The new mise en pratique for the metre—a review of approaches for the practical realization of traceable length metrology from 10⁻¹¹ m to 10¹³ m

The direct measurement of the travel time of light requires modulation of the light intensity in order to generate fiducial features. The most vivid modulation is represented by a light pulse, e.g., a laser light pulse which is formed from a wave packet. The incident light pulse shown in figure 1 is split into two pulses, one of which travels a short reference path, the other one travels the measuring path. The mirrors in both pathways are arranged such that the light is retroreflected. After the second passage through the beam splitter the light pulse originating from the short reference path first hits a light detector and sets a first trigger at a defined threshold, defining a reference point in time. A second trigger is then generated by the delayed light pulse originating from the measurement path.

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From the time delay Δt between both triggers, the length difference between measurement and reference pathways is derived, which represents the length ${\Delta}z=l=\frac{1}{2}{c}_{\mathrm{g}}\cdot {\Delta}t$. For example, if the length, l, of 1 m, is traversed twice by the light pulse (outwards and return directions), then the time delay is approximately 6 ns. Instead of utilizing light pulses, representing a wave packet, amplitude-modulated light can be used to achieve higher resolutions. With this approach, the reference mirror (see figure 1) is omitted and the phase of the intensity modulation is determined electronically from the detector signal. The travel time for the light along the measuring path is then the difference ${\Delta}{\varphi }^{\mathrm{m}\mathrm{o}\mathrm{d}}={\varphi }^{{\mathrm{m}\mathrm{o}\mathrm{d}}_{\mathrm{d}}}-{\varphi }^{{\mathrm{m}\mathrm{o}\mathrm{d}}_{\mathrm{i}}}$ between the measured intensity modulation phases at the detector, ${\varphi }^{{\mathrm{m}\mathrm{o}\mathrm{d}}_{\mathrm{d}}}=2\pi {f}^{\enspace \mathrm{mod}\enspace }\cdot \left(t+{\Delta}t\right)$, and before travel, ${\varphi }^{\mathrm{m}\mathrm{o}{\mathrm{d}}_{\mathrm{i}}}=2\pi {f}^{\enspace \mathrm{mod}\enspace }\cdot t$, which leads to:

$$\begin{equation}{\Delta}t=\frac{{\Delta}{\varphi }^{\mathrm{m}\mathrm{o}\mathrm{d}}}{2\pi {f}^{\mathrm{m}\mathrm{o}\mathrm{d}}},\end{equation} \tag{ 5 }$$

in which f^mod is the modulation frequency. High modulation frequencies of the order of several gigahertz are used in sophisticated devices using external electro-optic [7] or electro-absorption modulators [8]. Using femtosecond lasers as a light source, multiple highly stable modulation frequencies from several hundred megahertz up to several tens of gigahertz can be generated in parallel [9].

The direct measurement of the light's travel time is largely used for measurements at geodetic scales. Here, the group refractive index of air, its homogeneity and invariance, are the limiting factors of the attainable measurement uncertainty. It is also used for even larger distances, e.g., from the earth to the Moon [10]. For the realization of short lengths interferometrical methods are more effective and generally more accurate.

For the realization of lengths below a few metres, as well as for the most accurate realization of length in general, interferometric techniques are far superior. Monochromatic light can be considered as an electromagnetic wave, the electric field of which is propagating along the measurement pathway (defined as z-direction) ³ :

$$\begin{equation}E(z,t)=A\cdot \mathrm{cos}\left(\varphi \right)=A\cdot \mathrm{cos}\left(k\cdot z-\omega \cdot t+\delta \right)\end{equation} \tag{ 6 }$$

in which A is the amplitude, φ the phase, k the wave number, ω the angular frequency, t the time, and δ the initial phase. The relationship between the parameters k and ω with wavelength λ and frequency f is given by k = 2π/λ and ω = 2π ⋅ f. Wavefronts travel the distance of a single wavelength during a single oscillation period T (T = 1/f). The frequency of visible light waves lies in the range from 300 THz to approximately 600 THz. The respective oscillation period is therefore extremely short and, because the upper frequency limit of detectors is too low, the oscillation period could not be recorded directly by any real detector. The only measurable parameter of a single light wave is a mean intensity ${\left\langle {E}^{2}\right\rangle }_{t}$. The infinite average (t → ∞) of the latter, ${\mathrm{lim}}_{t\to \infty }\enspace \frac{1}{t}{\int }_{0}^{t}{\left(E\left(t,z\right)\right)}^{2}\enspace \mathrm{d}t$ results in the constant value of the light intensity: $I=\frac{1}{2}{A}^{2}$.

Two waves can be brought into interference with the simplest kind of interferometer shown in figure 2, left, which is basically the same arrangement as in figure 1. The amplitude ratio of the two waves, which interfere when leaving the interferometer, depends on the reflectivity of the beam splitter and it can be assumed that both waves have the same initial phase and frequency. However, the phase of the light wave ${E}_{2}(z,t)={A}_{2}\cdot \mathrm{cos}\left[k\cdot \left(z+2{\Delta}z\right)-\omega \cdot t+\delta \right]$ which double passes the measurement path is enlarged by k ⋅ 2Δz compared to the wave ${E}_{1}(z,t)={A}_{1}\cdot \mathrm{cos}\left[k\cdot z-\omega \cdot t+\delta \right]$ traversing the reference path. The sum of both waves represents the interference wave which can be written as ⁴ :

$$\begin{equation}{E}_{12}={E}_{1}+{E}_{2}={A}_{12}\cdot \mathrm{cos}\left(k\cdot z-\omega \cdot t+{\delta }_{12}\right).\end{equation} \tag{ 7 }$$

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The amplitude of the interference wave is given by ${A}_{12}=\sqrt{{{A}_{1}}^{2}+{{A}_{2}}^{2}+2{A}_{1}\cdot {A}_{2}\enspace \mathrm{cos}\left(2k\cdot {\Delta}z\right)}$ and the phase shift by ${\delta }_{12}=\mathrm{arctan}\enspace 2\left({A}_{1}\cdot \mathrm{sin}\left(2k\cdot {\Delta}z\right),{A}_{1}\cdot \mathrm{cos}\left(2k\cdot {\Delta}z\right)+{A}_{2}\right)$.

