4.3.1 Light as a wave

This chapter will introduce the aspects of optics that are required to understand interferometry. For a more thorough treatment of optics, the reader is referred to Ref. [8].

For the treatment of light, we will restrict ourselves to electromagnetic waves of optical frequencies, usually called ‘visible light’. From Maxwell’s equations, it follows that the electric field of a plane wave, with speed, c, frequency, f, and wavelength, λ, travelling in the z-direction, is given by

$E(z,t)=\left(\begin{array}{c}{E}_x\\ E_y\end{array}\right){\mathrm{e}}^{\mathrm{i}(kz-\omega t)},$ (4.1)

(4.1)

where ω=2πf=2πc/λ is the circular frequency and k is the circular wave number, k=2π/λ.

Here we use the convention that a measurable quantity, for example the amplitude, E_x, can be obtained by taking the real part of Eq. (4.1) and we assume that E_y=0, that is the light is linearly polarised in the x-direction. At the location z=0, the electric field E=E_x cos ωt. This means that the momentary electric field oscillates with a frequency f. For visible light, for example green light (λ=500 nm), this gives, with the speed of light defined as c=299 792 458 m·s⁻¹, a frequency of f=6×10¹⁴ Hz. No electric circuit can directly follow such a high frequency; therefore, light properties are generally measured by averaging the cosine function over time. The intensity is given by the square of the amplitude, thus

$I(z)=⟨E\cdot E⟩=(E_x^2)⟨{\cos }^2\omega t⟩.$ (4.2)

(4.2)

A distortion at t=0, z=0, for example of the amplitude E_x in Eq. (4.1), will be the same as at time, t, at location z=ωt/k=ct, so the propagation velocity is c indeed.

In Eq. (4.1), the amplitudes E_x and E_y can both be complex. In that general case, we speak of elliptical polarisation; the E-vector describes an ellipse in space. If E_x and E_y are both real, the light is called linearly polarised. Another special case is when E_y=iE_x, in which case the vector describes a circle in space; for that reason this case is called circular polarisation.

When light beams from different sources, or from the same source but via different paths, act on the same location, their electric fields can be added. This is called the principle of superposition and causes interference. Visible, stable interference can appear when the wavelengths are the same and there is a determined phase relationship between the superimposed waves. If the wavelengths are not the same, or the phase relationship is not constant, the effect is called beating, which means that the intensity may vary with a certain frequency.

A fixed phase relationship can be achieved by splitting light, coming from one source, into two beams and recombining the light again. An instrument that accomplishes this is called an interferometer. An example of an interferometer is shown in Figure 4.3.

Consider the fields E₁(t) and E₂(t) in the interferometer in Figure 4.3, which travel paths to and from M₁ and M₂, respectively and combine at the detector, D. According to the principle of superposition, we can write

$E(t)=E_1(t)+E_2(t).$ (4.3)

(4.3)

Combining Eqs. (4.1)–(4.3), with some additional assumptions, gives finally

$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \left(\frac{4\pi \mathrm{\Delta }L}{\lambda }\right),$ (4.4)

(4.4)

where ΔL is the path difference between the two beams and I are intensities, that is the squares of the amplitudes.

Equation (4.4) is the essential equation of interference. Depending on the term 4πΔL/λ, the resultant intensity on a detector can have a minimum or a maximum, and it depends with a (co)sine function on the path difference or the wavelength.

From Eq. (4.4), it is evident that the intensity has maxima for 4pΔL/λ=2pπ, with p=0, ±1, ±2, …, so that ΔL=pλ/2 and minima for ΔL=(p+0.5)λ/2.

4.3.2 Beat measurement when ω₁ ≠ ω₂

If either E₁ or E₂ is shifted in frequency, or if E₁ and E₂ originate from sources with a different frequency, we can write analogous to Eq. (4.4)

$I=I_1+I_2+2\sqrt{I_1{I}_2}\cos \left(\frac{4\pi L}{{\lambda }_2}+({\omega }_2-{\omega }_1)t\right).$ (4.5)

(4.5)

We obtain an interference signal that oscillates with the difference frequency, which can readily be measured by a photodetector if ω₁ and ω₂ are not significantly different.

