Richard Leach - Fundamental prinsiples of engineering nanometrology (778895), страница 20
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The interferometer can also be madewith spherical mirrors. In this case the equation for the finesse changessomewhat. This and other details of the Fabry-Pérot interferometer areextensively treated in [13].Fabry-Pérot interferometers have many applications in spectroscopy.However, in engineering nanometrology they are used as the cavity in lasersand they can be used to generate very small, very well defined displacements,either as part of a laser (the so-called ‘measuring laser’) or as an externalcavity. This is treated in more detail in section 5.7.1.2.7172C H A P T ER 4 : Length traceability using interferometry4.5 Gauge block interferometry4.5.1 Gauge blocks and interferometryAs discussed in section 4.2 the length of a gauge block wrung to a platen canbe measured using interferometry.
The ISO definition of a gauge block lengthhas a two-fold purpose: (1) to ensure that the length can be measured byinterferometry, and (2) to ensure that there is no additional length due towringing. An issue that is not obvious from the definition is whether the twosided length of a gauge block after calibration by interferometry coincideswith the mechanical length, for example as measured by mechanical probescoming from two sides. Up to now no discrepancies have been found thatexceed the measurement uncertainty, which is in the 10 nm to 20 nm range.Figure 4.15 shows a possible definition for a mechanical gauge blocklength.
A gauge block with length L is probed from both sides with a perfectlyround probe of diameter d, being typically a few millimetres in diameter. Themechanical gauge block length, L, is the probe displacement, D, in the limitof zero force, minus the probe diameter, or L ¼ D d.4.5.2 Gauge block interferometryIn order to measure gauge blocks in an interferometer, a first requirement forthe light source is to have a coherence length that exceeds the gauge blocklength. Gauge block interferometers can be designed as a Twyman-Green ora Fizeau configuration, where the former is more common. For the majorityof the issues discussed in this section either configuration can be considered.Figure 4.16 is a schema of a gauge block interferometer containing a gaugeblock.
The observer sees the fringe pattern that comes from the platen asshown in Figure 4.10. If the platen has a small tilt this will be a set of straight,FIGURE 4.15 Possible definition of a mechanical gauge block length.Gauge block interferometryFIGURE 4.16 Schema of a gauge block interferometer containing a gauge block.parallel interference fringes. However, at the location of the gauge block,a parallel plate can also be observed, but the fringe pattern may be displaced(see Figure 4.17).If the fringes are not distorted, then an integer number of half wavelengths will fit in the length of the gauge block.
In general this will not be thecase, and the shift of fringes gives the fractional length of the gauge block.The length of the gauge block is given by½4block ðtopÞ 4ref ðtop areaÞ ½4platen ðbaseÞ 4ref ðbase areaÞlnL¼Nþ2p2nðlÞlnðN þ fÞ¼2nðlÞ(4.17)where N is the number of half wavelengths between the gauge block top andthe position on the platen for wavelength l, n is the air refractive index and fis the fraction f ¼ a/b in Figure 4.17. fblock (top) is the phase on top of thegauge block, fref (top area) is the phase at the reference plate at the location ofthe top area, fplaten (base) is the phase on the platen next tothe gauge block and fref (base area) is the phase at the reference plate at the7374C H A P T ER 4 : Length traceability using interferometryFIGURE 4.17 Theoretical interference pattern of a gauge block on a platen.location next to the image of the gauge block.
For a flat reference surface, thephase for the areas corresponding to the base and top of the gauge block arethe same (fref (top area) ¼ fref (base area)) and equation (4.17) simplifiesaccordingly. Equation (4.17) is the basic equation that links an electromagnetic wavelength, l, to a physical, mechanical length, L.Some practical issues that are met when applying equation (4.17) aretreated in the next sections.4.5.3 Operation of a gauge block interferometer4.5.3.1 Fringe fraction measurement – phase steppingAs indicated in Figure 4.17, the fringe fraction can be estimated visually. Forthis purpose, as a visual aid some fiducial dots or lines can be applied to thereference mirror. Experienced observers can obtain an accuracy of 5%, corresponding to approximately 15 nm.However, more objective and accurate methods for determining the fringefraction are possible by phase shifting; this means that the optical distance ofeither the reference mirror or the platen–gauge block combination is changedin a controlled way [14].
Established methods for phase shifting include:-displacing the reference mirror or the gauge block with platen usingpiezoelectric displacement actuators;-positioning an optical parallel in the beam. Giving the optical parallela small rotation generates a small controllable phase shift.Having the possibility of shifting the phase, the fraction can be derived ina semi-manual way. For example, the fringes on the platen can be adjusted toa reference line then on the gauge block, and then the next fringe on theplaten can be adjusted to this reference line. 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.
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