall Overall view; view; and and (b) (b) partial partial enlarged 3.2. Relationship between Fiber-Grating Distance and Received Optical Power 3.2. Relationship between Fiber-Grating Distance and Received Optical Power 3.2. Relationship between Fiber-Grating Distance and Received Optical Power The relationship between the fiber-grating distance ZFG and the fiber received optical power The relationship between the fiber-grating distance ZFG and the fiber received optical power The relationship between thenow fiber-grating the fiber received optical power FG and has been derived above, and can be verifieddistance throughZthe following simulations. Assuming the has been derived above, and can now be verified through the following simulations. Assuming the has been fiber derived above,on and nowof bethe verified through thearea, following simulations. Assuming the emitting is located thecan center highly-reflective and ZFG increases from 0 to 4 mm emitting fiber is located on the center of the highly-reflective area, and ZFG increases from 0 to 4 mm emitting fiber on average the center of the area, and ZFG increases fromis0obtained to 4 mm with a step of is 0.1located mm, the value of highly-reflective optical power from the two receiving fibers with a step of 0.1 mm, the average value of optical power from the two receiving fibers is obtained with a step of 0.1 mm, the average value of optical power from the two receiving fibers is obtained and normalized, then the curve of ZFG and the received optical power can be fitted numerically,and as and normalized, then the curve of ZFG and the received optical power can be fitted numerically, as normalized, then 4. theThe curve of ZFG and the received power cancurves be fitted numerically, as shown shown in Figure comparison between fittingoptical and theoretical indicates that both agree shown in Figure 4. The comparison between fitting and theoretical curves indicates that both agree in Figure 4. The betweendistance fitting and curves indicates that both reasonably well.comparison The fiber-grating ZFGtheoretical and the received optical power areagree of a reasonably non-linear reasonably well. The fiber-grating distance ZFG and the received optical power are of a non-linear well. The fiber-grating distance ZFG and the received optical power are of a non-linear relationship, and the maximum value of optical power is reached when ZFG equals 600 µm. When the RGP moves Sensors 2016, 16, 808 Sensors 2016, 16, 808 Sensors 2016, 16, 808 5 of 15 5 of 15 5 of 15 relationship, and the maximum value of optical power is reached when ZFG equals 600 µm. When relationship, and the maximum value of optical power is reached when ZFG equals 600 µm. When the RGP moves with the measurand, the received optical power signal varies like a sinusoidal the moves withthe the measurand, the received optical power signal varies like aIf sinusoidal withRGP the measurand, optical power varies like a sinusoidal function. the optical function. If the optical received power gets larger, thesignal peak-peak value of the signal becomes larger, function. If the optical power gets larger, the peak-peak value of the signal becomes larger, power gets larger, the peak-peak value of the signal becomes larger, accordingly. The measurement accordingly. The measurement sensitivity is proportional to the optical power, theoretically. On the accordingly. The measurement sensitivity is proportional to the optical power, theoretically. On the sensitivity proportional the optical power, the the other hand, the light other hand,is the light spottocan be changed intheoretically. size linearlyOn with parementer ZFG, spot and can the other hand, in thesize light spot with can be changed in size the parementer ZFG, will andvary the be changed linearly parementer ZFG ,linearly andthere the with identifiable grating pitch identifiable grating pitch will varythe accordingly. Therefore, is an inverse relationship between identifiable grating pitch will vary accordingly. Therefore, there is an inverse relationship between accordingly. Therefore, there an inverse relationshipresolution between the distance ZFG and the the fiber-grating distance ZFGisand the measurement onfiber-grating the whole. In summary, among the fiber-grating distance Z FG and the measurement resolution on the whole. In summary, among measurement the whole. In summary, the points 1, 2, and and 3medium in Figure 4, point 1 the points 1, 2,resolution and 3 inon Figure 4, point 1 has the among maximum resolution sensitivity, the 1, 2, and 3 in Figure 4, point 1sensitivity, has the maximum resolution and medium has points the maximum resolution and medium pointand 2 has medium andsensitivity, maximum point 2 has medium resolution and maximum sensitivity, point 3 has resolution the minimum resolution point 2 has and medium and maximum sensitivity, and point 3 has theTominimum resolution sensitivity, pointresolution 3 has minimum resolution medium sensitivity. resolution and medium sensitivity. Tothe improve resolution and and sensitivity, we choose the Zimprove FG of point 1 in the and sensitivity. To improve resolution and sensitivity, we choose theexperiments. ZFG of point 1 in the and medium sensitivity, we choose ZFG of point 1 in the remaining simulations and remaining simulations andthe experiments. remaining simulations and experiments. Figure Figure 4. 