4.3.3 Visibility and contrast

If the intensities I₁ and I₂ are equal, Eq. (4.4) reduces to

$I=2I_1\left(1+\cos \left[\frac{4\pi \mathrm{\Delta }L}{\lambda }\right]\right)=4I_1\cos \left(\frac{2\pi \mathrm{\Delta }L}{\lambda }\right).$ (4.6)

(4.6)

This means that the minimum intensity is zero and the maximum intensity is 4I₁. Also it is clear that if I₁ or I₂ is zero, the interference term in Eq. (4.4) vanishes and a constant intensity remains. The relative visibility, V, of the interference can be defined as

$V=\frac{I_{\max }-I_{\min }}{I_{\max }+I_{\min }}=\frac{2\sqrt{I_1{I}_2}}{I_1+I_2}.$ (4.7)

(4.7)

The effect of visibility is illustrated in Figure 4.4, for the cases I₁=I₂=0.5 (V=1); I₁=0.95, I₂=0.05 (V=0.44) and I₁=0.995, I₂=0.005 (V=0.07).

Figure 4.4 illustrates that, even with very different intensities of the two beams, the fringes can be easily distinguished. Also note that increasing a single intensity whilst leaving the other constant diminishes the contrast but increases the absolute modulation depth.

4.3.4 White light interference and coherence length

Equation (4.4) suggests that the interference term will continue to oscillate up to infinite ΔL. However, there is no light source that emits a single wavelength λ; in fact every light source has a finite bandwidth, Δλ. Figure 4.5 shows the general case; if Δλ/λ<0.01, we can speak of a monochromatic light source. However, for interferometry over a macroscopic distance, light sources with a very small bandwidth are needed.

From Eq. (4.4), it is evident that an interference maximum appears for ΔL=0, independent of the wavelength, λ. This phenomenon is called white light interference. If the light source emits a range of wavelengths, in fact for each wavelength a different interference pattern is formed and where the photodetector measures the sum of all of these patterns, the visibility, V, may deteriorate with increasing path difference, ΔL.

In Figure 4.6, the effect of a limited coherence length is illustrated for a number of different light sources:

Note that for each wavelength (colour), a different pattern is formed. In practical white light interferometry, these colours can be visibly distinguished over a few wavelengths. White light interference is only possible in interferometers where the path difference can be made approximately zero.

The path length, ΔL, over which the interference remains visible, that is the visibility decreases by less than 50%, is called the coherence length and is given by

$\mathrm{\Delta }L=\frac{{\lambda }_0}{\mathrm{\Delta }\lambda }{\lambda }_0=Q{\lambda }_0,$ (4.8)

(4.8)

where λ₀ is the wavelength of the light source and Q is the quality factor which determines over how many wavelengths interference is easily visible. Table 4.2 gives a few characteristics of known light sources.

In the early twentieth century, the cadmium spectral lamp was used for interference over macroscopic distances. Michelson’s determination of the cadmium lamp wavelength related to the metre standard was a breakthrough to a metre definition based on physical constants. The orange-red line of the ⁸⁶Kr spectral lamp was used as the metre definition from 1963 until 1983. This definition was possible as, with some effort, interference over a metre length difference was possible and a length up to 1 m could be measured using interferometry.

4.4.1 The Michelson and Twyman–Green interferometer

Where Michelson was a major pioneer in interferometry and carried out experiments that achieved major breakthroughs in physics, one often refers to a Michelson interferometer where in fact a Twyman–Green interferometer is intended. The original Michelson interferometer does not operate with collimated light, but with a point source, S, as shown in Figure 4.7.

A beam splitter, A, with a 50% coating splits the input beam. The interference fringes are detected from B. The compensator, C, is a glass plate with the same thickness as A which makes the optical path length through glass equal for both beams. This ensures that chromatic effects in the glass plate, A, are compensated and white light interferometry is possible.

