Richard Leach - Fundamental prinsiples of engineering nanometrology (778895), страница 24
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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. The secondorder nature means that it quickly diminishes as the alignment is improved,but has the disadvantage that its magnitude is difficult to estimate once itbecomes relevant.5.2.8.4 Non-linearityBoth homodyne and heterodyne interferometers are subject to non-linearitiesin the relationship between the measured phase difference and the displacement.
The many sources of non-linearity in heterodyne interferometers arediscussed in [23], and further discussed, measured and extended in [24,25].These sources include misalignment of laser polarization axes with respectto the beam-splitter, ellipticity of the light from the laser source, differentialtransmission between the two arms of the interferometer, rotation of the planeof polarization by the retro-reflectors, leakage of light with the unwantedpolarization through the beam-splitter, and lack of geometrical perfectionof the wave plates used. For homodyne interferometers, the main source ofnon-linearity [26] is attributed to polarization mixing caused by imperfectionsin the polarizing beam-splitters, although there are several other sources [6].The various sources of non-linearity give rise to periodic errors – a first-orderphase harmonic having a period of one cycle per fringe and a second harmonicwith a period of two cycles per fringe.
Errors due to non-linearities are usuallyDisplacement interferometryof the order of a few nanometres but can be reduced to below a nanometre withcareful alignment and high-quality optics. There have been many attempts tocorrect for non-linearity in interferometers with varying degrees of success(see for example [27,28]). Recently researchers have developed a heterodyneinterferometer for which a zero periodic non-linearity is claimed [29].5.2.8.5 Heydemann correctionWhen making displacement measurements at the nanometre level, the sineand cosine signals from interferometers need to be corrected for dc offsets,differential gains and a quadrature angle that is not exactly 90 . The methoddescribed here is that due to Birch [5] and is a modified version of thatoriginally developed by Heydemann [30].
There are many ways to implementsuch a correction in both software and hardware but the basic mathematics isthat presented here. The full derivation is given, as this is an essentialcorrection in many MNT applications of interferometry. This method onlyrequires a single-frequency laser source (homodyne) and does not requirepolarization optics.
Birch [5] used computer simulations of the correctionmethod to predict a fringe-fractioning accuracy of 0.1 nm. Other methods,which also claim to obtain sub-nanometre uncertainties, use heterodynetechniques [31] and polarization optics [32].Heydemann used two equations that describe an ellipseU1d ¼ U1 þ p(5.6)U2 cos a U1 sin aþqG(5.7)andU2d ¼where U1d and U2d represent the noisy signals from the interferometercontaining the correction terms p, q and a as defined by equations (5.14),(5.15) and (5.12) respectively, G is the ratio of the gains of the two detectorsystems and U1 and U2 are given byU1 ¼ RD cosd(5.8)U2 ¼ RD sind(5.9)where d is the instantaneous phase of the interferograms.
If equations (5.6) to(5.7) are combined they describe an ellipse given byR2D ¼ ðU1d pÞ2 þ½ðU2d qÞG þ ðU1d pÞsina2:cosa(5.10)9596C H A P T ER 5 : Displacement measurementIf equation (5.10) is now expanded out and the terms are collectedtogether, an equation of the following form is obtained22AU1dþ BU2dþ CU1d U2d þ DU1d þ EU2d ¼ 1(5.11)withA ¼ ½R2D cos2 a p2 G2 q2 2Gpq sina1B ¼ AG2C ¼ 2AG sinaD ¼ 2A½p þ Gq sinaE ¼ 2AG½Gq þ p sinaEquation (5.11) is in a form suitable for using a linearized least squaresfitting routine [33] to derive the values of A through E, from which thecorrection terms can be derived from the following set of transforms"#C1a ¼ sin(5.12)ð4ABÞ1=2 1=2BG ¼ARD ¼(5.13)p ¼2BD ECC2 4AB(5.14)q ¼2AE DCC2 4AB(5.15)½4Bð1 þ Ap2 þ Bq2 þ CpqÞ1=2:5AB C2(5.16)Consequently, the interferometer signals are corrected by using the twoinversions0U1 ¼ U1d p(5.17)and0U2 ¼00ðU1d pÞsina þ GðU2d qÞcosa(5.18)where U1 and U2 are now the corrected phase quadrature signals and,therefore, the phase of the interferometer signal is derived from the arctan00gent of ðU2 =U1 Þ.Displacement interferometryThe arctangent function varies from p/2 to þp/2, whereas for ease offringe-fractioning a phase, q, range of 0 to 2p is preferable.
This is satisfied byusing the following equationq ¼ tan1 ðU1 =U2 Þ þ p=2 þ L(5.19)where L ¼ 0 when U1d > p and L ¼ p when U1d < p.The strong and weak points of a Heydemann-corrected system are that itappears correct in itself and refers to its own result to predict residual deviations (for example, deviations from the ellipse). However, there are uncertainty sources that still give deviations even when the Heydemann correctionis applied perfectly, for example, so-called ghost-reflections.5.2.8.6 Random error sourcesThere are many sources of random error that can affect an interferometer.Anything that can change the optical path or mechanical part of themetrology loop can give rise to errors in the measured displacement.Examples include seismic and acoustic vibration (see section 3.9), airturbulence (causing random fluctuations of the air refractive index) andelectronic noise in the detectors and amplifier electronics.
Random errors areusually non-cumulative and can be quantified using repeated measurements.Homodyne systems measure phase by comparing the intensities of twosinusoidal signals (sine and cosine). By contrast, modern heterodyne systemsmeasure phase by timing the arrival of zero crossings on a sinusoidal signal.Because the signal slope at the zero crossings is nominally 45 , phase noise isapproximately equal to intensity noise.
Therefore, the influence of noise onboth systems is effectively the same.5.2.8.7 Other sources of error in displacement interferometersThere are many sources of error that only have a significant effect whentrying to measure to accuracies of nanometres or less using interferometry.Due to the very high spatial and temporal coherence of the laser source, straylight can interfere with beams reflected from the surfaces present in thereference and measurement arms of the interferometer. The dominanteffects are usually due to unwanted reflections and isolated strong pointscatterers, both leading to random and non-random spatial variations in thescattered phase and amplitude [13].
These effects can be of the order ofa nanometre (see for example [22]). To minimize the effects of stray reflections all the optical components should be thoroughly cleaned, the retroreflectors (or mirrors) should be mounted at a non-orthogonal angle to thebeam propagation direction (to avoid reflections off the front surfaces) and all9798C H A P T ER 5 : Displacement measurementthe non-critical optical surfaces should be anti-reflection coated. It isextremely difficult, if not impossible, to measure the amplitude of the straylight, simply because it propagates in the same direction as the main beams.Also due to the laser source, the shift of the phase and changes in thecurvature of the wavefronts lead to systematic errors and diffraction effects[34].
There will also be quantum effects [35] and even photon bounce [36].These effects are very difficult to quantify or measure but are usuallysignificantly less than a nanometre.5.2.9 Angular interferometersIn the discussion on angle in section 2.6 the possibility of determining anangle by the ratio of two lengths was discussed.
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