5.3.1 Basics of displacement interferometry

Displacement interferometry is usually based on the Michelson configuration or some variant of that basic design. In Chapter 4, we introduced the Michelson and Twyman–Green interferometers for the measurement of static length, and most of the practicalities in using such interferometers apply to displacement measurement. Displacement measurement, being simply a change in length, is usually carried out by counting the number of fringes as the object being measured (or reference surface) is displaced. Just as with gauge block interferometry, the displacement is measured as an integer number of whole fringes and a fringe fraction. Displacement interferometers are typically categorised as either homodyne systems (single frequency) or heterodyne systems (two frequencies). Homodyne displacement interferometers require at least two fringe patterns that are 90° out of phase (referred to as phase quadrature) to allow bidirectional fringe counting and to simplify the fringe analysis. Heterodyne displacement interferometers use a frequency modulation method where the two optical frequencies produce a nominal heterodyne frequency, typically in the megahertz regime. The phase of the measurement signal is then tracked relative to an optical reference that is detected at the heterodyne frequency. Photodetectors and digital electronics are used to count the fringes, and the fraction is determined by electronically sub-dividing the fringe [7]. With this method, fringe subdivisions of λ/1000 are common, giving sub-nanometre resolutions for both heterodyne and homodyne systems. There are many homodyne and heterodyne interferometers commercially available, and the realisation of sub-nanometre accuracies in a practical setup is an active area of research [8,9]. Many of the modern advances in high-accuracy interferometry come from the community searching for the effects of gravitational waves [10].

5.3.2 Homodyne interferometry

Figure 5.1 shows a homodyne interferometer configuration. The homodyne interferometer uses a single frequency, f₁, laser beam. Often this frequency is one of the modes of a two-mode stabilised laser (see Section 2.9.3.1). The beam from the stationary reference is returned to the beam splitter with a frequency f₁, but the beam from the moving measurement path is returned with a Doppler-shifted frequency of f₁±δf. These beams interfere in the beam splitter and enter the photodetector. The Doppler-shifted frequency gives rise to a count rate, dN/dt, which is equal to f (2v/c), where v is the velocity of the retro-reflector and c is the velocity of light. Integration of the count over time, t, leads to a fringe count, N=2d/λ, where d is the displacement being measured.

In a typical homodyne interferometer using a polarised beam, the measurement arm contains a quarter-wave plate, which results in the measurement and reference beams having a phase separation of 90° (for bidirectional fringe counting). In some cases, where an un-polarised beam is used [11], a coating is applied to the beam splitter to give the required phase shift [12]. After traversing their respective paths, the two beams re-combine in the beam splitter to produce an interference pattern.

Homodyne interferometers have an advantage over heterodyne interferometers (see Section 5.3.3) because the reference and measurement beams are split at the interferometer and not inside the laser (or at an acousto-optic modulator). This means that the light can be delivered to the interferometer via a standard fibre optic cable. In the heterodyne interferometer, a polarisation-preserving (birefringent) optical fibre has to be employed [13]. Therefore, fibre temperature or stress changes alter the relative path lengths of the interferometer’s reference and measurement beams, causing drift. A solution to this problem is to employ a further photodetector that is positioned after the fibre optic cable [14]; doing so makes it redundant to measure the optical reference prior to the fibre.

Homodyne interferometers can have sub-nanometre resolutions and nanometre-level accuracies, usually limited by their non-linearity (see Section 5.3.8.4). Their speed limit depends on the electronics and the detector photon noise; see also Section 5.3.4. For a speed of 1 m·s⁻¹ and four counts per 0.3 μm cycle, a 3 MHz signal must be measured within 1 Hz. Maximum speeds of 4 m·s⁻¹ with nanometre resolutions are claimed by some instrument manufacturers.

