Lecture Notes on Metrology and Interferometry

Fundamentals of Metrology and Interferometry What is ISO-GPS? Geometrical Product Specification An internationally accepted concept covering all of the different require- ments indicated on a technical drawing Implemented through...

Fundamentals of Metrology and Interferometry What is ISO-GPS? Geometrical Product Specification An internationally accepted concept covering all of the different require- ments indicated on a technical drawing Implemented through a series of standards American version: ”Geometrical Dimensioning and Tolerancing”  ISO 1 ISO Standard No 1 - The Standard Reference Temperature The standard temperature for geometrical product specification and verification is fixed at 20° C All dimensions and tolerances on the drawing apply to a component that is at 20° C We have to measure at this temperature or correct for thermal expansion The GPS Tolerancing Model GPS contains fundamental technical rules that define technical drawings. Dimensional vs. Geometric Tolerances The position of a hole will have different meaning whether it is decided by dimensional tolerances or by positional tolerances. Geometrical Tolerances Symbols for Geometrical Tolerances What does the tolerance mean? Which is the datum? The surface should be contained between two parallel planes separated by 0.1 mm What does the tolerance mean? The tolerance zone is a cylinder with diameter 0.1 mm perpendicular to the top surface of the object. CMM – Coordinate Measuring Machine  Geometric Elements 1. part of sphere derived: centre line 2. cylinder 3. plane 4. part of torus 5. cone 6. plane... from SIS handbook 545:2005 Current Definition of the Metre Since 1983 the metre is defined as the distance that light travels in vacuum in 1/299 792 458 s. Laser setup for realizing a metre normal.  Prerequisites for a universal measurement system An international coherent system for units (SI) The possibility to compare measurement from different places and times (traceability) A unified way of expressing uncertainties of measurement results (GUM – Guide to the expression of uncertainty in measurement)  Traceability BIPM Bureau international des poids et mesures Certified laboratory Certified laboratory Certified laboratory RISE (Research Institutes of Sweden) (earlier SP Sveriges Tekniska Forskningsinstitut Workshop earlier Sveriges Provnings- och Forskningsinstitut earlier Statens Provningsanstalt) Gauge Blocks Carl Edvard Johansson (1864 – 1943) Calipers Pierre Vernier (1580 – 1637) The Micrometer Abbe’s Principle A length measuring instrument should be constructed so that the measured length is in line with the measuring scale.  Abbe’s Principle: A length measuring instrument should be constructed so that the measured length is in line with the measuring scale. Ernst Abbe (1840 - 1905) Go/NoGo gauges Taylor’s Principle (Envelope Principle) A go-gauge should have such a shape that it can check the form of the object in its entire length A nogo-gauge should have such a shape that it can make pointwise check of the object  No, not that Taylor. Taylor Swift (1989 -) Brook Taylor (1685–1731) Frederick Winslow Taylor (1856–1915)  Go No go William Taylor (1865-1937)  Three types of errors Error is the difference between the measured value and the ”true value” ”Error is an idealized concept and errors cannot be known exactly” Known systematic errors Unknown systematic errors Random errors  Cosine Error The measuring scale is not aligned with the intended measured distance. Sine Error  Uncertainty – is different from error Uncertainty is a quantification of the doubt about the measurement result. Formal definition [GUM]: uncertainty (of measurement) – parameter, associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand. Conformance of Non-conformance?  Uncertainties Generally we do not talk about random and systematic uncertainty Instead we divide the uncertainties after the method by which they were evaluated in either: Type A which are estimated using statistics or Type B which are estimated from any other information  Type A uncertainties: Can be described with statistical methods.They are random. Type B uncertainties: Has to be decided subjectively, for example by experience. Can be systematic errors that cannot be determined or corrected  We want to add uncertainties as: We usually use the standard deviations as the u:s. u  u A21  u A2 2    u B21  u B2 2 So we will guess a standard deviation for the type B uncertainties. Gaussian (Normal) Distribution Central Limit Theorem Why is the Gaussian distribution especially useful? The sum of many independent and identically distributed random variables tend to have a Gaussian distribution, no matter what the individual distribution is. Rectangular Distribution  Type A Evaluation of Standard Uncertainty If we have n independent observations (measurements ) q k obtained under the same conditions the best estimate in most cases is n 1 q̄= ∑ q k the arithmetic mean or average n k =1 The variations of the individual observations is characterized by √ n 1 s(q k )= ∑ n−1 j =1 (q j −q̄ )2 the experimental standard deviation The best estimate of the standard deviation of the mean is s (q k ) s( q̄ )= the experimental standard deviation of the mean √n s (q k ) This is taken as the standard uncertainty u=s( q̄ )= √n Document the degree of freedom used, n-1 Type B Evaluation of Standard Uncertainty Previous measurement data Experience or general knowledge of the behaviour and properties of relevant materials and instruments Manufacturer’s specifications Data provided in calibration and other certificates Uncertainties assigned to reference data taken from handbooks Type B Evaluation of Standard Uncertainty The associated estimated standard uncertainties u(x i) is evaluate by scientific judgement. For example: If we have the information that a certain proportion of the values is within a certain interval, it is probably