The infinite average of ${\left\langle {{E}_{12}}^{2}\right\rangle }_{t\to \infty }$, which represents the intensity, is thus given by:

$$\begin{equation}{I}_{12}=\frac{1}{2}{{A}_{12}}^{2}={I}_{0}\left(1+\gamma \enspace \mathrm{cos}\left(2\pi \cdot \frac{{\Delta}z}{\lambda /2}\right)\right),\end{equation} \tag{ 8 }$$

where γ denotes the interference contrast $\gamma =2\sqrt{{I}_{1}{I}_{2}}/\left({I}_{1}+{I}_{2}\right)=\left({I}_{\mathrm{max}}-{I}_{\mathrm{min}}\right)/\left({I}_{\mathrm{max}}+{I}_{\mathrm{min}}\right)$ and I₀ = I₁ + I₂ = (A₁ ² + A₂ ²)/2.

As shown in figure 2, there is a relationship between the length of the measurement path and the number of oscillations, the so-called interference order, due to the phase change:

$$\begin{equation}\frac{{\Delta}\varphi }{2\pi }=\frac{{\Delta}z}{\lambda /2}.\end{equation} \tag{ 9 }$$

Viewed from the other side, measurement of Δφ/2π allows determination of Δz based on the cumulative phase change. In the simplest case the integer interference order, i.e. the integer part of Δφ/2π, is obtained by counting the number of intensity periods while continuously shifting the measurement mirror. In any case, the size of shift of the measurement mirror, i.e. a length, is measured as an arithmetic product of half of the light wavelength and the interference order:

$$\begin{equation}l={\Delta}z=\frac{1}{2}\lambda \cdot \frac{{\Delta}\varphi }{2\pi }=\frac{1}{2}\frac{c}{n}\cdot \frac{{\Delta}\varphi }{\omega }.\end{equation} \tag{ 10 }$$

Accordingly, equation (10) reveals (see equation (2)) that the travel time of light is obtained from:

$$\begin{equation}{\Delta}t=\frac{{\Delta}\varphi }{\omega }=\frac{{\Delta}\varphi }{2\pi f}.\end{equation} \tag{ 11 }$$

Again (for comparison see equation (5)), the travel time of light results from the quotient of a phase difference and a frequency.

Equation (11) clearly reveals that the indirect measurement of the travel time of light requires measurement of the following quantities:

• (a)  
  The frequency f of the light;
• (b)  
  The phase difference Δφ between the two interfering waves resulting from the observation of the intensity of interference using an interferometer.

As equations (10) and (11) reveal, knowledge of the frequency of the light, f, is an essential requirement for the realization of the unit of length. It provides the scaling factor between a measured phase difference and the length that is realized by interferometry. For the highest demands on the accuracy of the light frequency, a light source could be synchronized to the primary frequency standards by an appropriate technique.

Generally, the amount of the frequency difference $\left\vert {\Delta}f\right\vert$ of a light source under test (wave ${E}_{2}\left(t\right)={A}_{2}\cdot \mathrm{cos}\left(\left(k+{\Delta}k\right)\cdot z-\left(\omega +{\Delta}\omega \right)\cdot t+{\delta }_{2}\right)$) with respect to the frequency of a reference source (wave ${E}_{1}\left(t\right)={A}_{1}\cdot \mathrm{cos}\left(k\cdot z-\omega \cdot t+{\delta }_{1}\right)$), is measured also by the interference detection technique. In this approach the two light waves are brought into coincidence along the same path direction of propagation. In analogy to equation (7), the associated interference wave can be written as ${E}_{12}\left(t\right)={A}_{12}\left(t\right)\cdot \mathrm{cos}\left(k\cdot z-\omega \cdot t+\delta \left(t\right)\right)$, in which ${A}_{12}\left(t\right)$ represents the oscillating amplitude ${A}_{12}\left(t\right)=\sqrt{{{A}_{1}}^{2}+{{A}_{2}}^{2}+2{A}_{1}\cdot {A}_{2}\enspace \mathrm{cos}\left({\Delta}k\cdot z-{\Delta}\omega \cdot t+{\delta }_{2}-{\delta }_{1}\right)}$ and z the position of the detector. Again, the high frequency term of ${E}_{12}\left(t\right)$, which includes the light frequency itself, cannot be acquired with any detector. Therefore, the measurable interference intensity is given by ${I}_{12}\left(t\right)=\frac{1}{2}{A}_{12}{\left(t\right)}^{2}={I}_{0}\left(1+\gamma \enspace \mathrm{cos}\left({\Delta}k\cdot z-{\Delta}\omega \cdot t+{\delta }_{2}-{\delta }_{1}\right)\right)$. For the purpose of further considerations, made in the following paragraphs, Δk = k₁ − k₂ is expressed as $\frac{1}{c}\left({n}_{1}\cdot {\omega }_{1}-{n}_{2}\cdot {\omega }_{2}\right)$ and n₂ as ${n}_{2}={n}_{1}+\left({\omega }_{2}-{\omega }_{1}\right)\cdot \frac{\mathrm{d}n}{\mathrm{d}\omega }$. This results in:

$$\begin{align}{I}_{12}\left(t\right)& =\frac{1}{2}{A}_{12}{\left(t\right)}^{2}\\ & ={I}_{0}\left( 1 + \gamma \enspace \mathrm{cos}\left( -2\pi \cdot {\Delta}f \cdot \left( t - \frac{1}{c}{n}_{\mathrm{g}}\cdot z \right) + {\delta }_{2}-{\delta }_{1 }\right), \right),\end{align} \tag{ 12 }$$

in which n_g is the group refractive index. Equation (12) represents the so-called beat signal from which $\left\vert {\Delta}f\right\vert$ can be derived electronically. The frequency of a light source (E) under test (f_EUT) is therefore related to the frequency of a standard: f_EUT = f_standard ± f_beat. To achieve unambiguity, it is necessary to have more than one standard frequency light source available, or to be able to vary the standard frequency in a known direction, e.g. by reducing the frequency by a small amount and observing the change in f_beat.