5.3.3 Heterodyne interferometry

Figure 5.2 shows a heterodyne interferometer configuration. The output beam from a dual-frequency laser source contains two orthogonal polarisations, one with a frequency of f₁ and the other with a frequency of f₂ (separated by about 3 MHz using the Zeeman effect [15] or some other means – see Section 2.9.4). A polarising beam splitter reflects the light with frequency f₁ into the reference path. Light with frequency f₂ passes through the beam splitter into the measurement path where it strikes the moving retro-reflector causing the frequency of the reflected beam to be Doppler shifted by ±δf. This reflected beam is then combined with the reference light in the beam splitter and returned to a photodetector with a beat frequency of f₂−f₁±δf. This signal is mixed with the reference signal that continuously monitors the frequency difference, f₂−f₁. The beat difference, δf, gives rise to a count rate, dN/dt, which is equal to f (2v/c), where v is the velocity of the retro-reflector and c is the velocity of light. Integration of the count over time, t, leads to a fringe count, N=2d/λ, where d is the displacement being measured.

With a typical reference beat of around 3 MHz, it is possible to monitor δf values up to 3 MHz before introducing ambiguities due to the beat crossing through zero. This limits the target speed possible in this case to less than 2 m·s⁻¹ [16], which could be a constraint in some applications. Practical signal processing, and filter roll-off in the electronics, further limits this velocity range. An alternative method of producing a two-frequency laser beam is to use an acousto-optic frequency shifter [17]. This method has the advantage that the frequency difference can be much higher, so that higher count rates can be handled [18].

Many variations on the theme in Figure 5.2 have been developed which improve the speed of response, measurement accuracy and resolution. Modern commercial heterodyne interferometers can be configured to measure both displacement and angle (see, e.g. the xy interferometers in Ref. [19]).

5.3.4 Fringe counting and subdivision

There are two main types of optical fringe counting methods: hardware fringe counting and software fringe counting [20]. Hardware fringe counting utilises hardware circuits to subdivide and count interference fringes [7]. Its principle of operation is as follows. Two interference signals (sine and cosine) with π/2 phase difference are converted into two square waves by means of a zero crossing detector. It is important to employ hysteresis in the zero crossing detectors to ensure spurious noise in the signal does not cause false triggers [21]. Activated by the rising edge of the sine-equivalent square wave, a reversible counter adds or subtracts counts according to the moving direction of the measured object, which is determined by the level of the cosine-equivalent square wave that corresponds to the rising edge of the sine-equivalent square wave. The advantages of the hardware fringe counting method are good real-time performance and relatively simple realisation. However, the electronically countable shift of π/2 corresponds to a phase shift of λ/4 (or λ/8 in a double-pass interferometer – see Section 5.3.5), which defines the resolution limit for most existing hardware fringe counting systems.

Software fringe counting mainly uses digital processing to subdivide and count interference fringes [22]. Its basic principle is that the sine and cosine interference signals, when properly amplified, can be converted by an analogue-to-digital converter (ADC) and then processed by a digital computer to give the number of counts. Compared with hardware fringe counting, software fringe counting can overcome the effect of counting results that are due to random interference signal oscillation and has better intelligence in discriminating the direction of movement. Modern signal processing systems, often called phase meters, typically employ 10–14 bit ADCs with tens of megahertz of measurement bandwidth. The detected signals are then typically processed in a field programmable gate array (FPGA), which can perform massively parallel computations. Commercial phase meters that employ FPGAs typically achieve a further 1:2000 to 1:4000 interpolation factors in addition to the optical resolution. Custom systems have shown even higher interpolations factors (e.g. Refs. [23–25]).

5.3.5 Double-pass interferometry

The simple Michelson interferometer requires a high degree of alignment and requires that alignment to be maintained. Using retro-reflectors relaxes the alignment requirements but it may not always be possible to attach a retro-reflector (usually a cube corner or a cat’s eye) to the target. The Michelson interferometer may be rendered insensitive to mirror tilt misalignment by double-passing each arm of the interferometer and inverting the wavefronts between passes. An arrangement is shown in Figure 5.3, where double passing is achieved with a polarising beam splitter and two quarter-wave plates, and wavefront inversion by a cube-corner retro-reflector. Note that the beams are shown as laterally separated in Figure 5.3. This separation is not necessary but may be advantageous to stop light travelling back to the source. Setting up the components appropriately [26] allows a high degree of alignment insensitivity. Note that such an arrangement has been used in the differential interferometer in Section 5.3.6. The added polarisation components can also increase periodic error that may be present in the measured phase [27,28].