best to assume a normal distribution If we only know that the value can vary within a certain interval [-a,a], we can assume a rectangular distribution which have a standard uncertainty of a 3 If we know that the values near the bounds are less likely, we can assume a triangular distribution with a standard uncertainty (this is tighter than rectangular, so rectangular is safer) a 6  Addition in Quadrature If the evaluated uncertainties of Type A and Type B are uncorrelated they can then be added as u tot  u A21  u A2 2    u B21  u B2 2    Expanded Uncertainty Coverage factor When presenting measurement results or the accuracy of measurement instruments, the data is most of the times given with a coverage factor k. An " expanded uncertainty" can be expressed by multiplying the obtained standard uncertainty by a coverage factor k : U kuc ( y ) The result is the expressed as Y  y U It is important to also state the coverage factor. Confidence Interval/Level The coverage faktor is usually used in order to give the user a level of confidence. If the uncertainty has a normal distribution these levels are, for example: k=1 for a confidence of approx. 68 percent k=1.960 for 95 percent k=2 for 95.45 percent k=2.58 for 99 percent k=3 for 99.7 percent Other distributions have different values for the levels of confidence. If we have an addition of several uncertainties and we don’t have one non-normally distributed uncertainty dominating, the total uncertainty will be approx. normally distributed according to the Central Limit Theorem.  Expressing the Answer Give the measurement result, together with the uncertainty figure, e.g. ”The length of the stick was 20 cm +- 1 cm The statement of the coverage factor and the level of confidence. A recommended wording is: ”The reported uncertainty is based on a standard uncertainty multiplied by a coverage factor k=2, providing a level of confidence of approximately 95% How the uncertainty was estimated (you could refer to a publication where the method is described)  References Free literature about metrology can for example be found in the publications from National Physical Laboratory in Great Britain, http://www.npl.co.uk/publications/good-practice-online-modules/. (NPL 11) Measurement Good Practice Guide No 11, “A Beginner's Guide to Uncertainty of Measurement”. (NPL 79) A National Measurement Good Practice Guide No 79, “Fundamental Good Practice in the Design and Interpretation of Engineering Drawings for Measurement Processes”. (NPL 80) A National Measurement Good Practice Guide No 80, “Fundamental Good Practice in Dimensional Metrology”. (GUM) “Evaluation of Measurement Data — Guide to the Expression of Uncertainty in Measurement”, JCGM 100:2008.  Interference (Superposition principle for linear systems) constructive interference destructive interference  Reference mirror beam splitter moving mirror counter photo sensor The light on the detector will go from bright to dark when the path difference change by half a wavelength i.e. when mirror has been moved a quarter of a wavelength  Interference between two plane waves A useful analogy is to represent two interferring light waves by striped patterns, where the stripes represent the wave fronts and the distance from stripe to stripe represent the wavelength of the light. The moiré pattern formed then represent the interference pattern between the waves. Although the stripe patterns are moved in the direction of the ”wave fronts” the moiré pattern will remain stationary.  C Wavelength Fabry-Pérot Interferometer Often used to resolve light of different wavelength e.g. identifying atomic spectral lines. Flatness of a Gauge Block Measured with an Optical Flat  Reference mirror beam splitter moving mirror counter photo sensor Michelson’s interferometer used for length measurement. A problem with using a simple Michelson interferometer for measurements. Heterodyning interferometer (AC interferometer) 21 measurements!  With the RenishawXL or Hewlett-Packard 5528A interferometer the following measurements can be made: Distance Velocity Angular displacement Flatness Straightness Squareness Parallelism Length and velocity measurements  Some typical specifications Distance measurement:  Resolution: 0.00001mm to 1.0 mm  Length of travel: +- 40 m  Distance between laser head and reflector: +- 61m  Max velocity: +-18 m/min  Some typical specifications Velocity measurement: Accuracy +-0.1% of displayed value Resolution: 0.1 mm/min or 1.0 mm/min  Rule-of-thumb for speed of light compensations Wavelength changes by about on part per million for every: Atmospheric-pressure change of 2.5 mmHg Air-temperature change of 10 C Relative-humidity change of 100%  Deadpath error The deadpath is the distance where no wavelength compensation can be made.  Cosine Error A misalignment will always result in measuring a too short distance Abbe Error Measurement of Angle  Some typical specifications Angular measurement: Accuracy +-0.2% of displayed value +- 0.05 arc-sec per metre of travelled distance between Interferometer and Reflector Resolution: 0.1 or 1.0 arc-seconds Range: +-3600 arc-seconds = 1 degree Straightness Measurement Straightness measurement  Some typical specifications Straightness measurements: Accuracy: 0.4 m per metre of travel Resolution: 0.01 m Straightness deviation range: +-1.5 mm Axial separation range 0.1 – 3 m or 1 – 30 m Squareness Measurement Straightness measurement not corrected for slope 14 12 10 8 6 Series1 4 2 0 -2 0 2 4 6 8 10 12  Corrected for slope 4 3.5 3 2.5 2 1.5 Series1 1 0.5 0 -0.5 0 2 4 6 8 10 12 -1 -1.5 LIGO – Laser Interferometer Gravitational-Wave Observatory Nobel Prize in physics 2017 www.kva.se www.kva.se www.kva.se www.kva.se  https://www.ligo.caltech.edu/video/ligo20160211v6 1 min https://www.ligo.caltech.edu/video/ligo20160215v1 3:36 https://www.ligo.caltech.edu/video/ligo20170216v 11:48 https://www.ligo.caltech.edu/video/LIGO-a-discover-that-shook-the-world 16:24