4. Comparisons Comparisons between between simulation simulation fitting fitting curve curve and and theoretical theoretical curve. curve. Figure 4. Comparisons between simulation fitting curve and theoretical curve. 3.3. Relationship between Lateral Displacement and Received Optical Power 3.3. Relationship Relationship between between Lateral Lateral Displacement Displacement and and Received Received Optical Optical Power Power 3.3. The simulation on the relationship between the lateral displacement and received optical The simulation between the the lateral displacement and received opticaloptical power The simulation on onthe therelationship relationship between lateral displacement and received power is conducted with the fiber-grating distance ZFG = 0.3 mm, the pitch of RGP p1 = p2 = 0.5p = is conducted with the fiber-grating distance Z = 0.3 mm, the pitch of RGP p = p = 0.5p = 150 µm. FG 2 p1 = p2 = 0.5p power is conducted with the fiber-grating distance ZFG = 0.3 mm, the pitch of1 RGP = 150 µm. In order to get a phase difference of 90° between the two receiving fiber signals, the center ˝ In order a phase of 90 between two receiving signals, thesignals, center distance of 150 µm. to In get order to getdifference a phase difference of 90°the between the twofiber receiving fiber the center distance of two receiving fibers D as shown in Figure 2, should be approximately equal to (k + 1/4)p; two receiving D as fibers shownDinasFigure equal to (kequal + 1/4)p; k is distance of twofibers receiving shown2,inshould Figurebe 2, approximately should be approximately to (khere, + 1/4)p; here, k is an integer. Due to space limitations, the center distance D takes the value 1275 µm an integer. Due to space limitations, center distance D takes the value 1275the µmvalue (namely, 4). here, k is an integer. Due to space the limitations, the center distance D takes 1275k =µm (namely, k = 4). The center of a low reflective is set as a starting point. The two received optical The center a low a startingispoint. two received power signals are, (namely, k of = 4). Thereflective center ofisa set lowasreflective set asThe a starting point. optical The two received optical power signals are, respectively, obtained and normalized, and the curve of lateral displacement and respectively, obtained and normalized, andand thenormalized, curve of lateral and optical power can be power signals are, respectively, obtained anddisplacement the curve of lateral displacement and optical power can be fitted, as shown in Figure 5. The optical power signals vary like a sinusoid, fitted, as shown in be Figure 5. The opticalinpower likepower a sinusoid, with thelike cycle the optical power can fitted, as shown Figuresignals 5. Thevary optical signals vary a being sinusoid, with the cycle being the same as the pitch of RGP, 300 µm. From Figure 5, the signals from two same as the pitch of RGP, 300 µm. From Figure 5, the signals from two receiving fibers have a phase with the cycle being the same as the pitch of RGP, 300 µm. From Figure 5, the signals from two receiving fibers ˝have a phase difference of 90°, which can be used to realize the subdivision and differencefibers of 90 have , which can bedifference used to realize subdivision and discerning direction in order to receiving a phase of 90°,the which can be used to realize the subdivision and discerning direction in order to improve measurement system performance. improve measurement systemtoperformance. discerning direction in order improve measurement system performance. Figure 5. Simulation on relationship between received optical power and lateral displacement. Figure 5. Simulation on relationship between received optical power and lateral displacement. Figure 5. Simulation on relationship between received optical power and lateral displacement. Sensors 2016, 16, 808 6 of 15 Sensors 2016, 16, 808 6 of 15 4. Signal Subdivision and Error Analysis 4. Signal When Subdivision and Error the measured objectAnalysis moves, the reflecting grating and the optical fiber probe start relative motion at the same time, and the receiving optical fiber will receive a periodic signal modulated When the measured object moves, the reflecting grating and the optical fiber probe start by the reflecting grating. As shown in