Optically, the system as viewed from B consists of two sources, M₁ and M₂, behind each other. If the two image planes, M₁ and M₂, are parallel, this is equivalent to sources in line behind each other and one detects circular fringes. If M₁ and M₂ intersect, the crossover is the position of zero path difference and, as this region is a straight line of intersection, white light fringes will appear on the straight line of the intersection. The fringes appear to be localised at the front mirror, M₁, that is the detector must be focused on this surface in order to obtain the sharpest fringes. With increasing displacement, the fringes become spherical because of the divergent light source.

4.4.1.1 The Twyman–Green modification

In the Twyman–Green modification to the Michelson interferometer, the source is replaced by a point source, S, at the focus of a well-corrected concave lens (Figure 4.8). The lens B collects the emerging light and the detector observes the interference pattern at the focal plane, D.

Consider the case where the mirror and its image are parallel. Now the collimated point source leads to a field of uniform intensity. Variations of this interferometer are the Köster gauge block interferometer [9], displacement measuring interferometers (see Section 5.2) and the Linnik- and Mirau-type interference microscopes (see Section 6.7.3.2).

An important characteristic of the Twyman–Green interferometer is that the paths in both beams can be made equal so that white light interference occurs. A disadvantage is that both beams have a macroscopic path length and can be sensitive to turbulence and vibration. The reflectivity of both mirrors can be up to 100%. If the reflectivity of the mirrors is different, the visibility decreases, as is illustrated in Figure 4.4. In the interferogram, the difference between the two mirrors is observed. For example, if both mirrors are slightly convex, and one mirror is slightly tilted, the interferogram will consist of straight lines (the same as with perfectly flat mirrors).

4.4.2 The Fizeau interferometer

In Fizeau interferometry, the reference surface and the surface to be measured are brought close together. Compared with Figure 4.3, mirror M₁ is transparent and partially reflecting, and the partial reflecting side is positioned close and almost parallel to mirror M₂. This gives a configuration as shown in Figure 4.9.

For a wedge angle, α, and perfectly flat mirrors, the intensity of the interference pattern between the mirrors is given by

$I(x)=I_1+I_2+2\sqrt{I_1{I}_2}\cos 2k(\mathrm{\Delta }L+x\alpha ),$ (4.9)

(4.9)

where x is the position of the interference pattern from the left edge of the mirrors. In two dimensions, with circular mirrors, this gives a characteristic interference pattern consisting of straight lines (Figure 4.10).

The Fizeau interferometer gives a direct way of observing geometrical features in an interferogram. If the distance ΔL is increased, the fringes will move from left to right (or right to left). If the tilt angle is changed, the distance between the fringes changes. If either of the mirrors is not flat, this is observed as distortions in the straightness of the fringes.

If the interference term in Eq. (4.9) can be changed in some controlled manner, the phase φ=2kΔL can be determined by making intensity measurements in one location (x, y). The phase can be changed by a small displacement, ΔL, or by a wavelength change. If ΔL is changed in four steps of λ/8 each, and the intensities are labelled as I_A, I_B, I_C and I_D, then it can be shown that

$\phi (x,y)=\arctan \frac{I_{\mathrm{B}}-I_{\mathrm{D}}}{I_{\mathrm{A}}-I_{\mathrm{C}}}.$ (4.10)

(4.10)

This is an example of deriving the phase, and ΔL, by phase stepping. This can only give an estimate of ΔL within an unknown integer number, N, of half wavelengths.

Considered over the surface, the distance between the surfaces S₁ and S₂ can be expressed as

$\mathrm{\Delta }L(x,y)=\frac{\lambda }{2}\left(N+\frac{{\phi }_{{\mathrm{S}}_2}(x,y)-{\phi }_{{\mathrm{S}}_1}(x,y)}{2\pi }\right).$ (4.11)

(4.11)

If the upper surface deviations of both S₁ and S₂ are to be considered positive in the glass–air interface direction then, apart from a constant term and a constant tilt, the deviations can be expressed as

${\mathrm{S}}_2(x,y)={\phi }_2(x,y)\lambda /4\pi \mathrm{and}{\mathrm{S}}_1(x,y)=-{\phi }_1(x,-y)\lambda /4\pi .$ (4.12)

(4.12)

In S₁, the coordinates can be (x, −y) or (−x, y), depending on the definition and the (flipping) orientation of the (optical) surface. However, in a Michelson interferometer for S₁ the equivalent of S₂ holds.

