Принципы нанометрологии (1027506), страница 21
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The reason that it isnecessary to specify a temperature is because all gauge blocks will change sizewhen their temperature changes due to thermal expansion. The amount byGauge block interferometrywhich the material changes length per degree temperature change is thecoefficient of thermal expansion, a. For a typical steel gauge block, thecoefficient of thermal expansion is about 11.5 106 K1 and for a tungstencarbide gauge block it is nearer 4.23 106 K1.In order to correct for the change in length due to thermal expansion, it isnecessary to measure the temperature of the gauge block at the same time asthe length is being measured. The correction factor can be derived fromLðTÞ ¼ Lð20Þ ð1 þ a½T 20Þ(4.20)where L(T) is the length at temperature, T (in degrees celsius), and L(20) isthe length at 20 C.Equation (4.20) indicates that an accurate temperature measurement ismore critical when a is large and that knowledge of a is more critical if thetemperature deviates from 20 C.
For example, for a one part per millionper degree celsius error, Da, in the expansion coefficient, the error is 100nm for a 100 mm gauge block at 21 C. For a ¼ 10 106 K1 a 0.1 Cuncertainty in the temperature gives 100 nm uncertainty in a 100 mmgauge block.4.5.3.5 Refractive index measurementThe actual wavelength depends on the frequency and the refractive index ofthe air in the path adjacent to the gauge block.
In very accurate interferometers, for long gauge blocks, the refractive index is measured directly bya refractometer that may effectively be described as a transparent gauge blockcontaining a vacuum.The refractive index of air is directly related to the air density, which itselfis influenced by:-air temperature;-air pressure;-air humidity;-other gases in the air (for example, carbon dioxide).The last of these influences, other gases, has a negligible effect and canusually be ignored.
So we need to measure the air temperature, air pressureand humidity. We then use well-known equations to calculate the airrefractive index from these measured parameters.These equations go by several names, depending on the exact equationsused, and are known by the names of the scientists who derived them.Examples include Edlén [16], Birch and Downs (also known as the modified7778C H A P T ER 4 : Length traceability using interferometryTable 4.3Effect of parameters on refractive index: RH is relative humidityEffectSensitivityAir pressureAir temperatureAir humidityWavelength2.79.31.02.0107107108108Variation neededfor change of 10 nm in 100 mmL/mbarL/ CL/%RHL/nm0.37 mbar0.11 C10% RH–Edlén equation) [17,18], Ciddor [19] and Bönsch [20].
NIST has published allthe equations and considerations on their website, including an on-linecalculator: see emtoolbox.nist.gov/Wavelength/Abstract.asp. It may be usefulto note the sensitivity of the refractive index to these various parameters, asshown in Table 4.3.From Table 4.3 it can be seen that if one wishes to reduce the contributionof these potential error sources to below 1 107 L (i.e. 10 nm in 100 mmlength) then one needs to make air pressure measurement with an uncertainty below 0.4 mbar, air temperature measurement to better than 0.1 Cand air humidity measurement to better than 10 % RH (relative humidity).Such measurements are not trivial, but well achievable with commercialinstruments. The wavelength also needs to be known accurately enough –within small fractions of a nanometre (it is mentioned here for completeness,as the refractive index is also wavelength-dependent).4.5.3.6 Aperture correctionA subtle optical effect that is less obvious than the previous uncertaintyinfluences is the so-called aperture correction.
The figures in section 4.4show the light sources as point sources, but in reality a light source hasa finite aperture. This means that light does not only strike the gauge blockand reference plane exactly perpendicular, but also at a small angle. Thismakes the gauge block appear shorter than it really is. The correction for thiseffect for a circular aperture is given byDL ¼LD216f 2(4.21)where D is the aperture and f is the focal length of the collimating lens.Taking some typical numbers, D ¼ 0.5 mm, f ¼ 200 mm, L ¼ 100 mm, wefind DL ¼ 0.04 mm. This correction is much larger in the case of an interference microscope, where it may amount to up to 10% of the measuredheight (see section 6.7.1).Gauge block interferometryIn some interferometer designs there is a small angle between theimpinging light and the observed light that gives rise to a similar correctionknown as the obliquity correction.4.5.3.7 Surface and phase change effectsAs indicated in the gauge block definition, the gauge block has to be wrung onto a platen having the same material and surface roughness properties as thegauge block.
In practice this can be approached, but never guaranteed.Sometimes glass or quartz is preferred as a platen because the wringingcondition can be checked through the platen. Because of the complexrefractive index of metals, light effectively penetrates into the material beforebeing reflected, so a metal gauge block on a glass platen will be measured astoo short.