Figure 6, the measured displacement can be obtained by the relative motion at the same time, and the receiving optical fiber will receive a periodic signal computational expression x = N¨ p = 3p, with the period number N = 3 and the resolution p (namely modulated by grating the reflecting grating. As in Figure 6, the measured displacement can be the known pitch). Assuming that shown the subdivision number is 10, then the resolution increase obtained thetimes computational expression = N·p = 3p,xwith period number Ncotangent = 3 and the of upby to 10 can be achieved, and thexdisplacement = 30τ.the Here the tangent and Sensors 2016, 16, 808 6 of 15 resolution p (namely the grating pitch). Assuming that the subdivision is 10, functions were used to known realize signal subdivision for the modulated intensity from thenumber RGP, which canthen the resolution increase of up(7)toand 10 shown times in can be achieved, and the displacement x = 30τ. Here the be expressed as Equation Figure 7. 4. Signal Subdivision and Error Analysis tangent and cotangent functions were $ used to realize signal subdivision for the modulated intensity & |u1 | “ | Asinθ | “ |tanθ| , |u1 | ď |u2 | from When the RGP, can beobject expressed (7) andgrating shown and in Figure 7. Acosθreflecting thewhich measured the optical fiber probe | Equation |u2 as |the | w “ moves, (7)start | Acosθ | |u2 | % “ “ |cotθ| | ą |u | , |u relative motion at the same time, and receiving optical 2 will receive a periodic signal 1 fiber |u1 | the | Asinθ | modulated by the reflecting grating. As shown in Figure 6, the measured displacement can be Errors analysis of subdivision signals for displacement measurement is carried out for the obtained by the computational expression N·pamplitude = 3p, with the period number N harmonic = 3 and the following aspects: (a) the DC signal error Ud ; x(b)= the variation coefficient ε; (c) the resolution p (namely known grating pitch). the subdivision is 10, then component Ah ; andthe (d) the non-orthogonal phaseAssuming error δ. For that example, the change in number the fiber-grating the resolution increaseabove of up to cause 10 times can be achieved, and thedegree. displacement x = 30τ.curves Here the distance discussed may the amplitude variation to some The subdivision andand signal subdivision processing given in Figures 8 and 9, and thefor error shown in tangent cotangent functions wereare used to realize signal subdivision theanalysis modulated intensity 1 [36].which can be expressed as Equation (7) and shown in Figure 7. fromTable the RGP, Figure 6. Schematic diagram of signal subdivision in displacement measurement. u1 A sin tan , A cos u2 w u 2 A cos cot , u1 A sin u1 u 2 (7) u1 u 2 Figure 6. Schematic diagram displacement measurement. Figure 6. Schematic diagramofofsignal signal subdivision subdivision inindisplacement measurement. u1 A sin tan , A cos u2 w u 2 A cos cot , u1 A sin u1 u 2 (7) u1 u 2 Figure 7. Ideal subdivision curve andcotangent cotangent functions. Figure 7. Ideal subdivision curvebased based on on tangent tangent and functions. Errors analysis of subdivision signals for displacement measurement is carried out for the following aspects: (a) the DC signal error Ud; (b) the amplitude variation coefficient ε; (c) the harmonic component Ah; and (d) the non-orthogonal phase error δ. For example, the change in the fiber-grating distance discussed above may cause the amplitude variation to some degree. The subdivision curves and signal subdivision processing are given in Figures 8 and 9, and the error analysis shown in Table7. 1Ideal [36].subdivision curve based on tangent and cotangent functions. Figure Sensors 2016, 16, 808 7 of 15 Sensors 2016, 16, 808 Sensors 2016, 16, 808 7 of 15 7 of 15 (a) (a) (b) (b) (c) (c) (d) (d) Figure analysis of subdivision signals. (a) Error DC the signal; error from the signal Figure 8.8.Error Error analysis of subdivision signals. (a) from Errorthe from DC(b) signal; (b) error from Figure 8. Error analysis of subdivision signals. (a) Error from the DC signal; (b) error from the signal amplitude error from(c)the harmonic component; (d) error from the signal variation; amplitude(c)variation; error from the harmonicand component; and the (d) non-orthogonal error from the amplitude variation; (c) error from the harmonic component; and (d) error from the non-orthogonal phase. non-orthogonal phase. phase. (a) (a) (b) (b) (c) (c) (d) (d) Figure 9. Signal subdivision processings. (a) Original signal (above) and tangent subdivision signal Figure 9. 