5.3.6 Differential interferometry

Figure 5.4 is a schema of a differential plane mirror interferometer developed at NPL [29]. 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 polarisation optics can be used to convert the Michelson part of the interferometer into a plane mirror configuration, but a retro-reflecting configuration could just as easily be employed. After a double passage through the wave plate, the beams are transmitted back to the Jamin beam splitter where they re-combine and interfere. The design of the Jamin beam splitter coating is such that the two signals captured by the photodetectors are in phase quadrature and so give the optimum signal-to-noise conditions for fringe counting and subdivision. In this configuration only the differential motion of the mirrors is detected.

The differential nature of this interferometer means that many sources of 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 paths are orthogonal, changes in the air refractive index in one path can be different from those in the other path.

Differential interferometers can have sub-nanometre accuracies, as has been confirmed using X-ray interferometry [30]. When a Heydemann correction is applied (see Section 5.3.8.5), such interferometers can have non-linearities of a few tens of picometres.

5.3.7 Swept-frequency absolute distance interferometry

Swept-frequency (or frequency scanning) interferometry using laser diodes or other solid-state lasers is becoming popular due to the versatility of its sources and its ability to measure length absolutely. Currently, such interferometers achieve high resolution but relatively low accuracies and tend to be used for applications over metres. Consider the case of a laser diode aligned to an interferometer of free spectral range, ν_R. If the output of the laser is scanned through a frequency range ν_s, N fringes are generated at the output of the interferometer [31]. 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–Pérot interferometer in vacuum, the optical path length is given by

$L=\frac{c}{2{\nu }_{\mathrm{R}}}=\frac{Nc}{2{\nu }_{\mathrm{s}}}.$ (5.1)

(5.1)

It is generally convenient to use feedback control techniques to lock the laser to particular fringes at the start and finish of the scan and so make N integral. 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 by ν_s, and the difference frequency counted directly as a beat by means of a fast detector with several gigahertz of frequency response. This, together with the number 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 and range of the interferometer [32].

Swept-frequency interferometers have been used in applications where accurate alignment of components over relatively large distances is required, for example when aligning detectors for particle accelerators [33], for large coordinate measuring machines (CMMs) [34] and for remote sensing applications [35] – in the latter case using frequency combs as the source (see Section 2.6.9).

5.3.8 Sources of error in displacement interferometry

Many of the sources of uncertainty discussed in Section 4.5.4 also apply to displacement interferometry. There will be two types of error sources that will lead to uncertainties. Firstly, there will be error sources that are proportional to the displacement being measured, L, commonly referred to as cumulative errors. Secondly, there will be error sources that are independent of the displacement being measured, commonly referred to as non-cumulative errors. When calculating the measurement uncertainty, the standard uncertainties due to the cumulative and non-cumulative error sources must be combined in an appropriate manner (see Section 2.8.3), and an expanded uncertainty calculated. An example of an uncertainty calculation for the homodyne displacement interferometers on a traceable surface texture measuring instrument is given elsewhere [36], and the most prominent error sources are discussed here.

The effects of the variation in the vacuum wavelength and the refractive index of the air will be the same as described in Section 4.5.4, and the effect of the Abbe error is described in Section 3.4.

5.3.8.1 Thermal expansion of the metrology frame

All measuring instruments have thermal and metrology loops (see Section 3.6). In the case of a Michelson interferometer, with reference to Figure 4.7, both loops run from the laser, follow the optical beam paths through the optics and travel back to the laser via the mechanical base used to mount the optics. Any thermal expansion in these components, for example due to changes in the ambient temperature or conduction into the system from motors, will cause an error in the length measured by the interferometer. Such errors can be corrected for as described in Section 3.7.1 and must be considered in the instrument uncertainty analysis. Thermal expansion errors are cumulative. The change in length due to thermal expansion, Δl, of a part of length, l, is given by

$\mathrm{\Delta }l=\alpha l\mathrm{\Delta }T,$ (5.2)

(5.2)

where α is the coefficient of linear thermal expansion and ΔT is the change in temperature.