Additional to this is the gauge block roughness effect. Typical totalcorrection values are 0.05 mm for a steel gauge block on a glass platen and0.01 mm for a tungsten carbide gauge block on a glass platen.A very practical way of determining the surface effects is wringing a stackof two (or more) gauges together on a platen and comparing the length of thestack to the sum of the individually measured gauge blocks.This is illustrated for two gauge blocks in Figure 4.18, where g and p arethe apparent displacements of the optical surface from the mechanicalsurface, f is the wringing film thickness and Li are the defined (mechanical)lengths of individual gauges.
It can be shown that the measured length of thecombined stack minus the individual measured length is the correction pergauge. This method can be extended to multiple gauge blocks to reduce theuncertainties. Here it is assumed that the gauge blocks are from the samematerial and have nominally the same surface texture.Other methods for measuring corrections for surface effects of the gaugeblock and platen have been proposed and are used in some NMIs (see forexample [21]).
Such methods can offer a slightly reduced uncertainty for theFIGURE 4.18 Method for determining a surface and phase change correction.7980C H A P T ER 4 : Length traceability using interferometryphase correction, but are often difficult to set up and can give results that maybe difficult to interpret.4.5.4 Sources of error in gauge block interferometryIn this section, some more detailed considerations are given on the errorsgenerated by the different factors mentioned in section 4.5.3 (and see [22] fora more thorough treatment).4.5.4.1 Fringe fraction determination uncertaintyThe accuracy of the fringe fraction determination is governed by therepeatability of the measurement process, the quality of the gauge block, andthe flatness and parallelism of the end faces.
With visual fringe fractiondetermination, an uncertainty of 5 %, corresponding to approximately 15 nm,is considered as a limit. With photoelectric determination this limit can bereduced to a few nanometres; however, the reproducibility of the wringingprocess is of the same order of magnitude.4.5.4.2 Multi-wavelength interferometry uncertaintyAs previously mentioned, the determination of the correct interference orderis the main issue when using multiple wavelength interferometry. For thispurpose it is absolutely necessary that the fringe fractions are determinedwithin 10 % to 15 %.
Once the fringe fractions are less sure, the measurement becomes meaningless. Also a correct pre-determination of the gaugeblock length, for example by mechanical comparison, is essential if the gaugeblock is being calibrated for a first time.4.5.4.3 Vacuum wavelength uncertaintyThe uncertainty in the wavelength used is directly reflected in the calculatedlength, so long as the fringe order is uniquely defined. Stabilized lasers needperiodic re-calibration, preferably against a primary standard.
If one laser iscalibrated, other lasers can be calibrated using gauge blocks – from a knownlength and measured fraction, the real wavelength of a light source can bemeasured. An unknown fringe order now leads to a possible number ofwavelengths. By repeating the procedure for different gauge block lengths,a wavelength can be uniquely determined [23].4.5.4.4 Temperature uncertaintyThe temperature measurement is essential, and, if the temperature isdifferent from 20 C, the expansion coefficient must also be known.
Mosttemperature sensors can be calibrated to low uncertainties – calibration ofGauge block interferometrya platinum-resistance thermometer to 0.01 C is not a significant problem fora good calibration laboratory. The problem with temperature measurementof a material is that the temperature of the material must be transferred tothe sensor. This depends on thermal conductivity, thermal equilibrium withthe environment, self-heat of the sensor and other factors. For this reasonlong waiting times and multiple sensors attached to longer gauge blocks(L > 100 mm) are common.An approach that was already used in the first gauge block interferometers is to have a larger thermally conductive block near the gauge block.This block is measured with an accurate absolute sensor and the differenceof the gauge block with this reference block is determined bya thermocouple.For the highest uncertainties the uncertainty in the temperature scalebecomes relevant; for example when ITS-90 was introduced in 1990 [24], thelongest gauge blocks made a small but significant jump in their length.4.5.4.5 Refractive index uncertaintyIf the refractive index is established by indirect measurement of the airparameters, it is dependent on the determination of these parameters and inaddition a small uncertainty of typically 2 108 in the equation itself mustbe taken into account.
The air temperature measurement may be mostproblematic because of possible self-heating of sensors that measure airtemperature. Also, when the air temperature is different from the gauge blocktemperature it is questionable exactly what is the air temperature near thegauge block that is measured.4.5.4.6 Aperture correction uncertaintyAs the aperture correction is usually small, an error in this correction doesnot necessarily have dramatic consequences. If possible, the aperture can beenlarged or reduced to check whether the estimate is reasonable.
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