9. (b) Signal subdivision processings. (a)signal Original signal (above) and tangent tangent subdivision signal (below); filtered signal processings. by FIR; (c)(a) bysignal noise(above) processing; and subdivision (d) displacement Figure Signal subdivision Original and signal (below); (b) (b)filtered filtered signal by (c)bysignal by noise and processing; and (d) measurements displacement measurements with signal de-noising process. (below); by FIR; (c)FIR; signal noise processing; (d) displacement measurements with de-noising process. with de-noising process. Sensors 2016, 16, 808 8 of 15 Table 1. Error analysis for signal subdivision (subdivision 100). Sensors 2016, 16, 808 Parameter Index8 of 15 Ensuring Measures for Accumulation Quantitative Before Noise After Noise Requirements Interval Precision Subdivision Processing Processing Table 1. Error analysis for signal subdivision (subdivision 100). Error Type DC signal Error Type Accumulation π/4 Interval DC signal π/4 Ud < 3.68% π/2 ε < ±11.8% Amplitudes variation Amplitudes variation Harmonic component Harmonic component π/2 Non-orthogonal phase ε < ˘11.8% Ah < 0.038A π Ah < 0.038A π Non-orthogonal phase Quantitative Ud < 3.68% Requirements Original signals are Ensuring Measures for subtracted from Precision Subdivision average value Original signals are subtracted signal peak-to-peak from averageare value amplitudes used to signal peak-to-peak correct theamplitudes error are used to correct the error FIR low-pass filtering FIR low-pass filtering Calculation of phase δ < 0.0133π Calculation difference based on the of phase difference position of signal peak based on the position π π δ < 0.0133π Parameter Index Ud =Noise 0.0304A After Processing Processing U d = 0.1373A Before Noise Ud = 0.1373A Ud = 0.0304A ε = 0.1884A ε = 0.0086A ε = 0.1884A Ah = 0.0523A Ah = 0.0523A δ = 0.0741π δ = 0.0741π ε = 0.0086A Ah = 0.0351A Ah = 0.0351A δ = 0.0286π δ = 0.0286π of signal peak 5. Experimental Results and Discussion 5. Experimental Results and Discussion 5.1. Self-Made Fiber Optic Grating Ruler and Signal Processing 5.1. Self-Made Fiber Optic Grating Ruler and Signal Processing A fiber optic grating ruler was set up based on the homemade RGP, as shown in Figure 10. A fiber optic grating ruler was set up based on the homemade RGP, as shown in Figure 10. The The RGP is made up of a printed circuit board (PCB), using gilded and striped pads as high reflective RGP is made up of a printed circuit board (PCB), using gilded and striped pads as high reflective areas,areas, the the for low areas.areas. The measuring probe,probe, including emitting and receiving thesubstrate the substrate forreflective low reflective The measuring including emitting and fibers,receiving spring groove, and guiding ball bearings, can move laterally above the RGP. fibers, spring groove, and guiding ball bearings, can move laterally above the RGP. (a) (b) (c) (d) Figure 10. Fiber optic grating ruler. (a) Overall composition chart; (b) measuring probe; (c) overall Figure 10. Fiber optic grating ruler. (a) Overall composition chart; (b) measuring probe; (c) overall ruler appearance and measuring probe; and (d) engineering drawings (left) and physical appearance ruler (right) appearance of RGP.and measuring probe; and (d) engineering drawings (left) and physical appearance (right) of RGP. 5.2. Displacement Measurement Experiments 5.2. Displacement Measurement Experiments To verify the effectiveness of the proposed method, an experimental platform was set up, as shown in Figure 11. Some system parameters, as thean signal to noise ratio (SNR), sensitivity, To verify the effectiveness of the proposedsuch method, experimental platform was set up, as and resolution, have close relationship to the grating parameters. For comparison, three kinds of shown in Figure 11. Some system parameters, such as the signal to noise ratio (SNR), sensitivity, RGPs with different grating pitches of 12 mil (304.8 µm), 20 mil (508 µm), and 30 mil (762 µm), were and resolution, have close relationship to the grating parameters. For comparison, three kinds of fabricated. In order to simplify the analysis, the highly-reflective area width p1 is set equal to p2. A RGPs with different grating pitches of 12 mil (304.8 µm), 20 mil (508 µm), and 30 mil (762 µm), were Sensors 2016, 16, 808 9 of 15 fabricated. In order to simplify the analysis, the highly-reflective area width p1 is set equal to p2 . Sensors 2016,glass 16, 808 fiber is adopted as the emitting fiber (M15L02, Thorlabs, Newton, NJ, USA), 9 of 15 and A multimode plastic fibers as the receiving fibers (HFBR-RNS001Z, AVAGO, San Jose, CA, USA). A fiber-coupled multimode glass fiber is adopted as the emitting fiber (M15L02, Thorlabs, Newton, NJ, USA), and high-power LED is selected as the light source (M660F1, Thorlabs, central wavelength 660 nm, FWHM plastic fibers as the receiving fibers (HFBR-RNS001Z, AVAGO, San Jose, CA, USA). A fiber-coupled 20 nm). A dual-channel power meter used (PM320E, Thorlabs) and, correspondingly, twonm, optical high-power LED is selected as the is light source (M660F1, Thorlabs, central wavelength 