5.3.8.2 Deadpath length

Deadpath length, d, is defined as the difference in distance in air between the reference and measurement reflectors and the beam splitter when the interferometer measurement is initiated. Deadpath error occurs when there is a non-zero deadpath and environmental conditions change during a measurement. Equation (5.3) yields the displacement, D, for a single-pass interferometer [37]

$D=\frac{N{\lambda }_{\mathrm{vac}}}{n_2}-\frac{\mathrm{\Delta }nd}{n_2},$ (5.3)

(5.3)

where N is half the number of fringes counted during the displacement, n₂ is the refractive index at the end of the measurement, Δn is the change in refractive index over the measurement time: that is n₂=n₁+Δn and n₁ is the refractive index at the start of the measurement. The second term on the right-hand side of Eq. (5.3) is the deadpath error, which is non-cumulative (although it is dependent on the deadpath length). Deadpath error can be eliminated by presetting counts at the initial position to a value equivalent to d.

5.3.8.3 Cosine error

Figure 5.5 shows the effect of cosine error on an interferometer. The moving stage is at an angle to the laser beam (the scale) and the measurement will have a cosine error, Δl, given by

$\mathrm{\Delta }l=l(1-\cos \theta ),$ (5.4)

(5.4)

where l and θ are defined in Figure 5.5. The cosine error will always cause a measurement system to measure short and is a cumulative effect. The obvious way to minimise the effect of cosine error is to correctly align the interferometer. However, despite how perfectly aligned the system appears to be, there will always be a small, residual cosine error. This residual error must be taken into account in the uncertainty analysis of the system. For small angles, Eq. (5.4) can be approximated by

$\mathrm{\Delta }l=\frac{l{\theta }^2}{2}.$ (5.5)

(5.5)

Due to Eq. (5.5), cosine error is often referred to as a second-order effect, contrary to the Abbe error, which is a first-order effect. The second-order nature means that cosine error quickly diminishes as the alignment is improved, but it has the disadvantage that its magnitude is difficult to estimate once it becomes relevant.

Cosine error for plane mirror targets can also have another uncertainty contributor, even when the input beam propagation direction and the target motion axis are aligned. This occurs when the plane mirror target is not aligned normal to the input beam propagation direction. Thus, the reflected beam from the target propagates off at an angle, causing errors. Equation (5.5) can be expanded to include this error [38],

$\mathrm{\Delta }l=\frac{l}{2}({\theta }^2+{\theta }_N^2),$ (5.6)

(5.6)

where θ_N is the plane mirror misalignment angle between the target surface normal vector at the input beam propagation direction. A schema of this is depicted in Figure 5.6.

5.3.8.5 Heydemann correction

When making displacement measurements at the nanometre level, the sine and cosine signals from interferometers must be corrected for dc offsets, differential gains and a quadrature angle that is not exactly 90°. The method described here is that due to Birch [20] and is a modified version of that originally developed by Heydemann [48]. There are many ways to implement such a correction in both software and hardware but the basic mathematics is that presented here. The full derivation is given, as this is an essential correction in many micro- and nanotechnology (MNT) applications of interferometry. This method only requires a single-frequency laser source (homodyne) and does not require polarisation optics. Birch [20] used computer simulations of the correction method to predict a fringe-fractioning accuracy of 0.1 nm. Other methods, which also claim to obtain sub-nanometre uncertainties, use heterodyne techniques [49] and polarisation optics [50].

Heydemann used two equations that describe an ellipse

$U_{1d}=U_1+p$ (5.7)

(5.7)

and

$U_{2d}=\frac{U_2\cos a-U_1\sin a}{G}+q,$ (5.8)

(5.8)

where U_1d and U_2d represent the noisy signals from the interferometer containing the correction terms p, q and a as defined by Eqs. (5.13), (5.15) and (5.16) respectively, G is the ratio of the gains of the two detector systems and U₁ and U₂ are given by

$U_1=R_D\cos \delta$ (5.9)

(5.9)

$U_2=R_D\sin \delta ,$ (5.10)

(5.10)

where δ is the instantaneous phase of the interferograms. If Eqs. (5.7) and (5.8) are combined, they describe an ellipse given by