As an alternative to direct measurement of the light frequency, one may use one of the values from the list of 'recommended values of standard frequencies for applications including the practical realization of the metre and secondary representations of the second' [11]. This list is updated periodically by recommendation of new candidate standard frequencies. Candidate frequencies are examined according to a published set of guidelines and procedures [12] and only those that pass the necessary checks are recommended to the CIPM for entry into the list.

From the calibrated frequency of the light source, the wavelength λ₀ can be calculated which in vacuum is generally assumed to be identical to the path length corresponding to a single order of interference ($l=\frac{1}{2}\lambda$). While this assumption is correct for an ideal interference of two plane waves, it does not apply if the light is not perfectly monochromatic, or for non-planar waves (see section 4.5 for the corresponding effect to the measurement uncertainty).

With the arrangement shown in figure 2, the direction in which the measuring mirror is moved cannot be detected (see equation (8)). In extreme cases, moving the measurement mirror forward and backward can even simulate uniform motion along the z-axis. For this reason, most homodyne interferometers are equipped with additional components, using the 'quadrature procedure' in order to establish a unique relationship to the position of the measuring mirror. As shown in figure 3, a linearly polarized laser beam whose polarization axis is tilted by 45° hits a polarizing beam splitter which splits the incident light into the two polarization components that are perpendicular to each other. While the component which is polarized parallel to the incident plane (p-beam) entirely passes through the beam splitter, the vertically polarized component (s-beam) is fully reflected. These two beams are reflected and then brought together at the polarizing beam splitter. For the generation of interference between the two mutually perpendicular components s and p polarizers are required, the axis of which is tilted by 45°.

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Additionally, the beam which is transmitted at the beam splitter traverses a λ/4 delay plate before the polarizer which causes a mutual phase shift of π/2. Thus, the signal at this detector is shifted by π/2 compared to the signal of the other detector. The quadrature detection thus provides two interference signals that are shifted by π/2 with respect to each other and are thus called sine and cosine signals, respectively. The signals ${~{I}}_{\mathrm{sin}}$ and ${~{I}}_{\mathrm{cos}}$ can be represented as shown in figure 3 on the right, e.g. by means of an oscilloscope in the xy mode. A shift of the measuring mirror then leads to a circle being displayed. Depending on the direction of displacement, the corresponding vector rotates in one or the other direction. Hereby, a full revolution corresponds to one integer interference order (Δφ = 2π) and thus to a shifting of the measuring mirror by Δz = λ/2, which can be clearly detected with this procedure. While the number of revolutions can be counted electronically, in this approach the fractional part of the phase difference Δφ is represented by the angle $\varphi = \mathrm{arctan}\enspace 2\left({~{I}}_{\mathrm{sin}},{~{I}}_{\mathrm{cos}}\right)$ (see figure 3, right). In reality unavoidable imperfections, e.g. unequal detector gains and offsets can cause distortions of the circle resulting in an ellipse that can be 'corrected' back to a circle in the post processing by a Heydemann or similar correction (see [13–15] and references therein). Incomplete separation of the differently polarized partial beams in a polarizing interferometer as well as multiple back reflections, so-called optical mixing errors also produce periodic non-linearities (see [16–19] and references therein).

Laser sources can be designed to emit two or more coherent light waves of slightly different frequency that are polarized perpendicular to each other. For He–Ne lasers, this is often achieved by external magnetic fields inducing the Zeeman effect [16], leading to frequency differences of typically 1 MHz to 3 MHz. Larger frequency differences can be generated by suitable laser resonator designs [20]. Furthermore, it is possible to shift the frequency of a light wave by a defined value, e.g. by means of acousto-optic modulators. Figure 4 shows a scheme of such an interferometer known as a heterodyne interferometer. The laser light source used here generates two waves with different frequencies: ${E}_{1}\left(t\right)={A}_{1}\cdot \mathrm{cos}\left({k}_{1}\cdot z-{\omega }_{1}\cdot t+{\delta }_{1}\right)$ and ${E}_{2}\left(t\right)={A}_{2}\cdot \mathrm{cos}\left({k}_{2}\cdot z-{\omega }_{2}\cdot t+{\delta }_{2}\right)$ that are polarized perpendicular to each other ('s' indicates vertical polarization with respect to the image plane and 'p' parallel polarization). First, both light waves are split using a non-polarizing beam splitter. The two reflected waves then hit a polarizer, the axis of which is inclined by 45° with respect to the two polarization directions.

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As a result, the now equally polarized components of the interference waves E₁ and E₂ lead to a beat signal at the reference detector (see derivation of equation (12) above):

$$\begin{equation}{I}_{\mathrm{r}\mathrm{e}\mathrm{f}}\left(t\right)\propto 1 + \gamma \enspace \mathrm{cos}\left( - \left({\omega }_{2} - {\omega }_{1}\right)\cdot \left( t - \frac{1}{c}{n}_{\mathrm{g}}\cdot {z}_{\mathrm{r}\mathrm{e}\mathrm{f}}\right) + {\delta }_{2}-{\delta }_{1 }, \right),\end{equation} \tag{ 13 }$$

in which z_ref is the length of pathway travelled by E₁ and E₂ to the reference detector. The two light waves that pass through the non-polarizing beam splitter hit a polarizing beam splitter where E₁ is reflected and E₂ is transmitted and pass the measurement path. After both waves have been reflected by the retroreflectors, they are combined after the second pass of the polarization beam splitter. After passing through a polarizer, the axis of which is inclined by 45° in relation to the two polarization directions, a second beat signal is generated at the measuring detector:

$$\begin{align}{I}_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}\left(t\right)& \propto 1 + \gamma \mathrm{cos} \left({ k}_{2}\cdot 2{\Delta}z - \left({\omega }_{2}-{\omega }_{1}\right)\cdot \left( t - \frac{1}{c}{n}_{\mathrm{g}}\cdot {z}_{1,\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s} }\right)\right.\\ & \quad \left.+{\delta }_{2}-{\delta }_{1}\right),\end{align} \tag{ 14 }$$