660 power sensors (S150C,power Thorlabs). the process of Thorlabs) the experiment, the measurand moves FWHM 20 are nm).adopted A dual-channel meter In is used (PM320E, and, correspondingly, two according to following five stages: A-keep static for 5 s; B-move forward for 3.125 mm at a speed optical power sensors are adopted (S150C, Thorlabs). In the process of the experiment, the of movesforward according following five stages: of A-keep forD-keep 5 s; B-move for 312.5measurand µm/s; C-move forto9.375 mm at a speed 937.5 static µm/s; static forward for 3 s; E-move 3.125 mm at a speed of a312.5 µm/s; C-move forward mmstages, at a speed 937.5 µm/s; D-keep backward for 12.5 mm at speed of 625 µm/s. Afterfor the9.375 above the of measured object returns 3 s;position E-move with backward for 12.5 mm at aofspeed of 625 µm/s. After the back static to its for initial the total movement 25 mm. Generally, therethe areabove somestages, measurement objectasreturns back to its initial position with the totalofmovement of 25 mm. Generally, errormeasured factors, such transmission clearance and losing steps the stepper motor; therefore, the there are some measurement error factors, such as transmission clearance and losing steps of the above-mentioned motion processes are monitored by a laser displacement sensor (LDS, LK-G400, stepper motor; therefore, the above-mentioned motion processes are monitored by a laser LK-G3001, KEYENCE, Osaka, Japan) synchronously. The comparative data by the RFODS (single displacement sensor (LDS, LK-G400, LK-G3001, KEYENCE, Osaka, Japan) synchronously. The channel) and LDSdata areby shown in Figure 12.channel) and LDS are shown in Figure 12. comparative the RFODS (single Figure 11. Experimental platform for lateral displacement measurement. Figure 11. Experimental platform for lateral displacement measurement. Figure 12 indicates that the output voltage matches the displacement data well. The velocity of the measured object can obtained processing either the the displacement displacement data LDS or velocity the Figure 12 indicates thatbethe outputbyvoltage matches datafrom well. The output voltage data from RFODS. The relative measurement error was computed, for example, of the measured object can be obtained by processing either the displacement data from LDSinor the stage B, where thefrom velocities fromThe the LDS and RFODS are 314.4error µm/swas and 313.7 µm/s, respectively, output voltage data RFODS. relative measurement computed, for example, in and the relative error is about −0.223%. The measurement errors comparison under different pitches stage B, where the velocities from the LDS and RFODS are 314.4 µm/s and 313.7 µm/s, respectively, and different stages are listed in Table 2. and the relative error is about ´0.223%. The measurement errors comparison under different pitches and different stages are listed Table 2. error between proposed method and LDS (1 mil = 25.4 µm). Table 2. Comparison of in measurement Stage 1 Stage 2 Stage 3 Stage 1 Stage B Stage 2 Stage C Stage 3 Stage E Stage Stage BB Stage C Stage C Stage E Stage E Stage Stage BB Stage C Stage C Stage E Stage E Stage B Stage C Stage E Table 2. Comparison of measurement betweenbyproposed andMeasurement LDS (1 mil =Error 25.4 µm). Velocity by LDS error VL Velocity R RFODS Vmethod Relative Pitch Pitch 12 mil 12 mil 20 mil 20 mil 30 mil 30 mil (µm·s−1) 315.8 Velocity by LDS V L (µm¨ 936.0s´1 ) −626.9 315.8 315.6 936.0 933.4 ´626.9 −628.2 315.6 314.4 933.4 935.9 ´628.2 −627.6 314.4 935.9 ´627.6 (µm·s−1) 316.9 Velocity by RFODS V R (µm¨ s´1 ) 937.8 −627.6 316.9 315.9 937.8 933.6 ´627.6 −628.9 315.9 313.7 933.6 938.7 ´628.9 −625.6 313.7 938.7 ´625.6 (VR − VL)/VL 0.348% Relative Measurement Error (V R ´ V L )/V L 0.192% 0.111% 0.095%0.348% 0.192% 0.021% 0.111% 0.111% 0.095% −0.223% 0.021% 0.299% 0.111% −0.319% ´0.223% 0.299% ´0.319% 10.5, 6.3, and 4.2 cycles for RGP pitches of 12 mil, 20 mil, and 30 mil, respectively, and their relative errors are about 2.4%. However, smaller pitch is easier to be influenced by the machining process. As shown in Figure 13, in stages A and D, the measured object remains static, so the output voltage can be treated theoretically as noise signals of the RFODS. Then, the signal to noise ratios under different Sensors 2016,pitches 16, 808 can be calculated and listed in Table 3. We can see that the pitch is generally 10 of 15 inversely related to the SNR. (a) (b) Figure 12. Comparisons between proposed RFODS (RGP pitch = 30 mil) and LDS. (a) Output voltage Figure 12. Comparisons between proposed RFODS (RGP pitch = 30 mil) and LDS. (a) Output voltage verse displacement; and (b) displacement from the RFODS and LDS. verse displacement; and (b) displacement from the RFODS and LDS. Table 3. SNR with different grating pitches (1 mil = 25.4 µm). The whole output from