Standard Small Angle Generator Using Laser Interferometer

Transcription

1 40 Kasetsart J. (Nat. Sci.) 40 : (2006) Kasetsart J. (Nat. Sci.) 40(5) Standard Small Angle Generator Using Laser Interferometer Kittisak Nugkim 1, Kanokpoj Areekul 1 * and Bancha Panacharoensawad 2 ABSTRACT Small angle generator is equipment used for making an adjustable small inclined angle. This equipment is use for calibration of gradual scale of level measurement instrument, such as, bubble level, electronic level, inclinometer, survey scope, collimator, etc. In the research, the possible method to make the low cost high accuracy small angle generator is studied. The materials employed in this project are available in Thailand. The accuracy of the inclined angle measurement relies on the sine principle of triangle. Hypotenuse length (l) of mm is measured by steel ruler which uncertainty about ±0.5 mm. The small displacement of the opposite side ( h) is determined by the Michelson s interferometer. The shift of one fringe is half wavelength of 650 nm laser diode. The constructed system is capable to measure inclined angle of resolution about 325 nm /500 mm or about an order of 1 nm/m. Key words: small angle generator, incline angle, Michelson s Interferometer INTRODUCTION 1. Sine principle According to Naval Plant Representative Office [NPRO] (1977). Considering the basic right angle triangle illustrated in Figure 1. Any angle which is formed between the adjacent and hypotenuse sides of the right angle triangle. There exists one and only one ratio between the lengths of the opposite side against the length of the hypotenuse. Thus, if the certain length of hypotenuse and adjacent side are fixed, and the length of the opposite side is precisely adjustable ( h), the variation of h can generate standard inclined level. The standard inclined level is capable in calibrating several kind of the level measurement equipment. An accuracy of the level measurement depends on how accurate of the Dh measurement. Figure 1 The right angle triangle used to describe sine principle. 1 Metrology and Physics of Instrumentation Research Unit. Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand. 2 Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand.. * Corresponding author, Kanokpoj.A@ku.ac.th

2 Kasetsart J. (Nat. Sci.) 40(5) The incline level standard There are several methods to design reference standard which is used to calibrate the level measurement equipment. The reference standards commonly used to calibrate the level are elaborated as follows. 2.1 Sine Bar, Sine Plate, and Small Angle Generator According to NPRO (1977), a practical small angle calibration with sine bar setting on two stacks of gauge block set is illustrated in Figure 2. In order to obtain accurate measurement, two criteria have to be fulfilled. First, for any sine bar size, the diameters of the two cylinders have to be exactly the same. Second, the distance between centers of both cylinders is precisely known. The triangle illustrated in Figure 2 (a) is an equivalent triangle as defined by the sine bar illustrated in Figure 2 (b). Since the diameters of both cylinders are the same, the angle opposite side is equal to the height difference between the two stacks of gauge blocks. The hypotenuse (l) of the triangle is an accurately known constant. It is the distance between axes os two cylinders. The angle (θ) generated by the sine bar is calculate as follow. sin θ = = difference in gage block stack heights length of sinebar side opposite hypotenuse 2.2 Michelson Interferometer A schematic of the Michelson interferometer is shown in Figure 3 (a). A beam from an extended source S, (1), is split by the beam splitter (BS). A transmitted beam 2 and reflected beam 3 which equal amplitude reflected by mirrors Figure 2 (a) Sine bar for angle level calibration, (b) Sine bar and sine plate Figure 3 (a) Michelson Interferometer, which designed optical path for the system. (b) Hardware design.

3 42 Kasetsart J. (Nat. Sci.) 40(5) M 1 and M 2, respectively. On returning to the beam splitter, beam 2 is now transmitted, and beam 3 is reflected so that they come together to detector. Thus beam 4 includes ray that have traveled difference optical path and will demonstrate interference. At least one of the mirrors is equipped with tilting adjustment screws that allow the surface of M 1 to be made perpendicular to that of M 2. One of the mirrors is also movable along the direction of the beam by means of an accurate track and micrometer screw. The actual interferometer in Figure 3 possesses two optical axes at right angle to one another. The optical path difference between the two beams emerging from the interferometer is p = 2d cos θ, (1) The optical system of Figure 3 (b) is now equivalent to the case of interference due to a plane parallel air film, illuminated by an extended source. Assuming that the two interfering beams are of equal amplitude, the irradiance of the fringe system of circles concentric with the optical axis is given by the equation 2 d I = 4I0 cos (2) 2 δ is the phase difference, defined by equation. δ = k = 2π λ (3) The net optical path difference is = p + r. A relative phase shift π between the two beams occurs because beam 3 experiences one reflections, but beam 2 experiences only one. For dark fringe then, p + r = 2d cos θ + λ 2 = 1 m + 2 λ (4) Or more simply, 2d cosθ = mλ (5) Note: the path differences with m = 1, 2, 3 correspond to dark fringes Equation (5) requires an increase in angular separation of θ of a given small fringe interval m as the mirror spacing d becomes smaller, since λ m θ = (6) 2 d sinθ This means that the fringes are more widely separated when optical path differences are smaller. In fact, if d = λ/2, then from the equation (5), m = cosθ, and the entire field of view encompasses no more than one fringe. For a mirror translation d, and the number m of fringe passing a point at or near the center of the pattern is according to the equation (5). m = 2 d (7) λ As described above, the Michelson interferometer is capable in several kinds of measurement, especially the length measurement. The length measurement for the h of opposite side is designed as follow MATERIAL AND METHODS The experimental design follows the Guidelines for Designing Experiments. This guideline provides steps to carry out the experimental design with Tagushi s method. 1. Choice of factors and levels The temperature factor can be controlled by adjusting air condition, this parameter is divided into 3 levels, i.e. 20 C (TC-1), 23 C (TC-2), and 25 C (TC-3). The ground vibration is an uncontrollable parameter. According to the usual working time, thus the experiment should be performed in separate times defined in 3 levels as 09:00 to 10:00 on the days off (Saturday or Sunday; VC-1), 17:00 to 18:00 on working day (VC-2), and 11:00 to 13:00 on working day (VC- 3). The third parameter is battery life of laser diode, also divide into 3 levels. The first pack is a brand new battery (BC-1), the second is a 2-3 hour used battery (BC-2), and the third is the 5-6 hour used battery (BC-3).