$R_D^2={(U_{1d}-p)}^2+\frac{{[(U_{2d}-q)G+(U_{1d}-p)\sin a]}^2}{\cos a}.$ (5.11)

(5.11)

If Eq. (5.11) is now expanded out and the terms are collected together, an equation of the following form is obtained

$A{U}_{1d}^2+B{U}_{2d}^2+C{U}_{1d}{U}_{2d}+D{U}_{1d}+E{U}_{2d}=1$ (5.12)

(5.12)

with

$\begin{array}{cc}A& ={[R_D^2{\cos }^{2}a-p^2-G^2{q}^2-2Gpq\sin a]}^{-1}\\ B& =A{G}^2\\ C& =2AG\sin a\\ D& =-2A[p+Gq\sin a]\\ E& =-2AG[Gq+p\sin a]\end{array}.$

Equation (5.12) is in a form suitable for using a linearised least squares fitting routine [51] to derive the values of A through E, from which the correction terms can be derived from the following set of transforms

$a={\sin }^{-1}\left[\frac{C}{{(4AB)}^{1/2}}\right]$ (5.13)

(5.13)

$G={\left[\frac{B}{A}\right]}^{1/2}$ (5.14)

(5.14)

$p=\frac{2BD-EC}{C^2-4AB}$ (5.15)

(5.15)

$q=\frac{2AE-DC}{C^2-4AB}$ (5.16)

(5.16)

$R_D=\frac{{[4B(1+A{p}^2+B{q}^2+Cpq)]}^{1/2}}{5AB-C^2}.$ (5.17)

(5.17)

Consequently, the interferometer signals are corrected by using the two inversions

${U\prime }_1=U_{1d}-p$ (5.18)

(5.18)

and

${U\prime }_2=\frac{(U_{1d}-p)\sin a+G(U_{2d}-q)}{\cos a},$ (5.19)

(5.19)

where ${U\prime }_1$ and ${U\prime }_2$ are now the corrected phase quadrature signals and, therefore, the phase of the interferometer signal is derived from the arctangent of $({U\prime }_2/{U\prime }_1)$.

The arctangent function varies from −π/2 to +π/2, whereas, for ease of fringe fractioning, a phase, θ, range of 0–2π is preferable. This is satisfied by using the following equation

$\theta ={\tan }^{-1}(U_1/U_2)+\pi /2+\Lambda ,$ (5.20)

(5.20)

where θ=0 when U_1d>p and Λ=π when U_1d<p.

The strong and weak points of a Heydemann-corrected system are that it appears correct in itself and refers to its own result to predict residual deviations (e.g. deviations from the ellipse). However, there are uncertainty sources that still give deviations even when the Heydemann correction is applied perfectly, for example so-called ghost reflections [52] and beam shear [53]. These error sources result in periodic errors at harmonics other than the first or second orders, which is not captured in the underlying model for the Heydemann correction.

5.3.9 Latest advances in displacement interferometry

Most of the recent advances in displacement interferometry have been to reduce thermal and mounting errors, make heterodyne interferometry more practical for embedded applications and add inherent multi-axis measurement.

Many commercial interferometers use fused silica optics housed in Invar (a low-expansion steel alloy) mounts that are subsequently bolted together to assemble multi-part interferometers. Mounting the interferometers in this manner increases the overall optical paths within the interferometer, increasing the susceptibility to thermal variations and refractive index fluctuations. For high-accuracy applications, commercial interferometers are now available with all of the optical components mounted to one central optic, typically the beam splitter. Mounting optics in this manner eliminates air gaps in the interferometer assemblies, eliminates some ghost reflections due to reduced glass–air transitions and shortens optical paths, resulting in lower noise interferometers that are more stable over time. The drawback to these single interferometer assemblies is that they are dedicated to a single configuration and cannot be readily disassembled for use in other interferometer configurations.