RFODS under different pitches are given in Figure 13, which shows that Pitch changes (mil) with P-P of Noise Signal (V)For example, P-P of Effective Signal SNR the signal cycle the grating pitch. the actual cycles(V) in stage B are 10.5, 6.3, and 4.2 cycles for RGP pitches of 12 mil, 20 mil, and 30 mil, respectively, and their relative 12 0.050 1.191 21.655 errors are about 2.4%. However, smaller pitch is easier to be influenced by the machining process. As shown in 20 0.026 1.709 65.731 Figure 13, in stages A and D, the measured object remains static, so the output voltage can be treated 2.885ratios under different 137.381pitches can theoretically as30 noise signals of the 0.021 RFODS. Then, the signal to noise be calculated and listed in Table 3. We can see that the pitch is generally inversely related to the SNR. Table 3. SNR with different grating pitches (1 mil = 25.4 µm). Pitch (mil) P-P of Noise Signal (V) P-P of Effective Signal (V) SNR 12 20 30 0.050 0.026 0.021 1.191 1.709 2.885 21.655 65.731 137.381 Sensors 2016, 16, 808 11 of 15 Sensors 2016, 16, 808 11 of 15 Sensors 2016, 16, 808 11 of 15 Figure 13. Comparisons of the output voltages under different RGP pitches. Figure 13. Comparisons of the output voltages under different RGP pitches. In addition,Figure as shown in Figure of14, twovoltages receiving fibers collect two light signals, whose 13. Comparisons thethe output under different RGP pitches. phase difference is approximately equal to 90°. Based on phase difference, the measurement In addition, as shown in Figure 14, the two receiving fibers collect two light signals, whose phase ˝ resolution could be improved to to about µm with 400-fold subdivision. When RGP resolution moves In is addition, as shown in Figure the twoon receiving fibers collectthe two lightthe signals, whose difference approximately equal 9014,.0.76 Based phase difference, measurement forward, the phase of signal 1 is behind that of signal 2, and vice versa, then the function of phase difference is approximately equal to 90°. Based on phase difference, the measurement could be improved to about 0.76 µm with 400-fold subdivision. When the RGP moves forward, discerning could direction can be realized practically. Moreover, the subdivision. exit peak and valley errors mainly resolution be improved to about 0.76 µm with 400-fold When the RGP moves the phase of signal 1 is behind that of signal 2, and vice versa, then the function of discerning caused bythethephase reflectivity inhomogeneity of the lead then to the forward, of signal 1 is behind that of RGP, signal which 2, andwould vice versa, thereduction function of direction can be realized practically. Moreover, the exit peak and valley errors mainly caused by the measurement accuracy Throughpractically. high-precision fabrication processes and valley assurance measures, discerning direction can[34]. be realized Moreover, the exit peak and errors mainly reflectivity inhomogeneity of the RGP, which would lead to the reduction of measurement accuracy [34]. the measurement accuracy could be improved caused by the reflectivity inhomogeneity of accordingly. the RGP, which would lead to the reduction of Through high-precision processes and assurance accuracy measurement accuracyfabrication [34]. Through high-precision fabricationmeasures, processes the and measurement assurance measures, could improved accordingly. thebe measurement accuracy could be improved accordingly. Figure 14. Two receiving fiber signals with phase difference. 5.3. Large Range Measurement Figure Experiments 14. Two receiving fiber signals with phase difference. Figure 14. Two receiving fiber signals with phase difference. It can be known from the system structure and operating principle that the measurement range 5.3. Large Range Measurement Experiments is related to the length of the RGP, and a large range measurement could be obtained with a long 5.3. Large Range Measurement Experiments RGP.ItIncan these experiments, RGP length takes the value of 300 principle mm, and that the grating pitch distance of be known from the system structure and operating the measurement range Itrelated can be known from the system structure and operating principle thatbethe measurement range is 12 mil. The lengths of the emitting receiving fibers are both 40 m. is to the length of the RGP,fiber andand a large range measurement could obtained with a long related toInthe length of the RGP, and a large range measurement begrating obtained with a longofRGP. RGP. these experiments, RGP length takes the value of 300 mm,could and the pitch distance 12 mil. The lengths RGP of thelength emitting fiberthe and receiving fibers both m. In these experiments, takes value of 300 mm,are and the40grating pitch distance of 12 mil. The lengths of the emitting fiber and receiving fibers are both 40 m. Sensors 