4 Kasetsart J. (Nat. Sci.) 40(5) Choice of experimental design The Tagushi s method (Design of Experiment from the Minitab software) designs the (3 factors 3 levels) experiment as illustrated in table Performing the experiment The precision dial comparator which established at the opposite side of the sine arm is the standard equipment that measuring the displacement of Dh. This dial comparator indicates this displacement in mm. The movable mirror M 1 and the dial comparator are held, they move together. The fringe counter should count the number of fringe(s) proportional to M 1 displacement as recorded in table 3. Table 1 Factors and levels of experimentation. Number Parameter Level-1 Level-2 Level-3 1 Temperature TC-1 TC-2 TC-2 2 Ground vibration VC-1 VC-2 VC-3 3 Battery life BC-1 BC-2 BC-3 Table 2 Design of experimentation. Experiment No. Factor-1 (Temperature) Factor-2 (Vibration) Factor-3 (Battery life) 1 TC-1 VC-1 BC-1 2 TC-1 VC-2 BC-2 3 TC-1 VC-3 BC-3 4 TC-2 VC-1 BC-2 5 TC-2 VC-2 BC-3 6 TC-2 VC-3 BC-1 7 TC-3 VC-1 BC-3 8 TC-3 VC-2 BC-1 9 TC-3 VC-3 BC-2 Table 3 Experimental data with calculated average value, and standard deviation. Exp. Fringe counter indication along 10 mm displacement (Counts) Mean S.D. No. Run1 Run2 Run3 Run4 Run5 Run6 Run7 Run8 Run9 Run

5 44 Kasetsart J. (Nat. Sci.) 40(5) RESULT 1. Data analysis The Minitab software analyzes data from the experimental results, the main effect plot for mean values illustrate in Figure 4, and the main effect plot for standard deviation in Figure 5. The response variable is the fringe counter indication, which this experiment need to know the best factor effect from each variable. The three line graph showed the effect of data cause by varying the controllable factor and level (3 3 levels). 2. Statistical analysis The summary flowchart that used for calculation of uncertainty in measurement is illustrated in Figure The Statistical (Type A) Uncertainty Mathematical model of measurement is defined by f(x i ) = x i ± U (8) where x i = fringe counter measurement (counts). U = uncertainty of measurement (± counts) To calculate the real displacement length ( d) that measured by this system, use equation (7), calculation as follow m = 2 d λ (9) where m = number of fringe(s) those move pass detection point d = mjj or (M 2 ) displacement length (mm) λ = wave length of laser light ( ± 0.01 µm) d = mλ 2 = ( ) / 2 = µm 2.2 Systematic (Type B) Uncertainty. The length measurement which using equipment list above concern with 3 main parts those includes self systematic uncertainty source. Figure 4 Main effect plot for the experimental mean value. Figure 5 Main effect plot for the experimental standard deviation.