If S₁ is perfectly flat, or has a known flatness deviation, the form of the other surface can be derived, either by visually observing the interference pattern or by analysing the phase using Eq. (4.11). This method of surface interferometry is a research field of its own and is covered in several textbooks (see Refs. [10,11]). Because the lateral resolution is usually limited, this is form rather than surface texture measurement. Uncertainties can be in the nanometre region in the direction perpendicular to the surface. Limitations are in the roughness and the maximum angle that can be measured. For engineered surfaces, this method is applicable for polished, and precision turned, lapped and ground surfaces. For such surfaces, Fizeau interferometry is a very powerful tool to obtain very rapidly the complete geometry of the surface.

Some characteristics of Fizeau interferometers should be mentioned, also in comparison to Michelson set-ups:

4.4.3 The Jamin and Mach–Zehnder interferometers

The Jamin interferometer is depicted in Figure 4.11. The beams are split in A and recombine at D. A first important application of the Jamin interferometer was the measurement of the refractive index of gases (T₁ and T₂ represent gas cells in Figure 4.11). The Jamin arrangement can also be used to make an image interfere with itself, but slightly displaced, for example by tilting one mirror relative to the other. This is called shearing interferometry.

A modification of the Jamin arrangement is known as the Mach–Zehnder interferometer and is depicted in Figure 4.12. As in the Michelson interferometer, white light interference is possible and there is no limitation to the reflectance at, for example, points C and F.

The Mach–Zehnder interferometer can be used for refractometry, that is for measurement of the refractive index of a medium in either arm. It can also be modified in order to enable displacement measurement.

4.4.4 The Fabry–Pérot interferometer

If in the Fizeau interferometer in Figure 4.9 both mirrors are placed almost parallel and the reflectance of both mirrors is increased, a particular type of interferometer is obtained, called the Fabry–Pérot interferometer (Figure 4.13).

Light enters from the left, and B and B′ are the reflecting faces between which the interference occurs. P and P′ are spacers to put flats B and B′ as parallel as possible. Between B and B′ multiple reflections occur. Equation (4.4) no longer holds if the reflectance, R, of both plates becomes significantly large, for example R>0.1. Summation of all reflected and transmitted components leads to an infinite series, which can be expressed as

$T=\frac{F{\sin }^2(L/\lambda )}{1+F{\sin }^2(L/\lambda )},$ (4.13)

(4.13)

where F is defined as

$F=\frac{4R}{{(1-R)}^2}.$ (4.14)

(4.14)

The reflectance of the whole system is given by R=1−T, where T is given by Eq. (4.13) and where it is assumed that no absorption takes place. The transmittance as a function of the distance, L, between the plates, for a wavelength λ=600 nm, is shown in Figure 4.14.

Figure 4.14 shows (co)sine-like behaviour similar to that described in Eq. (4.4) for low reflectances, but for high reflectance of the mirrors there are sharp transmittance peaks. This has the disadvantage that in between the peaks the position is difficult to estimate, but it has the advantage that once a peak reflectance is achieved, one is very sure that a displacement of exactly an integer number of half wavelengths has taken place.

The reciprocal of the full width of a fringe at half of the maximum intensity expressed as a fraction of the distance between two maxima is given by

$N_R=\frac{\pi \sqrt{R}}{1-R}=\frac{\pi }{2}\sqrt{F}.$ (4.15)

(4.15)

The term N_R is called the finesse of the interferometer. For example, for R=0.9, N_R=30. This means that 1/30th of a half wavelength can readily be resolved by this interferometer; compare this to half of a half wavelength using the same criterion for the cosine function in Eq. (4.4).

