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Opt 35 1566–1573[20] Bönsch G, Potulski E 1998 Measurement of the refractive index of air andcomparison with modified Edlén’s formulae Metrologia 35 133–139[21] Leach R K, Jackson K, Hart A 1997 Measurement of gauge blocks by interferometry: measurement of the phase change at reflection NPL ReportMOT 11[22] Decker J E, Pekelsky J R 1997 Uncertainty evaluation for the measurementof gauge blocks by optical interferometry Metrologia 34 479–493[23] Haitjema H, Kotte G 1998 Long gauge block measurements based ona Twyman-Green interferometer and three stabilized lasers Proc.
SPIE 347725–34[24] Preston-Thomas H 1990 The International Temperature Scale of 1990(ITS-90) Metrologia 27 3–1083This page intentionally left blankCHAPTER 5Displacement measurement5.1 Introduction to displacement measurementAt the heart of all instruments that measure a change in length, or coordinates, are displacement sensors. Displacement sensors measure the distancebetween a start position and an end position, for example the verticaldistance moved by a surface measurement probe as it responds to surfacefeatures. Displacement sensors can be contacting or non-contacting, andoften can be configured to measure velocity and acceleration. Displacementsensors can be used to measure a whole range of measurands such asdeformation, distortion, thermal expansion, thickness (usually by using twosensors in a differential mode), vibration, spindle motion, fluid level, strain,mechanical shock and many more.
Many length sensors are relative in theiroperation, i.e. they have no zero or datum. For this type of sensor the zero ofthe system is some arbitrary position at power-up. An example of a relativesystem is a laser interferometer. Many encoder-based systems have a defineddatum mark that defines the zero position or have absolute position information encoded on the track. An example of an absolute sensor is a lasertime-of-flight system or certain types of angular encoder.There are many types of displacement sensor that can achieve resolutionsof the order of nanometres and less, and only the most common types arediscussed here.
The reader can consult several modern reviews and books thatdiscuss many more forms of displacement sensor (see for example [1–3]).Displacement sensors are made up of several components, including theactual sensing device, a transduction mechanism to convert the measurementsignal to an electrical signal, and signal-processing electronics. Only themeasurement mechanisms will be covered here, but there are severalcomprehensive texts that can be consulted on the transduction and signalprocessing systems (see for example [4]).Fundamental Principles of Engineering NanometrologyCopyright Ó 2010 by Elsevier Inc.
All rights reserved.CONTENTSIntroduction todisplacementmeasurementDisplacementinterferometryCapacitivedisplacement sensorsInductivedisplacement sensorsOptical encodersOptical fibre sensorsCalibration ofdisplacement sensorsReferences8586C H A P T ER 5 : Displacement measurement5.2 Displacement interferometry5.2.1 Basics of displacement interferometryDisplacement interferometry is usually based on the Michelson configurationor some variant of that basic design.
In chapter 4 we introduced the Michelsonand Twyman-Green interferometers for the measurement of static lengthand most of the practicalities in using such interferometers apply todisplacement measurement. Displacement measurement, being simplya change in length, is usually carried out by counting the number of fringes asthe object being measured (or reference surface) is displaced. Just as with gaugeblock interferometry the displacement is measured as an integer number ofwhole fringes and a fringe fraction. Most displacement interferometers requiretwo fringe patterns that are 90 out of phase (referred to as phase quadrature) toallow bi-directional fringe counting and to simplify the fringe analysis.Photodetectors and digital electronics are used to count the fringes and thefraction is determined by electronically sub-dividing the fringe [5]. With thismethod, fringe sub-divisions of l/1000 are common, giving sub-nanometreresolutions.
There are many homodyne and heterodyne interferometerscommercially available and the realization of sub-nanometre accuracies ina practical set-up is an active area of research [6]. Many of the modernadvances in high-accuracy interferometry come from the communitysearching for the effects of gravitational waves [7].5.2.2 Homodyne interferometryFigure 5.1 shows a homodyne interferometer configuration. The homodyneinterferometer uses a single frequency, f1, laser beam. Often this frequency isone of the modes of a two-mode stabilized laser (see section 2.9.3.1). Thebeam from the stationary reference is returned to the beam-splitter witha frequency f1, but the beam from the moving measurement path is returnedwith a Doppler shifted frequency of f1 df. These beams interfere in thebeam-splitter and enter the photodetector.
