Принципы нанометрологии (1027623), страница 21

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 21)

Reading the actuator signal orGauge block interferometrya rotary position of the optical parallel at these three settings gives thepossibility of deriving a fringe fraction, f.Recording complete images and applying equation (4.9) is probably themost objective and accurate method to determine f. This is similar to Fizeauinterferometry, although in this case it is usually done with multiple spectralor laser lines in a Michelson configuration.4.5.3.2 Multiple wavelength interferometry analysisIf just a single fraction of a single wavelength is known, the gauge block lengthmust be known beforehand within an uncertainty of 0.15 mm in order to defineN in equation (4.17) within one integer unit.

For gauge blocks to be calibratedthis level of prior knowledge is usually not the case – see Table 4.1 – and it iscommon practice to solve this problem by using multiple wavelengths. In theoriginal gauge block interferometers this was usually possible as spectrallamps were used that emitted several lines with an appropriate coherencelength. In modern interferometers laser sources are normally used, and thedemand for multiple wavelength operation is met with multiple laser sources.For multiple wavelengths, li (i ¼ 1, 2, .), equation (4.17) can be rewrittenasl1;nl2;nl3;nðN1 þ f1 Þ ¼ðN2 þ f2 Þ ¼ðN3 þ f3 Þ ¼ .L ¼2nðl1 Þ2nðl2 Þ2nðl3 Þ(4.18)However, because of a limited uncertainty in the fringe fraction determinations there is not a single length that can meet the requirements ofequation (4.18) for all wavelengths and fractions. There are several strategiesfor finding an optimal solution for the length; for example, for the longestwavelength a set of possible solutions around the nominal length can betaken and for each of these lengths the closest solution for the possiblelengths for the measurements at the other wavelengths can be calculated.The average of the length with the least dispersion is then taken as the finalvalue.

This method is known as the method of exact fractions and hassimilarities with reading a vernier on a ruler.More generally, equation (4.18) can be written as a least-squares problem.The error function is given byc2 ¼KXi¼1!2li;nðNi þ fi ÞLe 2nðli Þ(4.19)where K is the number of wavelengths used and Le is the estimated length.For ideal measurements, c2 ¼ 0 for Le ¼ L. For real measurements, the best7576C H A P T ER 4 : Length traceability using interferometryestimate for L is the value for Le where c2 is minimal. For any length, Le, firstthe value of Ni that gives a solution closest to L has to be calculated for eachwavelength before calculating c2.

As equation (4.19) has many local minima(every 0.3 mm), it must be solved by a broad search around the nominal value,for example as was implicitly done in the procedure described above. Todistinguish between two adjacent solutions the fringe fractions must bedetermined accurately enough.

This demand is higher if the wavelengthsare closer together. For example, for wavelength l1 ¼ 633 nm (red) andl2 ¼ 543 nm (green), the fractions must be determined within 15% in orderto ensure that a solution that is 0.3 mm in error is not found.Multiple wavelengths still give periodic solutions where c2 is minimal,but instead of 0.3 mm, these are further apart; in the example of the twowavelengths just given, this period becomes 2.5 mm. If two wavelengths arecloser together, the demand on the accuracy of the fringe fraction determination is increased accordingly, and the period between solutions increases.Using more than two wavelengths further increases the period of thesolutions; the wavelength range determines the demand on the accuracy ofthe fraction determination. A common strategy for obtaining an approximatevalue for Le, with an uncertainty at least within the larger periodicity, is tocarry out a mechanical comparison with a calibrated gauge block.4.5.3.3 Vacuum wavelengthThe uncertainty in the length of a gauge block measured by interferometrydirectly depends on the accuracy of the determination of the vacuum wavelength.

In the case of spectral lamps these are more or less natural constants,and the lines of krypton and cadmium are (still) even defined as primarystandards. Stabilised lasers must be calibrated using a beat measurementagainst a primary standard, as described in section 2.9.5. When usingmultiple wavelengths, especially for larger lengths up to 1 m, a small deviation of the vacuum wavelength can cause large errors because a solution isfound one or more fringe numbers in error.

For example, for a 1 m gauge blockan error of 4 108 in wavelength will result in the wrong calculated valuefor N such that the error is 3 107 (one fringe in a metre). This limits themaximum length that can be determined depending on the accuracy of thewavelengths.4.5.3.4 Thermal effectsThe reference temperature for gauge block measurements is defined in thespecification standard ISO 1 [15] to be 20 C, exactly. 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.

Характеристики

Список файлов учебной работы