One advantage that homodyne systems have over heterodyne systems is their ability to readily have the source fibre delivered to the interferometer. Homodyne systems do not use a so-called optical reference to establish the nominal detection frequency. Thus, only one fibre is needed to deliver light to the interferometer, as shown in Figure 5.7. Heterodyne interferometers can be fibre delivered but polarisation-maintaining fibre must be used, an optical reference is needed at each interferometer and periodic errors can still contribute nanometres of error [13]. Spatially separated interferometer configurations [25,46,60–62] offer the possibility of fibre delivery with limited or zero periodic error. In these interferometers, the heterodyne frequency is typically generated using two acousto-optic modulators driven at slightly different radio frequencies (RF). The Wu interferometer configuration has been widely adapted for the LISA programme (see, e.g. Ref. [53]) and can be viewed similarly to a differential interferometer, where the measured signal is the difference between the measurement and reference optical paths. Spatially separated configurations by Tanaka, Joo and Ellis offer the possibility of enhanced optical resolution; however, they cannot be configured as differential interferometers as their measured signal is the total path length changes from both the measurement and reference optical paths.

The spatially separated interferometers by Wu et al. [61] and Ellis et al. [25] (shown in Figure 5.8) also use differential wavefront sensing to measure the displacement simultaneously with target tip and tilt. Differential wavefront sensing uses the spatially varying phase from a target’s angle changes incident on a quadrant photodiode to measure four interference patterns [63,64] (Figure 5.9). Then, by knowing the beam size and detector geometry, the measurement target’s angle change can be determined by differencing matched pairs of measured phase from the quadrant photodiode (while the displacement is determined from the average phase over the four quadrants). Differential wavefront sensing has several advantages over traditional single-axis interferometers: (i) the beam size is smaller and only a plane mirror is needed as the target; (ii) the measurement axis location is more readily determined than in traditional plane mirror interferometers; (iii) interferometers for calibration need fewer set-ups to determine linear and angular errors and (iv) the measured phase on the four quadrants has a small spatial location that limits refractive index errors for the angle measurement (the linear measurement is still susceptible to refractive index changes). The downsides to differential wavefront sensing are that it requires extra measurement channels, the wavefront of the interfering beams must be known and aberrations must be limited. Aberrations in the interfering beams can lead to inaccuracies in the sensitivity coefficient, creating uncertainty in the angle measurement [65].

5.3.10 Angular interferometers

In the discussion on angle in Section 2.6, the possibility of determining an angle by the ratio of two lengths was discussed. This method is applicable in interferometry.

Figure 5.10 shows a typical optical arrangement of an interferometer set-up for angular measurements. The angular optics is used to create two parallel beam paths between the angular interferometer and the angular reflector. The distance between the two beam paths is found by measuring the separation of the retro-reflectors in the angular reflector. This measurement is made either directly or by calibrating a scale factor against a known angular standard.

The beam that illuminates the angular optics contains two frequencies, f₁ and f₂ (heterodyne). A polarising beam splitter in the angular interferometer splits the frequencies, f₁ and f₂, that travel along separate paths.

At the start position, the angular reflector is assumed to be approximately at a zero position (i.e. the angular measurements are relative). At this position, the two paths have a small difference in length. As the angular reflector is rotated relative to the angular interferometer, the relative lengths of the two paths will change. This rotation will cause a Doppler-shifted frequency change in the beam returned from the angular interferometer to the photodetector. The photodetector measures a fringe difference given by (f₁±Df₁)−(f₂±Df₂).

The returned difference is compared with the reference signal, (f₁−f₂). This difference is related to velocity and then to distance. The distance is then converted to an angle using the known separation of the reflectors in the angular interferometer.

Other arrangements of angular interferometer are possible using plain mirrors, but the basic principle is the same. Angular interferometers are generally used for measuring small angles (less than 10°) and are commonly used for measuring guideway errors in machine tools and measuring instruments.

5.10.1 Calibration using optical interferometry

In order to characterise the performance of a displacement sensor, a number of interferometers can be used (provided the laser source has been traceably calibrated; see Section 2.9.5). A homodyne or heterodyne set-up (see Sections 5.3.2 and 5.3.3, respectively) can be used by rigidly attaching or kinematically mounting an appropriate reflector so that it moves collinearly with the displacement sensor. One must be careful to minimise the effects of Abbe offset (see Section 3.4) and cosine error (see Section 5.3.8.3) and to reduce any external disturbances. A differential interferometer (see Section 5.3.6) can also be used but over a reduced range.