2016, 16, 808 12 of 15 Sensors 2016, 16, 808 12 of 15 In order order to to investigate investigate the the system system accuracy, accuracy, the In the measured measured object object moves moves by by the the following following stages: stages: (a)-move forward for 50 mm at a speed of 2 mm/s; (b)-move forward for 150 mm at a speed (a)-move forward for 50 mm at a speed of 2 mm/s; (b)-move forward for 150 mm at a speed of of 66 mm/s; mm/s; and (c)-move (c)-move backward backward for for 200 200 mm mm at at aa speed Thetotal totalmeasurement measurementrange rangeisis400 400mm. mm. and speed of of 44 mm/s. mm/s. The Figure 15 has shown the accuracy errors between the proposed RFODS and commercial LDS under Figure 15 has shown the accuracy errors between the proposed RFODS and commercial LDS under different velocities movement time to 10 in seach stage.stage. The peak-peak accuracy errors different velocitiesfrom fromthe theinitial initial movement time to s10 in each The peak-peak accuracy are ˘0.0205 mm, ˘0.0393 mm, mm, ˘0.0303 mmmm in (a), respectively. It isItobvious thatthat the errors are ±0.0205 mm, ±0.0393 ±0.0303 in (b), (a), (c) (b),stages, (c) stages, respectively. is obvious measurement system could maintain its accuracy well over a large measurement range. the measurement system could maintain its accuracy well over a large measurement range. Figure Figure 15. 15. Accuracy Accuracy error error between between the the proposed proposed RFODS RFODS and and LDS LDS under under different different velocities. velocities. 5.4. Temperature Test 5.4. Temperature Test Due to the capability of remote measurement by optical fiber, only the fiber probe is needed to Due to the capability of remote measurement by optical fiber, only the fiber probe is needed to be be put near the measurand; the other components could be placed in a favorable environment away put near the measurand; the other components could be placed in a favorable environment away from from the measurand. Therefore, the proposed RFODS has the capability to work in some harsh the measurand. Therefore, the proposed RFODS has the capability to work in some harsh conditions, conditions, such as high to low temperature. To obtain system performance in high to low such as high to low temperature. To obtain system performance in high to low temperature working temperature working environments, an incubator is used to set up the experimental platform. As environments, an incubator is used to set up the experimental platform. As shown in Figure 16, the shown in Figure 16, the measurand, fiber probe, RGP, and stages are placed inside the incubator, measurand, fiber probe, RGP, and stages are placed inside the incubator, while the other modules while the other modules are put outside. In the experimental process, the incubator temperature are put outside. In the experimental process, the incubator temperature change from ´20 to 80 ˝ C by change from −20 to 80 °C by every 10 °C. At each temperature, the measurand moves forward for every 10 ˝ C. At each temperature, the measurand moves forward for 20 mm at a speed of 625 µm/s 20 mm at a speed of 625 µm/s three times. At the same time the light signals are acquired by the three times. At the same time the light signals are acquired by the device outside the incubator. device outside the incubator. The movement velocities under different temperatures can be The movement velocities under different temperatures can be obtained, as shown in Table 4, where V 1 , obtained, as shown in Table 4, where V1, V2, and V3 are the velocities for each time, respectively.˝The V 2 , and V 3 are the velocities for each time, respectively. The average velocity of three times at 20 C is average velocity of three times at 20 °C is treated as the nominal value. Accordingly, the relative treated as the nominal value. Accordingly, the relative errors α1 , α2 , and α3 could also be computed errors α1, α2, and α3 could also be computed and shown in Table 4. and shown in Table 4. It can be seen that the proposed RFODS works well in the temperature range from −20 to 80 °C, and the relative error is less than ±0.5%. When the temperature is higher than 80 °C, the system performance is influenced by softening and deformation of the fibers. With glass or special optical fibers, the measurement system has the potential to work at higher temperature. Sensors 2016, 16, 808 13 of 15 Sensors 2016, 16, 808 13 of 15 (a) (b) Figure 16. Experimental system for temperature test. (a) Overall system appearance; and (b) system Figure 16. Experimental system for temperature test. (a) Overall system appearance; and (b) system architecture architecture schematic. schematic. Table 4. Analysis of measurement error between proposed RFODS and LDS. Table 4. Analysis of measurement error between proposed RFODS and