in which z_1,meas is the length of pathway of E₁ via the reference mirror to the measuring detector and Δz is again the length difference between the distances from measuring mirror and reference mirror, respectively, to the beam splitter ($2{\Delta}z={z}_{2}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}-{z}_{1}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}$). Considering equations (13) and (14), the difference between the phases of I_meas with respect to I_ref, which can be measured electronically, is represented by:

$$\begin{equation}{\varphi }_{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}-{\varphi }_{\mathrm{r}\mathrm{e}\mathrm{f}}={k}_{2}\cdot 2{\Delta}z+{\varphi }_{0},\end{equation} \tag{ 15 }$$

in which ${k}_{2}=\frac{1}{c}{\omega }_{2}\cdot {n}_{2}$ represents the wave number of E₂ and ${\varphi }_{0}=\left({\omega }_{2}-{\omega }_{1}\right)\cdot \frac{1}{c}{n}_{\mathrm{g}}\cdot \left({z}_{1}^{\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}}-{z}^{\mathrm{r}\mathrm{e}\mathrm{f}}\right)$ can be considered as a constant value, provided that positions of detectors and the reference mirror, but also n_g are unchanged. Therefore, when considering a movement of the measuring mirror in a distance measurement, the length of the distance is again obtained from equation (10). The effect of non-linearities in heterodyne displacement interferometry and how to reduce them is widely discussed in the literature (e.g. see [16, 21] and references therein).

The larger the length to be measured, the more likely that beam interruptions can occur, the more impractical the movement of a measuring mirror over a distance becomes. The relationship between a path length and the phase difference via equation (10) can be considered inversely, i.e. ${\Delta}\varphi =\frac{2}{c}\omega \cdot n\cdot l=\frac{4{\pi}}{c}f\cdot n\cdot l$. When the frequency of the light source can be altered, e.g. when using a suitable tuneable laser, that allows a time dependent frequency $f\left(t\right)$, phase differences are observed as a function of the light frequency. In this approach, from the measured slope of $\frac{\mathrm{d}{\Delta}\varphi }{\mathrm{d}f}=\frac{4\pi }{c}\cdot l\cdot \left(n+f\cdot \frac{\mathrm{d}n}{\mathrm{d}f}\right)=\frac{4{\pi}}{c}\cdot l\cdot {n}_{\mathrm{g}}$ the length is determined:

$$\begin{equation}l=\frac{1}{4\pi }\cdot \frac{c}{{n}_{\mathrm{g}}}\cdot \frac{\mathrm{d}{\Delta}\varphi }{\mathrm{d}f},\end{equation} \tag{ 16 }$$

in which n_g is the group refractive index of air (see equation (4)). While various concepts for frequency sweeping interferometer (FSI) [22–26] form the basis of versatile length measurements, the uncertainty in the resulting length is typically much larger than one-quarter of a wavelength. Critical to the traceability of FSIs is the determination of the rate of frequency change with a link to the SI second. This can be performed by simultaneously observing the laser output through an absorption medium which offers quantum transitions at well-known frequencies, such as the spectra of acetylene and HCN in the infra-red part of the spectrum. Currently the CIPM has approved a set of 111 recommended frequencies for transitions in various species of the acetylene molecule [11] at around 19 THz (1.5 µm wavelength) with uncertainties at the level of a few parts in 10¹¹, and work is underway internationally to secure agreement on similar transitions in HCN [27].

Figure 5 shows a diagram for a multi-beam interferometer, in which the interference of the light is observed in reflection. After passing through the beam splitter, the light passes through a semi-transparent optical plate. Some of the light reflected by the mirror is reflected by the semi-transparent surface and goes back to the mirror. The higher the reflectivity R of the semi-transparent surface, the more often this sequence is repeated.

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After passing through the beam splitter, the interference wave resulting from the multi-beam interference hits a detector. The amplitude of the interference wave can be derived from the infinite sum of partial light waves (see, for example, [28]). As in the case of the two-beam interference, the associated interference intensity is a periodic function of ${\Delta}\varphi ={\Delta}z/\left(\frac{1}{2}\lambda \right)$, i.e. the periodicity is given by half the wavelength of the light used. The interference intensity, the formula of which is shown in figure 5, leads to an increasingly sharper structure with increasing reflectivity R of the semi-transparent plate (see insert in figure 5).

A multiple-beam interferometer is called a 'Fizeau interferometer' if the distance between the two flat surfaces on which the reflection takes place is large compared to the wavelength of the light. An advantage of such interferometers is the spatial narrowness of the fringe minima which can be accurately localised using e.g. CCD detectors. On the other hand, the dependency of the interference signal on the path difference is more complicated for Fizeau interferometers. With an absorbing film used as a beam-splitter, the interference profile of the reflected light will even be asymmetric.

A Fabry–Pérot interferometer (FPI) is a Fizeau interferometer operated with two semi-transparent mirrors whose reflectivity R is typically large. The so-called 'finesse' serves to characterize the resonator. It is defined as the ratio of the so-called free spectral range Δλ to the full width at half maximum δλ of an individual maximum of the interference intensity: $\mathcal{F}={\Delta}\lambda /\delta \lambda =\pi \sqrt{R}/\left(1-R\right)$. Extremely stable resonators with high finesse can be used for short-term frequency stabilization of laser sources. FPI resonators made of monocrystalline silicon, for example, enable frequency stabilization of commercial laser systems of better than 10⁻¹⁶ [29]. In a limited range of the length FPIs can also function as displacement sensors [30]. Suitable FPIs can also be used to accurately determine the refractive index of air [31].