6 Kasetsart J. (Nat. Sci.) 40(5) 45 The 3 main parts are: (1) The granite set of small angle generator; (2) The reference standard (precision dial gage); (3) The interferometer set (includes fringe counter). Each of parts perform the same measuring function is (vertical) length measurement, so the sensitivity coefficient c i should equal to 1. Figure 6 Flowchart for the determination of measurement uncertainty. Table 4 Calculated experimental data. Exp. Measurement result report as counts number Mean S.D. Type A No. 10 µm displacement (counts) Run 1-2 Run 3-4 Run 5-6 Run 7-8 Run Note: Type A uncertainty is ± 1 count due to the counter resolution is 1 count. Table 5 Experimental data corrected from fringes number to length (mm). Exp. Result of 10 µm displacement (µm) Mean S.D. Type A No. Run 1-2 Run 3-4 Run 5-6 Run 7-8 Run 9-10 (µm) (µm) (±µm)

7 46 Kasetsart J. (Nat. Sci.) 40(5) Systematic uncertainty budget from the granite set. The small angle generator granite set consists of steel, aluminum, and granite bar, those possible to provide uncertainty to the system. The temperature deviation in photonic laboratory controlled approximately 20 to 23 ± 1 C (for an experiment control at 20 ± 0.5 C) Systematic uncertainty budget from the reference standard. The reference standard used in this thesis is the dial comparator, Mahr, model 1002, which measurements range of ± 25 µm, scale resolution 0.5 µm, and accuracy of 0.6 µm (This standard equipment comply with standard DIN 879-1) Systematic uncertainty budget from the interferometer set. 2.3 Combined uncertainty (U c (l)) The estimate uncertainty list in section 1 (Type A uncertainty) and section 2 (Type B uncertainty) summarize in table 11 in purpose of combined uncertainty calculation as follow M3003 (1997) The uncertainty budgets listed in table 12 should calculate the combined uncertainty for length measurement U c (l) by the Root Sum Square (R.S.S.) calculation as follow n 2 U c (l) = Ui () l = ± µm i= 1 This calculated result is the combined uncertainty which represent variant of ±1σ. 2.4 Expanded Uncertainty (U c (l)) The following step is method to expand the confidence level to 95 % of confidence level. The effective degree of freedom, ν eff 4 u l ν eff = c () = N 4 ui () l i= 1 νi (10) M3003 (1997) In this case, the calculated ν eff is approximate 13295, compare this value in table 10, the coverage factor value k 95 should approximately 2. The expanded uncertainty can calculate as following U 95 = k 95 U c (l) (11) = ± µm Table 6 Uncertainty budget for material thermal expansion. Symbol Source of uncertainty Value ( C) -1 Prob. distribution U i (µm) U tea Thermal expansion for aluminum 23 Rectangular 1.15 U tes Thermal expansion for steel 12 Rectangular 0.3 U teg Thermal expansion for granite base and granite bar 8.5 Rectangular 0.2 Table 7 Uncertainty budget for precision dial gage. Symbol Source of uncertainty Value (mm) Prob. Distribution U i (µm) U rsa Reference standard accuracy Rectangular 0.6 U rsr Reference standard resolution Semi-Rectangular 0.5 Table 8 Uncertainty budget for the interferometer set. Symbol Source of uncertainty Value Prob. Distribution U i U rfc Resolution of fringe counter ±1 count Semi-Rectangular 650 nm U lsr Laser source λ uncertainty ± 10 nm Rectangular 10 nm U teb Thermal expansion for Al base 23 ( C) -1 Rectangular 0.86 µm

8 Kasetsart J. (Nat. Sci.) 40(5) 47 Table 9 Uncertainty budget table. Symbol Source of uncertainty Value (µm) Prob. dist. Divisor c I U i (µm) v i U a Type A Uncertainty 0.16 Normal U tea T.E. for aluminum part 1.15 Rectangular U tes T.E. for steel part 0.3 Rectangular U teg T.E. for granite bar 0.2 Rectangular U rsa Reference std. accuracy 0.6 Rectangular U rsr Reference std. resolution 0.5 Semi-Rectan U rfc Fringe counter resolution 0.65 Semi-Rectan U lsr Laser λ uncertainty 0.01 Rectangular U teb T.E. for aluminum base 0.86 Rectangular U c (l) Combined uncertainty Normal (k=1) Table 10 Uncertainty budget with combined and expanded uncertainty. Symbol Source of uncertainty Value Prob. Divisor C i U i v i ±[µm] distribution ±[µm] U a Type A Uncertainty 0.13 Normal U tea T.E. for aluminum part 1.15 Rectangular U tes T.E. for steel part 0.3 Rectangular U teg T.E. for granite bar 0.2 Rectangular U rsa Reference std. accuracy 0.6 Rectangular U rsr Reference std. resolution 0.5 Semi-Rectan U rfc Fringe counter resolu n 0.65 Semi-Rectan U teb T.E. for aluminum base 0.86 Rectangular U c (l) Combined uncertainty Normal (k=1) U 95 Expanded uncertainty Normal (k=2) LITERATURE CITED Busch, T Fundamental of Dimensional Metrology. 2nd ed. Delmar Publisher Inc. Albany, New York. Eom, T.B Precision Angle Measurement. Ministry of Science and Technology, Korea Standard Research Institute. Taedok Science Town, Korea. Metrology Engineering Center, Dimensional Measurement. 2nd rev. Naval Plant Representative Office. California. United Kingdom Accreditation Service The Expression of Uncertainty and Confidence in Measurement, M st ed. UKAS Teddington, Middlesex, England. Anonymous MINITAB Statistical Software, Release Minitab Inc. Available Source: June 15, 2005.