At a fixed distance, L, the possible frequencies that fit in the cavity can be calculated as follows

$L=m\frac{\lambda }{2}=m\frac{c}{2nf}\to f_m=m\frac{c}{2nL}\to \mathrm{\Delta }f=f_{m+1}-f_m=\frac{c}{2nL}.$ (4.16)

(4.16)

Here m=0, 1, 2, … and n is the air refractive index, which is approximately 1. The frequency difference between two successive possible frequencies is called the free spectral range. For example, for a cavity length L=100 mm, Δf=1.5 GHz. Clearly, in a Fabry–Pérot interferometer, white light interferometry is not possible. The interferometer can also be made with spherical mirrors. In this case, the equation for the finesse changes somewhat. This and other details of the Fabry–Pérot interferometer are extensively treated in Ref. [14].

Fabry–Pérot interferometers have many applications in spectroscopy. However, in engineering nanometrology, they are used as the cavity in lasers and 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 external cavity. This is treated in more detail in Section 5.10.1.

4.5.1 Gauge blocks and interferometry

As discussed in Section 4.2, the length of a gauge block wrung to a platen can be measured using interferometry. The ISO definition of a gauge block length has a twofold purpose: (1) to ensure that the length can be measured by interferometry and (2) to ensure that there is no additional length due to wringing. An issue that is not obvious from the definition is whether the two-sided length of a gauge block after calibration by interferometry coincides with the mechanical length, for example as measured by mechanical probes coming from two sides. Up to now, no discrepancies have been found that exceed the measurement uncertainty, which is in the 10–20 nm range.

Figure 4.15 shows a possible definition for a mechanical gauge block length. A gauge block with length L is probed from both sides with a perfectly round probe of diameter d, being typically a few millimetres in diameter. The mechanical gauge block length, L, is the probe displacement, D, in the limit of zero force, minus the probe diameter, or L=D−d.

4.5.2 Gauge block interferometry

In order to measure gauge blocks in an interferometer, a first requirement for the light source is to have a coherence length that exceeds the gauge block length. Gauge block interferometers can be designed as a Twyman–Green or a Fizeau configuration, where the former is more common. For the majority of the issues discussed in this section, either configuration can be considered.

Figure 4.16 is a schema of a gauge block interferometer containing a gauge block. The observer sees the fringe pattern that comes from the platen as shown in Figure 4.10. If the platen has a small tilt, this will be a set of straight, 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 (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 the case, and the shift of fringes gives the fractional length of the gauge block. The length of the gauge block is given by

$\begin{array}{cc}L& =\frac{{\lambda }_{\nu }}{2n(\lambda )}\left\{N+\frac{[{\phi }_{\mathrm{block}}(\mathrm{top})-{\phi }_{\mathrm{ref}}(\mathrm{top}\mathrm{area})]-[{\phi }_{\mathrm{platen}}(\mathrm{base})-{\phi }_{\mathrm{ref}}(\mathrm{base}\mathrm{area})]}{2\pi }\right\}\\ & =\frac{{\lambda }_{\nu }}{2n(\lambda )}(N+f)\end{array},$ (4.17)

(4.17)

where N is the number of half wavelengths between the gauge block top and the position on the platen for wavelength λ, n is the air refractive index and f is the fraction f=a/b in Figure 4.17. ϕ_block (top) is the phase on top of the gauge block, ϕ_ref (top area) is the phase at the reference plate at the location of the top area, ϕ_platen (base) is the phase on the platen next to the gauge block and ϕ_ref (base area) is the phase at the reference plate at the location next to the image of the gauge block. For a flat reference surface, the phase for the areas corresponding to the base and top of the gauge block is the same (ϕ_ref (top area)=ϕ_ref (base area)) and Eq. (4.17) simplifies accordingly. Equation (4.17) is the basic equation that links an electromagnetic wavelength, λ, to a physical, mechanical length, L.

Some practical issues that are met when applying Eq. (4.17) are treated in the next sections.

4.5.3 Operation of a gauge block interferometer

4.5.3.2 Multiple wavelength interferometry analysis

If just a single fraction of a single wavelength is known, the gauge block length must be known beforehand within an uncertainty of 0.15 μm in order to define N in Eq. (4.17) within one integer unit. For gauge blocks to be calibrated this level of prior knowledge is usually not the case – see Table 4.1 – and it is common practice to solve this problem by using multiple wavelengths. In the original gauge block interferometers, this was usually possible as spectral lamps were used that emitted several lines with an appropriate coherence length. In modern interferometers, laser sources are normally used, and the demand for multiple wavelength operation is met with multiple laser sources.