The interferometers described above have beams with small diameter, typically a few millimetres, however, there is another class of interferometers, namely large field interferometers which are equipped with optical components to generate beam diameters of about 50 mm or larger. Besides collimating lenses, with a focal length of typically several hundred millimetres, comparatively large beam splitter and mirrors are necessary. Such interferometers are often combined with imaging optics, which make it possible to locate objects (e.g. the front of a gauge block) in relation to the pixel coordinates of a camera sensor. These large-field imaging interferometers can be designed as two-beam interferometer or multi-beam interferometer (Fizeau interferometer). These are suitable for the measurement of the length of bar shaped material measures, e.g. gauge blocks [32] wrung to a flat platen forming a step height. Although the basic physical principles of modern large field interferometers are unchanged for more than a century [33–36], today's interferometers benefit from highly monochromatic stabilized lasers, high dynamic range cameras, computer-controlled equipment including phase stepping and sophisticated software analysis (e.g. see [37]). Further, modern light sources can be stabilized to frequency standards by using optical frequency comb techniques (e.g. see [38]). The basic scheme of a two-beam large field interferometers is shown in figure 6. A point light source located in the focus of a collimator creates a large bundle of parallel rays that covers the prismatic body, such as a gauges block, as shown in figure 6 along with its end platen. In this way, the interference intensities for each partial beam of the bundle are laterally resolved ⁵ :

$$\begin{equation}\begin{gathered}\qquad {\Delta}\varphi \left(x,y\right)\\ I={I}_{0}\left\{1-\gamma \enspace \mathrm{cos}\left[\overbrace{\frac{2\pi }{\lambda /2}\left({z}_{2}\left(x,y\right)-{z}_{1}\left(x,y\right)\right)}\right]\right\},\end{gathered}\end{equation} \tag{ 17 }$$

where ${z}_{1/2}\left(x,y\right)$ represents the distribution of the geometric paths (1: reference, 2: measurement) perpendicular to the optical axis. Recall γ from equation (8). If the reflecting surfaces were completely flat, the path difference ${z}_{2}\left(x,y\right)-{z}_{1}\left(x,y\right)$ would describe a plane whose tilt depends on the alignment of the surfaces with one another.

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The cosine function in the equation is the main term responsible for the appearance of the fringes with maxima for path differences of mλ/2 and minima for $\left(m+\frac{1}{2}\right)\lambda /2$, where m is an integer. The prismatic body in the measuring arm of the interferometer represented in figure 6 causes the occurrence of two offset fringe systems. The value of this offset in relation to the fringe spacing corresponds to the length, l, of the artefact, but only the fractional interference order, q, is visible. Together with the integer interference order, i, which must be determined separately, the length of a prismatic body can then be obtained:

$$\begin{equation}\begin{gathered}l=\frac{{\lambda }_{0}}{2n}\frac{{\Delta}\varphi }{2\pi }=\frac{\lambda }{2}\cdot \left(i+q\right).\end{gathered}\end{equation} \tag{ 18 }$$

The sharpness of the interferograms is affected by the finite size of the aperture, which is positioned in the focal plane of the focussing lens at the output of the interferometer. This aperture serves to suppress disturbing secondary reflections. The application of digital processing techniques [39, 40] to phase shift interferometry has significantly improved measurement efficiency and accuracy. The basic concept of phase shift interferometry is that one can determine the phase modulo 2π over a beam diameter by acquiring multiple interferograms, each phase shifted by a certain amount, wherein the phase shift might be generated by variation of the interferometer's reference pathway. Considerable efforts have been made to develop effective and error-compensating phase extraction algorithms [41–47]. The algorithm proposed by Tang [47] is based, for example, on five intensities ${I}_{k}={I}_{0}\left\{1-\gamma \enspace \mathrm{cos}\left[{\Delta}\varphi +\left(k-3\right)\alpha \right]\right\}$, k = 1...5, which are recorded successively with equally spaced phase steps α, which leads to the phase difference:

$$\begin{equation}{\Delta}\varphi =\mathrm{arctan} \left( \frac{\sqrt{\left[2\left({I}_{4} - {I}_{2}\right) + \left({I}_{5} - {I}_{1}\right)\right]\left[2\left({I}_{4} - {I}_{2}\right) - \left({I}_{5} - {I}_{1}, \right)\right]}}{{I}_{1}+{I}_{5}-2{I}_{3}}\right) ,\end{equation} \tag{ 19 }$$

as illustrated in figure 7.

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For Fizeau interferences there exists a dedicated phase determination algorithm [48] which takes into account the non-sinusoidal intensity of fringes in Fizeau interferometers.

The installing of a permanently evacuated cell in the measuring arm of a large-field interferometer allows accurate determination of the refractive index of air for the specific wavelength of the light used. Here, the refractive index of air is determined from the difference of the optical path lengths in vacuum and air, along a common cell length [49]. If the collimated beam of a large-field interferometer is large enough to contain both a gauge block and a vacuum cell next to it, the effective refractive index of the air can be determined from a common phase topography. A rough estimate of the air refractive index will suffice [50].

When the integer interference order cannot be determined, e.g. in the measurement of long distances, the ambiguity range is limited to about a quarter of the wavelength. The range of unambiguity can be enlarged by applying more than a single light source, with a different wavelength in a length measurement by interferometry.

Considering the simplest case of two different light sources, for a given length l the respective phase differences can each be written as ${\Delta}{\varphi }_{i}=2\pi \cdot l/\left(\frac{1}{2}{\lambda }_{i}\right)$ (see equation (10)). Thus, the difference Δφ₂ − Δφ₁, which is called the synthetic phase Φ_synth, corresponds to

$$\begin{equation}{{\Phi}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}={\Delta}{\varphi }_{2}-{\Delta}{\varphi }_{1}=2\pi \frac{l}{{{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}/2},\end{equation} \tag{ 20 }$$

in which Λ_synth denotes the so-called synthetic wavelength (${{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}={\lambda }_{1}\cdot {\lambda }_{2}/\left({\lambda }_{1}-{\lambda }_{2}\right)$).

Consequently, the length can be rewritten as

$$\begin{equation}l=\frac{1}{2}{{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}\cdot \frac{{{\Phi}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}}{2\pi }.\end{equation} \tag{ 21 }$$

Compared to equation (10), the unambiguity range in a length measurement that is based on equation (21) is greatly enlarged due to the fact that the synthetic wavelength is much larger than a single wavelength. On the other hand, the accuracy of Φ_synth is limited by the accuracies of the individual phase differences Δφ. Therefore, the uncertainty of the length is generally upscaled be a factor of approximately λ/Δλ.