The Doppler shifted frequencygives rise to a count rate, dN/dt, which is equal to f (2v/c), where v is thevelocity of the retro-reflector and c is the velocity of light. Integration ofthe count over time, t, leads to a fringe count, N ¼ 2d/l, where d is thedisplacement being measured.In a typical homodyne interferometer using a polarized beam, themeasurement arm contains a quarter-wave plate, which results in themeasurement and reference beams having a phase separation of 90 (for bidirectional fringe counting).
In some cases, where an un-polarized beam isused [8], a coating is applied to the beam-splitter to give the required phaseDisplacement interferometryFIGURE 5.1 Homodyne interferometer configuration.shift [9]. After traversing their respective paths, the two beams re-combine inthe beam-splitter to produce an interference pattern.Homodyne interferometers have an advantage over heterodyne interferometers (see section 5.2.3) because the reference and measurement beamsare split at the interferometer and not inside the laser (or at an acoustooptic modulator).
This means that the light can be delivered to the interferometer via a standard fibre optic cable. In the heterodyne interferometera polarization-preserving (birefringent) optical fibre has to be employed [10].Therefore, fibre temperature or stress changes alter the relative path lengthsof the interferometer’s reference and measurement beams, causing drift.
Asolution to this problem is to employ a further photo-detector that ispositioned after the fibre optic cable [11].Homodyne interferometers can have sub-nanometre resolutions andnanometre-level accuracies, usually limited by their non-linearity (seesection 5.2.8.4). Their speed limit depends on the electronics and thedetector photon noise; see also section 5.2.4. For a speed of 1 m$s1 and fourcounts per 0.3 mm cycle, a 3 MHz signal must be measured within 1 Hz.Maximum speeds of 4 m$s1 with nanometre resolutions are claimed bysome instrument manufacturers.5.2.3 Heterodyne interferometryFigure 5.2 shows a heterodyne interferometer configuration.
The output beamfrom a dual-frequency laser source contains two orthogonal polarizations, one8788C H A P T ER 5 : Displacement measurementFIGURE 5.2 Heterodyne interferometer configuration.with a frequency of f1 and the other with a frequency of f2 (separated by about3 MHz using the Zeeman effect [12] or some other means – see section 2.9.4).A polarizing beam-splitter reflects the light with frequency f1 into the referencepath. Light with frequency f2 passes through the beam-splitter into themeasurement path where it strikes the moving retro-reflector causing thefrequency of the reflected beam to be Doppler shifted by df.
This reflectedbeam is then combined with the reference light in the beam-splitter andreturned to a photodetector with a beat frequency of f2 f1 df. This signal ismixed with the reference signal that continuously monitors the frequencydifference, f2 f1. The beat difference, df, gives rise to a count rate, dN/dt,which is equal to f (2v/c), where v is the velocity of the retro-reflector and c isthe velocity of light. Integration of the count over time, t, leads to a fringecount, N ¼ 2d/l, where d is the displacement being measured.With a typical reference beat of around 3 MHz, it is possible to monitor dfvalues up to 3 MHz before introducing ambiguities due to the beat crossingthrough zero.
This limits the target speed possible in this case to less than1 m$s1, which could be a constraint in some applications. An alternativemethod of producing a two-frequency laser beam is to use an acousto-opticfrequency shifter. This method has the advantage that the frequency difference can be much higher, so that higher count rates can be handled [13].Many variations on the theme in Figure 5.2 have been developed whichimprove both the speed of response, measurement accuracy and resolution.Modern commercial heterodyne interferometers can be configured to measureboth displacement and angle (see for example the xy interferometers in [14]).Displacement interferometry5.2.4 Fringe counting and sub-divisionThere are two main types of optical fringe counting methods: hardware fringecounting and software fringe counting [15].
Hardware fringe counting [5]utilises hardware circuits to subdivide and count interference fringes. Itsprinciple of operation is as follows. Two interference signals (sine and cosine)with p/2 phase difference are converted into two square waves by means ofa trigger circuit. Activated by the rising edge of the sine-equivalent squarewave, a reversible counter adds or subtracts counts according to the movingdirection of the measured object, which is determined by the level of thecosine-equivalent square wave that corresponds to the rising edge of the sineequivalent square wave. The advantages of the hardware fringe countingmethod are good real-time performance and relatively simple realization.However, the electronically countable shift of p/2 corresponds to a phase shiftof l/4 (or l/8 in a double pass interferometer – see section 5.2.5), which definesthe resolution limit for most existing hardware fringe counting systems.Software fringe counting mainly uses software to subdivide and countinterference fringes [16].
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