For multiple wavelengths, λ_i (i=1, 2, …), Eq. (4.17) can be rewritten as

$L=\frac{{\lambda }_{1,\nu }}{2n({\lambda }_1)}(N_1+f_1)=\frac{{\lambda }_{2,\nu }}{2n({\lambda }_2)}(N_2+f_2)=\frac{{\lambda }_{3,\nu }}{2n({\lambda }_3)}(N_3+f_3)=\cdots$ (4.18)

(4.18)

However, because of a limited uncertainty in the fringe fraction determinations, there is not a single length that can meet the requirements of Eq. (4.18) for all wavelengths and fractions. There are several strategies for finding an optimal solution for the length; for example, for the longest wavelength, a set of possible solutions around the nominal length can be taken and for each of these lengths the closest solution for the possible lengths for the measurements at the other wavelengths can be calculated. The average of the length with the least dispersion is then taken as the final value. This method is known as the method of exact fractions and has similarities with reading a vernier on a ruler.

More generally, Eq. (4.18) can be written as a least-squares problem. The error function is given by

${\chi }^2={\sum _{i=1}^K\left(L_{\mathrm{e}}-\frac{{\lambda }_{i,\nu }}{2n({\lambda }_i)}(N_i+f_i)\right)}^2,$ (4.19)

(4.19)

where K is the number of wavelengths used and L_e is the estimated length. For ideal measurements, χ²=0 for L_e=L. For real measurements, the best estimate for L is the value for L_e where χ² is minimal. For any length, L_e, first the value of N_i that gives a solution closest to L has to be calculated for each wavelength before calculating χ². As Eq. (4.19) has many local minima (every 0.3 μm), it must be solved by a broad search around the nominal value, for example as was implicitly done in the procedure described above. To distinguish between two adjacent solutions, the fringe fractions must be determined accurately enough. This demand is higher if the wavelengths are closer together. For example, for wavelength λ₁=633 nm (red) and λ₂=543 nm (green), the fractions must be determined within 15% in order to ensure that a solution that is 0.3 μm in error is not found.

Multiple wavelengths still give periodic solutions where χ² is minimal, but instead of 0.3 μm, these are further apart; in the example of the two wavelengths just given, this period becomes 2.5 μm. If two wavelengths are closer 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 the solutions; the wavelength range determines the demand on the accuracy of the fraction determination. A common strategy for obtaining an approximate value for L_e, with an uncertainty at least within the larger periodicity, is to carry out a mechanical comparison with a calibrated gauge block.

4.5.3.4 Thermal effects

The reference temperature for gauge block measurements is defined in the specification standard ISO 1 [18] to be 20°C, exactly. The reason that it is necessary to specify a temperature is because all gauge blocks will change size when their temperature changes due to thermal expansion. The amount by which the material changes length per degree temperature change is the coefficient of thermal expansion, α. For a typical steel gauge block, the coefficient of thermal expansion is about 11.5×10⁻⁶ K⁻¹ and for a tungsten carbide gauge block it is nearer 4.23×10⁻⁶ K⁻¹.

In order to correct for the change in length due to thermal expansion, it is necessary to measure the temperature of the gauge block at the same time as the length is being measured. The correction factor can be derived from

$L(T)=L(20)\times (1+\alpha [T-20]),$ (4.20)

(4.20)

where L(T) is the length at temperature, T (in degrees Celsius), and L(20) is the length at 20°C.

Equation (4.20) indicates that an accurate temperature measurement is more critical when α is large and that knowledge of α is more critical if the temperature deviates from 20°C. For example, for a one part per million per degree Celsius error, Δα, in the expansion coefficient, the error is 100 nm for a 100 mm gauge block at 21°C. For α=10×10⁻⁶ K⁻¹, a 0.1°C uncertainty in the temperature gives 100 nm uncertainty in a 100 mm gauge block.

