Принципы нанометрологии (1027623), страница 24
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Its basic principle is that the sine and cosine interference signals, when properly amplified, can be converted by an analogue-todigital converter (ADC) and then processed by a digital computer to give thenumber of counts. Compared with hardware fringe counting, software fringecounting can overcome the effect of counting results that are due to randominterference signal oscillation, and has better intelligence in discriminatingthe direction of movement. However, a measurement system that uses software fringe counting can deal only with low-frequency interference signalsowing to the relatively slow conversion rate of ADCs.5.2.5 Double-pass interferometryThe simple Michelson interferometer requires a high degree of alignmentand requires that alignment to be maintained. The use of retro-reflectorsrelaxes the alignment requirements but it may not always be possible toattach a retro-reflector (usually a cube-corner or a cat’s eye) to the target.
TheMichelson interferometer may be rendered insensitive to mirror misalignment by double-passing each arm of the interferometer and inverting thewavefronts between passes. An arrangement is shown in Figure 5.3 wheredouble passing is achieved with a polarizing beam-splitter and two quarterwave plates, and wavefront inversion by a cube-corner retro-reflector.
Notethat the beams are shown as laterally separated in Figure 5.3. This separationis not necessary but may be advantageous to stop light travelling back to thesource. Setting up the components appropriately [17] allows a high degree of8990C H A P T ER 5 : Displacement measurementFIGURE 5.3 Optical arrangement to double pass a Michelson interferometer.alignment insensitivity. Note that such an arrangement has been used in thedifferential interferometer in section 5.2.6.5.2.6 Differential interferometryFigure 5.4 is a schema of a differential plane mirror interferometer developedat NPL [18]. The beam from the laser is split by a Jamin beam-splitter,creating two beams that are displaced laterally and parallel to each other.Figure 5.4 shows how polarization optics can be used to convert theMichelson part of the interferometer into a plane mirror configuration, buta retro-reflecting configuration could just as easily be employed.
Aftera double passage through the wave-plate, the beams are transmitted back tothe Jamin beam-splitter where they recombine and interfere. The design ofthe Jamin beam-splitter coating is such that the two signals captured by thephoto-detectors are in phase quadrature and so give the optimum signal-tonoise conditions for fringe counting and sub-dividing. In this configurationonly the differential motion of the mirrors is detected.The differential nature of this interferometer means that many sourcesof uncertainty are common to both the reference and measurement paths,essentially allowing for common noise rejection.
For example, with a conventional Michelson configuration, where the reference and measurement pathsDisplacement interferometryFIGURE 5.4 Schema of a differential plane mirror interferometer.are orthogonal, changes in the air refractive index in one path can be differentfrom those in the other path.Differential interferometers can have sub-nanometre accuracies, as hasbeen confirmed using X-ray interferometry [19]. When a Heydemanncorrection is applied (see section 5.2.8.5), such interferometers can havenon-linearities of a few tens of picometres.5.2.7 Swept-frequency absolute distance interferometrySwept-frequency interferometry using laser diodes or other solid-state lasersis becoming popular due to the versatility of its sources and its ability tomeasure length absolutely.
Currently such interferometers achieve highresolution but relatively low accuracies and tend to be used for applicationsover metres. Consider the case of a laser diode aligned to an interferometerof free spectral range, nR. If the output of the laser is scanned througha frequency range ns, N fringes are generated at the output of the9192C H A P T ER 5 : Displacement measurementinterferometer [20]. Provided the frequency scan range is accurately known,the free spectral range and hence the optical path length, L, may be determined from counting the number of fringes.
For a Michelson or Fabry-Pérotinterferometer in vacuum, the optical path length is given byL ¼cNc¼:2nR2ns(5.1)It is generally convenient to use feedback control techniques to lock thelaser to particular fringes at the start and finish of the scan and so make Nintegral. For scans of up to several gigahertz, two lasers are typically used,which are initially tuned to the same frequency. One laser is then scanned byns, and the difference frequency counted directly as a beat by means of a fastdetector with several gigahertz of frequency response.
This, together with thenumber of fringes scanned, enables the optical path length to be determined.The number and size of the sweeps can be used to improve the accuracy andrange of the interferometer [21].5.2.8 Sources of error in displacement interferometryMany of the sources of uncertainty discussed in section 4.5.4 also apply todisplacement interferometry.
There will be two types of error sources thatwill lead to uncertainties. Firstly, there will be error sources that areproportional to the displacement being measured, L, commonly referred to ascumulative errors. Secondly, there will be error sources that are independentof the displacement being measured, commonly referred to as non-cumulative errors. When calculating the measurement uncertainty, the standarduncertainties due to the cumulative and non-cumulative error sources needto be combined in an appropriate manner (see section 2.8.3), and anexpanded uncertainty calculated.
An example of an uncertainty calculationfor the homodyne displacement interferometers on a traceable surfacetexture measuring instrument is given elsewhere [22] and the most prominent error sources are discussed here.The effects of the variation in the vacuum wavelength and the refractiveindex of the air will be the same as described in section 4.5.4, and the effect ofthe Abbe error is described in section 3.4.5.2.8.1 Thermal expansion of the metrology frameAll measuring instruments have thermal and metrology loops (see section3.6).
In the case of a Michelson interferometer, with reference to Figure 4.7,both loops run from the laser, follow the optical beam paths though theoptics, and travel back to the laser via whatever mechanical base the opticsDisplacement interferometryare mounted on. Any thermal expansion in these components, due tochanges in the ambient temperature, will cause an error in the lengthmeasured by the interferometer. Such errors can be corrected for as describedin section 3.7.1 and need to be considered in the instrument uncertaintyanalysis.
Thermal expansion errors are cumulative. The change in lengthdue to thermal expansion, Dl, of a part of length, l, is given byDl ¼ alDq(5.2)where a is the coefficient of linear thermal expansion and Dq is the change intemperature.5.2.8.2 Deadpath lengthDeadpath length, d, is defined as the difference in distance in air between thereference and measurement reflectors and the beam-splitter when theinterferometer measurement is initiated. Deadpath error occurs when thereis a non-zero deadpath and environmental conditions change duringa measurement. Equation (5.3) yields the displacement, D, for a single passinterferometerD ¼Nlvac Dndn2n2(5.3)where N is half the number of fringes counted during the displacement, n2 isthe refractive index at the end of the measurement, Dn is the change inrefractive index over the measurement time: that is n2 ¼ n1 þ Dn, and n1 isthe refractive index at the start of the measurement.
The second term onthe right-hand side of equation (5.3) is the deadpath error, which is noncumulative (although it is dependent on the deadpath length). Deadpatherror can be eliminated by presetting counts at the initial position to a valueequivalent to d.5.2.8.3 Cosine errorFigure 5.5 shows the effect of cosine error on an interferometer. The movingstage is at an angle to the laser beam (the scale) and the measurement willhave a cosine error, Dl, given byDl ¼ lð1 cosqÞ(5.4)where l and q are defined in Figure 5.5. The cosine error will always causea measurement system to measure short and is a cumulative effect. Theobvious way to minimize the effect of cosine error is to correctly align theinterferometer.
However, despite how perfectly aligned the system appearsto be, there will always be a small, residual cosine error. This residual error9394C H A P T ER 5 : Displacement measurementFIGURE 5.5 Cosine error with an interferometer.needs to be taken into account in the uncertainty analysis of the system. Forsmall angles, equation (5.4) can be approximated byDl ¼lq2:2(5.5)Due to equation (5.5) cosine error is often referred to as a second-ordereffect, contrary to the Abbe error, which is a first-order effect.
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