Application of such 'synthetic wavelength interferometry' in air benefits from a separate consideration of the air refractive index. Continuing the above example of two wavelengths, each written as ${\lambda }_{k}={\lambda }_{k}^{0}/{n}_{k}$, in which ${\lambda }_{k}^{0}$ denote so called vacuum wavelengths (${\lambda }_{k}^{0}=c/{f}_{k}$), the synthetic phase corresponds to

$$\begin{equation}{{\Phi}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}=4\pi \cdot l\cdot \frac{{\lambda }_{2}^{0}-{\lambda }_{1}^{0}}{{\lambda }_{1}^{0}\cdot {\lambda }_{2}^{0}}\cdot \left({n}_{1}-{\lambda }_{1}^{0}\cdot \frac{{n}_{2}-{n}_{1}}{{\lambda }_{2}^{0}-{\lambda }_{1}^{0}}\right).\end{equation} \tag{ 22 }$$

For small differences ${\Delta}{\lambda }^{0}={\lambda }_{2}^{0}-{\lambda }_{1}^{0}$ it can be easily demonstrated that the term ${n}_{1}-{\lambda }_{1}^{0}\cdot \left({n}_{2}-{n}_{1}\right)/\left({\lambda }_{2}^{0}-{\lambda }_{1}^{0}\right)$ is close ⁶ to the group refractive index n_g (see equation (4)) considered at the average vacuum wavelength ${\lambda }_{1}^{0}+\frac{1}{2}{\Delta}{\lambda }^{0}$. Accordingly, the length can then be rewritten as

$$\begin{equation}l=\frac{1}{2}\frac{{{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}^{0}}{{n}_{\mathrm{g}}}\cdot \frac{{{\Phi}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}}{2\pi },\end{equation} \tag{ 23 }$$

in which ${{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}^{0}$ denotes the synthetic vacuum wavelength (${{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}\mathrm{h}}^{0}={\lambda }_{1}^{0}\cdot {\lambda }_{2}^{0}/\left({\lambda }_{2}^{0}-{\lambda }_{1}^{0}\right)$). Such two-colour synthetic wavelength interferometer is suitable for absolute measurement of even very large distances (e.g. see [51, 52] and references therein).

While measurement of the synthetic phase and consideration of the synthetic wavelength can provide appropriate estimates in a length measurement [53], the accuracy of the length according to equation (23) is typically larger than a quarter of a wavelength.

The exact total number of integer interference orders can by determined by the so-called 'method of exact fractions' which was first suggested by Benoit [54]. Here, the phase difference, $\frac{1}{2\pi }{\Delta}\varphi$ representing the total interference order, is split into an integer part, i, and a fractional part q, i.e. $l=\frac{1}{2}\lambda \cdot \left(i+q\right)$. When different wavelengths, λ_k , are available for each wavelength the length is expressed as:

$$\begin{equation}{l}_{k}=\left({i}_{k}+{\delta }_{k}+{q}_{k}\right){\lambda }_{k}/2,\end{equation} \tag{ 24 }$$

in which i_k represent estimate integer orders, calculated from an estimate of the length, l_est (${i}_{k}=\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\left[{l}_{\mathrm{e}\mathrm{s}\mathrm{t}}/\left(\frac{1}{2}{\lambda }_{k}\right)\right]$, where 'round[]' represents the operator for rounding to the nearest integer) and δ_k are variation integers. For a set of N different wavelengths the mean length, $\overline{l}$, and the average deviation from the mean length, Δ, can be considered as described in more detail in [50]:

$$\begin{equation}\overline{l}=\frac{1}{N}\sum\limits _{k=1}^{N}{l}_{k},\qquad {\Delta}=\frac{1}{N}\sum\limits _{k=1}^{N}\left\vert \overline{l}-{l}_{k}\right\vert .\end{equation} \tag{ 25 }$$

The resulting set $\left\{\overline{l},{\Delta}\right\}$ can be displayed as scatterplot which represents a coincidence pattern. Considering the case of two wavelengths, the value of Δ is exactly given by $\frac{1}{2}\left\vert {l}_{1}-{l}_{2}\right\vert$. Assuming that i₁ and i₂ correspond to the correct integer orders (δ₁ = 0, δ₂ = 0), the two lengths, written as ${l}_{1}=\left({i}_{1}+{q}_{1}\right){\lambda }_{1}/2$ and ${l}_{2}=\left({i}_{2}+{q}_{2}\right){\lambda }_{2}/2$, coincide (Δ = 0) when q₁ and q₂ are given 'exactly'. A relation for further variation integers (δ₁ ≠ 0, δ₂ ≠ 0) for which coincidence is obtained can be extracted by setting ${\Delta}=\frac{1}{2}\left\vert {~{l}}_{1}-{~{l}}_{2}\right\vert =\frac{1}{2}\left\vert \left({i}_{1}+{\delta }_{1}+{q}_{1}\right){\lambda }_{1}/2-\left({i}_{2}+{\delta }_{2}+{q}_{2}\right){\lambda }_{2}/2\right\vert =0$, in which ${~{l}}_{ 1}$ and ${~{l}}_{ 2}$ correspond to blunder lengths. This situation is expressed with equation (26):

$$\begin{equation}{\Delta}=0=\left\vert {\lambda }_{1}{\delta }_{1}-{\lambda }_{2}{\delta }_{2}\right\vert {\Rightarrow}{\Delta}=0\enspace \mathrm{f}\mathrm{o}\mathrm{r}\enspace \frac{{\delta }_{2}}{{\delta }_{1}}={\pm}\frac{{\lambda }_{1}}{{\lambda }_{2}}.\end{equation} \tag{ 26 }$$

Therefore, integer interference orders could be calculated incorrectly when their ratio is equal to the ratio of the wavelengths used in the measurements, i.e. a rational number. In particular, the issue occurs when the first wavelength is given by an integer multiple of the second wavelength, i.e. λ₁ = aλ₂ (a is an integer), coincidence (Δ = 0) is obtained when δ₂ = aδ₁.