4.5.3.6 Aperture correction

A subtle optical effect that is less obvious than the previous uncertainty influences is the so-called aperture correction. The figures in Section 4.4 show the light sources as point sources, but in reality a light source has a finite aperture. This means that light does not only strike the gauge block and reference plane exactly perpendicular, but also at a small angle. This makes the gauge block appear shorter than it really is. The correction for this effect for a circular aperture is given by

$\mathrm{\Delta }L=\frac{L{D}^2}{16f^2},$ (4.21)

(4.21)

where D is the aperture and f is the focal length of the collimating lens. Taking some typical numbers, D=0.5 mm, f=200 mm, L=100 mm, we find ΔL=0.04 μm. This correction is much larger in the case of an interference microscope, where it may amount to up to 10% of the measured height (see Section 6.7.1).

In some interferometer designs, there is a small angle between the impinging light and the observed light that gives rise to a similar correction known as the obliquity correction.

4.5.3.7 Surface and phase change effects

As indicated in the gauge block definition, the gauge block has to be wrung on to a platen having the same material and surface roughness properties as the gauge block. In practice, this can be approached but never guaranteed. Sometimes glass or quartz is preferred as a platen because the wringing condition can be checked through the platen. Because of the complex refractive index of metals, light effectively penetrates into the material before being reflected, so a metal gauge block on a glass platen will be measured as too short. Additional to this is the gauge block roughness effect. Typical total correction values are 0.045 μm for a steel gauge block on a glass platen and 0.01 μm for a tungsten carbide gauge block on a glass platen.

A very practical way of determining the surface effects is wringing a stack of two (or more) gauges together on a platen and comparing the length of the stack to the sum of the individually measured gauge blocks.

This is illustrated for two gauge blocks in Figure 4.18, where g and p are the apparent displacements of the optical surface from the mechanical surface, f is the wringing film thickness and L_i are the defined (mechanical) lengths of individual gauges. It can be shown that the measured length of the combined stack minus the individual measured length is the correction per gauge. This method can be extended to multiple gauge blocks to reduce the uncertainties. Here it is assumed that the gauge blocks are from the same material and have nominally the same surface texture.

Other methods for measuring corrections for surface effects of the gauge block and platen have been proposed and are used in some NMIs (see, for example Ref. [24]). Such methods can offer a slightly reduced uncertainty for the phase correction but are often difficult to set up and can give results that may be difficult to interpret.

4.5.4 Sources of error in gauge block interferometry

In this section, some more detailed considerations are given on the errors generated by the different factors mentioned in Section 4.5.3 (see Ref. [25] for a more thorough treatment).

4.5.5 Alternative approaches

The need for wringing gauge blocks to a platen and let this stabilise is labour- and time consuming. For this reason, alternative interferometric methods have been developed. The basic idea is that both sides of a gauge block can be measured interferometrically, either in a single triangular interferometer or by two Twyman–Green interferometers from both sides simultaneously. Special care has to be taken for both the length determination and the phase correction. The phase correction that is necessary as the gauge block length is not measured according to its definition is often considered as the essential drawback; however, this phase correction needs to be determined anyhow so this should not be considered as so essential.

A fully operational double-sided gauge block interferometer was developed elsewhere [28]. The schema is given in Figure 4.19. The light is reflected from two reference mirrors (RM1, RM2) to create interference patterns from both ends of the gauge block (GB), as well as combinations of the total cavity length with the gauge block length, from which the absolute gauge block length can be derived. Abdelaty [29] has shown for a similar set-up that there are challenges, especially as the phase correction has to be applied twice and its uncertainty contributes accordingly, and additional mirrors and beam splitters of high quality are needed. Another approach was taken by Winarno et al. [30], who combine a double-sided interferometer in a triangle configuration with a displacement measurement that can be carried out remotely. In this case, the uncertainty of traditional gauge block interferometry has not yet been achieved.

The absence of wringing will mean shorter measurement times, less risk of scratches and wringing errors, and may make the interferometric gauge-stabilisation block calibration far more economic.

Using a frequency comb (see Section 2.9.6) opens up some interesting new possibilities [16,17], where a beam full of well-defined frequencies can be used both as a white light source that defines an absolute position in space, and as an accurate distance and/or displacement measurement that can be combined to an absolute gauge block length in single- and double-sided set-ups.