As displacement sensor characteristics are very sensitive over short distances, the limits and limiting factors of interferometric systems for very small displacement become critical. For the most common interferometers, it is the periodic error within one wavelength that becomes critical. Even with the Heydemann correction (see Section 5.3.8.5) applied, periodic error can be the major error source.

5.10.2 Calibration using X-ray interferometry

The fringe spacing for a single-pass two-beam optical interferometer is equal to half the wavelength of the source radiation and this is its basic resolution before fringe subdivision is necessary. The fringe spacing in an X-ray interferometer is independent of the wavelength of the source; it is determined by the spacing of diffraction planes in the crystal from which X-rays are diffracted [103]. Due to its ready availability and purity, silicon is the most common material used for X-ray interferometers. The atomic lattice parameter of silicon can be accurately measured (by diffraction) and is regarded as a traceable standard of length. Therefore, X-ray interferometry allows a traceable measurement of displacement with a basic resolution of approximately 0.2 nm (0.192 nm for the (220) planes in silicon).

Figure 5.19 shows a schema of a monolithically manufactured X-ray interferometer made from a single crystal of silicon. Three, thin, vertical and equally spaced lamella are machined with a flexure stage around the third lamella (A). The flexure stage has a range of a few micrometres and is driven by a PZT. X-rays are incident at the Bragg angle [12] on lamella B, and two diffracted beams are transmitted. Lamella A is analogous to a beam splitter in an optical interferometer. The transmitted beams are incident on lamella M that is analogous to the mirrors in a Michelson interferometer. Two more pairs of diffracted beams are transmitted and one beam from each pair is incident on lamella A, giving rise to a fringe pattern. This fringe pattern is too small to resolve individual fringes, but when lamella A is translated parallel to B and M, a moiré fringe pattern between the coincident beams and lamella A is produced. Consequently, the intensity of the beams transmitted through lamella A varies sinusoidally as lamella A is translated.

The displacements measured by an X-ray interferometer are free from the periodic error in an optical interferometer (see Section 5.3.8.4). To calibrate an optical interferometer (and, therefore, measure its periodic error), the X-ray interferometer is used to make a known displacement that is compared against the optical interferometer under calibration. By servo-controlling the PZT, it is possible to hold lamella A in a fixed position or move it in discrete steps equal to one fringe period [104]. Examples of the calibration of a differential plane mirror interferometer and an optical encoder can be found in Refs. [30] and [81], respectively. In both cases, periodic errors with amplitudes of less than 0.1 nm were measured once a Heydemann correction (see Section 5.3.8.5) had been applied. More recently, a comparison of the performance of the next generation of optical interferometers was undertaken using X-ray interferometry [8]. X-ray interferometry can also be used to calibrate the characteristics of translation stages in two orthogonal axes [105] and to measure nanoradian angles [106].

One limitation of X-ray interferometry is its short range. To overcome this limitation, NPL, PTB and Instituto di Metrologia ‘G. Colonetti’ (now known as Instituto Nazionale di Recerca Metrologica – the Italian NMI) collaborated on a project to develop the Combined Optical and X-ray Interferometer [107] as a facility for the calibration of displacement sensors and actuators up to 1 mm. The X-ray interferometer has an optical mirror on the side of its moving mirror that is used in the optical interferometer (Figure 5.20). The optical interferometer is a double-path differential system with one path measuring displacement of the moving mirror on the X-ray interferometer with respect to the two fixed mirrors above the translation stage. The other path measures the displacement of the mirror (M) moved by the translation stage with respect to the two fixed mirrors either side of the moving mirror in the X-ray interferometer. Both the optical and X-ray interferometers are servo controlled. The X-ray interferometer moves in discrete X-ray fringes; the servo system for the optical interferometer registers this displacement and compensates by initiating a movement of the translation stage. The displacement sensor being calibrated is referenced to the translation stage and its measured displacement is compared with the known displacements of the optical and X-ray interferometers.