Assuming the two fictive wavelengths 532.3 nm and 548 nm, the example coincidence pattern shown in figure 8 demonstrates that at the length deviation $\overline{l}-{l}_{\mathrm{e}\mathrm{s}\mathrm{t}}=\delta l$ there exists a minimum of Δ that is exactly zero (marked by the dotted circle). Further local minima exist which are separated by the amount of half the synthetic wavelength, ${{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}}={\lambda }_{1}{\lambda }_{2}/\left\vert {\lambda }_{2}-{\lambda }_{1}\right\vert$, i.e. at positions $\overline{l}-{l}_{\mathrm{e}\mathrm{s}\mathrm{t}}=\delta l+m\cdot {{\Lambda}}_{\mathrm{s}\mathrm{y}\mathrm{n}\mathrm{t}}/2$ in which m is an integer. However, it should be noted that the coincidence pattern (grouped values of Δ belonging to a certain number of m) is always different. The next neighboured minima for which Δ is nearly zero can be found at specific multiples of Λ_synt/2. In the example of figure 8 these minima are localized at m = ±3. This number is strongly dependent on the wavelengths λ₁ and λ₂. The amount of m could be calculated in the same way as the 'range multiplier' suggested by de Groot (see equation (14) of [53]). The lengths which are closest to the correct minimum $\overline{l}-{l}_{\mathrm{e}\mathrm{s}\mathrm{t}}=\delta l$ (Δ = 0) are separate by a single integer interference order, i.e. at $\overline{l}-{l}_{\mathrm{e}\mathrm{s}\mathrm{t}}=\delta l{\pm}\left({\lambda }_{1}+{\lambda }_{2}\right)/2$. The corresponding values of Δ amount to $\left\vert {\lambda }_{1}-{\lambda }_{2}\right\vert /4$. This reveals a very important requirement for the applicability of the method of exact fractions: the wavelengths must be clearly separate. Otherwise, the value of Δ for the neighbouring lengths becomes too small. When, e.g., λ₁ and λ₂ would be just 2 nm apart from each other, for $\left\{{\delta }_{1}{\pm}1,{\delta }_{2}{\pm}1\right\}$ the values of Δ would result in 0.5 nm which is very close to zero (Δ = 0 is obtained for the correct variation numbers $\left\{{\delta }_{1},{\delta }_{2}\right\}$). It is therefore important to note that in such case, despite the small values of Δ, the corresponding lengths are apart by relatively large amounts, namely ${\pm}\left({\lambda }_{1}+{\lambda }_{2}\right)/2$. In other words, the ambiguity range critically depends on the uncertainty in the measurement of the fractional interference orders q_k , with the remark that uncertainty contributions to the measured length, which are the same for all wavelengths used, have no influence on this 'method of exact fraction'.

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As mentioned in section 1, for measurement in air, the refractive index is significant and must be measured directly (e.g. through comparison of equal optical path lengths in air and in vacuum in a refractometer) or determined from measurement of the parameters given in table 1 through use of empirical equations.

If one considers the typical daily changes in the parameters in table 1, the sensitivity coefficients allow calculation of the commensurate changes in refractive index; by far the largest variations come from pressure and temperature, leading to refractive index changes of several parts in 10⁶.

There are several equations which can be used to calculate refractive index [49, 57–60] which all separate the effects of dispersion (wavelength-dependency, K_λ ) from the density-dependent terms (D_tp )

$$\begin{equation}{\left(n-1\right)}_{tp}={K}_{\lambda }\cdot {D}_{tp}.\end{equation} \tag{ 27 }$$

The density term is modified for the effects of water vapour (humidity) and for the effects of increased CO₂ in the atmosphere. The empirical equations claim standard uncertainties of the order of 1 × 10⁻⁸ for readings from perfect sensors, however sensor errors (calibration, drift) and inhomogeneity of the air between the sensor location(s) and the air path to be measured by interferometry, lead to overall uncertainties of the order of several parts in 10⁸. Achievement of this level of accuracy requires metrology-class sensors costing around 15 thousand euro or more. Care must be taken to ensure all optical paths in air are suitably compensated to avoid deadpath errors (e.g. when the zero position of the measurement arm does not coincide with a zero length from the beamsplitter).

An angular misalignment between the measurand length and the light path length in the measuring arm of an interferometer contributes an error directly proportional to the cosine of the angle, θ, between the two lengths. For small misalignments, using the small angle approximation, the fractional error is approximately θ²/2, e.g. for θ = 1 second of arc misalignment, the fractional error, given by δl/l = d ⋅ cos θ, is 1.2 × 10⁻¹¹. Minimisation of cosine error requires careful alignment of the interferometer with respect to the measurand.

When the measured length and the light path length are parallel, but laterally offset from one another, angular error, α, in the motion of the moving reflector in the measurement arm causes a length measurement error proportional to both the tangent of the angle, and the lateral offset, d, between the two lengths, e.g. for α = 5 s of arc and d = 100 mm, the error, given by δl = d ⋅ tan α, is 2.4 µm. Note that this error is length independent—it depends on the offset, d, and the motion error, α, not on l. Abbe error can be minimised by either reducing rotational errors caused by imperfect motion axes, or by minimising the lateral offset between the measurand length and the light path.

Interferometers with collimated beams are subject to obliquity errors due to the finite (non-zero) size of the source. Some forms of interferometer require the source aperture to be offset from the focal point of the collimator lens (i.e. not on the principal axis). In such cases, the off-axis source causes a small angular deviation of the measurement beam with respect to the measurand axis, similar to a cosine error; for a negligible sized source, positioned a distance s off-axis with a collimator lens of focal length f, the error scales as δl/l = s²/2f² [61]. For an interferometer with the source perfectly on axis, if the source size is finite, then elements of the source will still be off-axis and the resulting effect may be calculated by integrating the effect of all such infinitesimally small elements each contributing a cosine error. The result shows that the length error, for a source of diameter d and a collimator lens of focal length f, is given by $\delta l/l={\left(d/4f\right)}^{2}$ [61], e.g. for d = 1 mm, f = 1000 mm, the relative error is 6.25 × 10⁻⁸.

For wide field interferometers optical aberrations, especially those unique to either the reference or measurement arm, lead to topographic errors in measured surfaces, with a one-to-one correspondence, as a minimum (some errors will accumulate during propagation). Typical commercial optics can achieve around λ/20 to λ/40 wavefront quality, leading to 15 nm to 30 nm measurement error across the aperture-limited surface. Most of this so-called optical error can be corrected by using an almost ideally flat surface (e.g. a flatness standard). The remaining optical error after correction is then of the order of magnitude of the flatness deviation of the standard.