LDS. Temperature (°C) ˝ Temperature −20 ( C) V1 (µm/s) V 1 (µm/s) 622.2 V2 (µm/s) V 2 621.9 (µm/s) ´20 −10 ´100 0 10 10 2020 3030 4040 50 50 60 7060 8070 622.2 622.8 622.8 623.4 623.4 623.8 623.8 623.7 623.7 624.3 624.3 624.3 624.3 625.5 625.5 622.2 622.2 625.1 625.1 624.5 621.9 623.3 623.3 623.7 623.7 622.7 622.7 622.7 622.7 624.0 624.0 625.8 625.8 626.3 626.3 623.9 623.9 624.7 624.7 625.5 80 624.5 625.5 V3 (µm/s) V 3621.8 (µm/s) 621.8 623.2 623.2 624.6 624.6 623.4 623.4 623.7 623.7 624.3 624.3 625.3 625.3 625.4 625.4 624.6 624.6 625.3 625.3 625.3 625.3 α1 α1 −0.188% ´0.188% −0.091% ´0.091% 0.005% 0.005% 0.069% 0.069% 0.053% 0.053% 0.149% 0.149% 0.149% 0.149% 0.342% 0.342% ´0.188% −0.188% 0.278% 0.278% 0.181% 0.181% α2 α3 α2 α3 −0.236% −0.252% ´0.236% −0.027% ´0.252% −0.011% ´0.011% ´0.027% 0.053% 0.197% 0.053% 0.197% −0.107% 0.005% ´0.107% 0.005% −0.107% ´0.107% 0.053% 0.053% 0.101% 0.149% 0.101% 0.149% 0.390% 0.310% 0.390% 0.310% 0.470% 0.326% 0.470% 0.326% 0.085% 0.197% 0.085% 0.197% 0.213% 0.310% 0.213% 0.310% 0.342% 0.310% 0.342% 0.310% It can be seen that the proposed RFODS works well in the temperature range from ´20 to 80 ˝ C, 6. Conclusions and the relative error is less than ˘0.5%. When the temperature is higher than 80 ˝ C, the system performance is influenced by softening and deformation of the fibers. With or special system optical In this paper, we demonstrate an optical fiber measurement method andglass experimental fibers, the measurement has athe potentialgrating to workpanel. at higher temperature. for lateral displacement system based on reflective Through theoretical derivation, the optimized distance between the emitting fiber ends and the reflective grating panel is obtained. By 6. Conclusions means of modeling and simulation, the efficiency parameters are verified, and the relationship between thepaper, displacement and the optical power ismethod summarized. Experiments under In this we demonstrate an received optical fiber measurement and experimental system for different grating pitches indicate that the proposed method can achieve a long-distance lateral displacement based on a reflective grating panel. Through theoretical derivation, the optimized measurement, especially for applications harsh working grating environments with high temperatures. distance between the emitting fiber ends in and the reflective panel is obtained. By means of Moreover, utilizing the two receiving fiber signals with phase difference, discerning direction modeling and simulation, the efficiency parameters are verified, and the relationship betweenand the subdivision be received realized. optical If larger multiples of subdivision are applied, and the reflective displacementcould and the power is summarized. Experiments under different grating grating fabricated is potential to reduce system uncertainties and greatly pitches panel indicate that the precisely, proposed there method can achieve a long-distance measurement, especially improve the measurement accuracy. for applications in harsh working environments with high temperatures. Moreover, utilizing the two receiving fiber signals with phase difference, discerning direction and subdivision could be realized. Acknowledgments: This research is supported by the State Key Laboratory of Precision Measurement If larger multiples of subdivision are applied, and the reflective grating panel fabricated precisely, Technology and Instruments, the National Natural Science Foundation of China (Grant No. 51275259 and there is potential to reduce system uncertainties and greatly improve the measurement accuracy. No. 61362035) and the National Important Scientific Instrument Development Program of China (Grant No. 2011YQ030134). Acknowledgments: This research is supported by the State Key Laboratory of Precision Measurement Technology and Instruments, the National Science Foundation of China measurement (Grant No. 51275259 andZhaohui No. 61362035) and Author Contributions: YuheNatural Li developed the displacement method; Hu was the National Important Scientific Instrument Development Program of China (Grant No. 2011YQ030134). responsible for the design and simulation of optical path; Kaisen Guan and Yanxiang Chen carried out the Author Contributions: Yuhe Li developed the displacement method; Hu was responsible intensity calculation; Zhaohui Hu involved in fabricatingmeasurement the reflective gratingZhaohui panel and seting up the for the design and simulation of optical path; Kaisen Guan and Yanxiang Chen carried out the intensity calculation; measurement system; Yuhe Li and Kaisen Guan participated in the experiments and the most of result analysis. 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Master’s Thesis, Tsinghua University, Beijing, China, 30 May 2014. © 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).