Even for 'perfect' optical components, the physical wavefronts are not perfectly planar due to diffraction and the distance travelled by a wavefront during one period of the oscillation of the electromagnetic wave differs from that of an ideal plane wave and will also depend on location. A typical exemplar of this is the well-known TEM₀₀ mode emitted by He–Ne lasers, which is Gaussian. For well-designed interferometers operating near the diffraction limit, the effect can be shown to be of the order of 1 or 2 nm [62, 63].

Both homodyne and heterodyne interferometers often use polarisation to separate the light for the two arms of the interferometer. The separated polarisation states are often achieved using polarisation-dependent optics. Polarisation jitter and imperfect extinction during splitting and recombination appear as phase errors, as do any extraneous rotations of the polarisation state during path length traversal, leading to length measurement errors [16–19], typically of the order of a few nanometres.

All interferometers, and especially those utilising polarization optics for beam steering and phase measurement are prone to phase errors caused by parasitic beams which cause optical fringe interpolation to become non-linear, leading to errors of a few nanometres. For some effects, compensation is possible through sampling across several fringe periods [13] or improving the design of interferometer, but for the most demanding applications, where multiple parasitic beams produce a spectrum of phase non-linearities, detailed modelling is required to achieve the necessary compensation [64].

Apart from Fizeau designs of interferometer, in which the reference path is of zero length, all interferometers have a finite length of the reference path (usually the separation between the beam splitter and the reference mirror). Erroneous changes in the reference path length directly contribute to length measurement error. Variation of the reference path length can be caused by e.g. vibrations or thermal changes in the mounting mechanism, e.g. consider a 1 m mechanical arm made of steel (CTE 10.7 × 10⁻⁶ K⁻¹), a 0.1 °C change in temperature would change the arm length by 1.07 µm, leading to a length error of the same value if the change occurred during the time the measurement arm was being monitored.

When using AC detection, the detector itself can influence the phase measurement, e.g. typical detectors contain a small active area with some focussing optics in which any local inhomogeneity will couple with beam wandering (due to turbulence or surface form errors in optics) to cause phase errors. Other effects such as amplitude to phase coupling may be present depending on the type of sensor. It is possible that errors up to 1/2 fringe may be encountered.

For length measuring interferometers where a physical motion is monitored using a laser beam (fringe counting) parasitic motions of the moving part such as roll, pitch, and even yaw may cause errors in the reflected beam which returns to the detector. If plane mirrors are used to reflect the beam, roll and pitch cause angular deflections in the returned beam, which can cause sine errors if large aperture detectors are used. Even with cube corner retro reflectors, roll, pitch, and yaw will contribute errors which can be as large as 0.5 µm for parasitic rotations of 0.36° [65].

An uncalibrated, unstabilized 633 nm He–Ne laser can be assumed to have a vacuum wavelength λ = 632.9908 nm with a relative standard uncertainty of 1.5 × 10⁻⁶, according to [66]. If this is insufficient, stabilized lasers are available commercially which can achieve frequency stability of around 10⁻⁹; frequency stabilized lasers can be calibrated by beat frequency against a reference laser or optical frequency comb with an uncertainty of a few parts in 10¹¹, thus for the majority of the length scale, the instability or calibration error of the light source is usually the least significant error source.

It is possible for a 633 nm He–Ne laser to have a small secondary mode lasing on the He–Ne 3s² → 2p² transition at 640 nm, which might cause a non-linearity in the fringe interpolation, so it may be desirable to check for the presence of such modes [66]. About 50 years ago, an NMI reported that a laser purporting to operate at 633 nm was found to actually operate almost exclusively on the 640 nm transition. Rather than a non-linearity, this extreme case causes a systematic length-proportional error of 1.1% due to the incorrect value of the wavelength. We are not aware of more recent occurrences of this extreme problem but there is still a small possibility of a small amount of 640 nm light contaminating the 633 nm output.

A certain amount of 'parasitic light', which comes from an undesired mode of the resonator of a laser, can even lead to a relative error of a few parts in 10⁷ when measuring a length, although the light frequency of the main mode was calibrated to a relative uncertainty of 10⁻¹². As shown in [67] for a single parasitic light mode, the maximum effect on a length measurement is given approximately by 0.16/η fractional interference orders, where η is the ratio between the intensity of the main light mode and that of the parasitic light mode. It therefore appears desirable to have information about the amount of parasitic light. An additional error source in stabilized lasers comes from parasitic reflections inside the laser system (often from the optics used in the stabilization scheme) [68]. The phase of such reflections varies with the optical path length and wavelength of the laser; for lasers employing modulation-based stabilization schemes, such as the less-often used Lamb dip (e.g. 32 kHz modulation to 16 MHz optical frequency depth), the phase difference modulates, giving rise to an intensity modulation at the same frequency as the modulation signal. This can bias the stabilization circuit, especially during warm-up. In two-mode stabilized and transverse Zeeman stabilized lasers which operate with two orthogonally polarized modes, optical feedback can cause changes in the quality factor of the optical cavity, causing reduction in the frequency stability performance [68].

Reflection and transmission of light can be described by the well-known Fresnel equations. If a light wave, written in complex notation as $E=\mathrm{R}\mathrm{e}\left[A\cdot {\text{e}}^{\text{i}\cdot \left(k\cdot z-\omega \cdot t\right)}\right]$, is reflected at perpendicular incidence at the surface of a material, the ratio between reflected (E_r) and incident light wave (E) is generally given by:

$$\begin{equation}{E}_{\mathrm{r}}/E=\left({\hat{n}}_{1}-{\hat{n}}_{2}\right)/\left({\hat{n}}_{1}+{\hat{n}}_{2}\right),\end{equation} \tag{ 28 }$$

in which ${\hat{n}}_{1}$ is the complex refractive index of the medium in which the wave propagates and ${\hat{n}}_{2}$ is the complex refractive index of the material from which the light is reflected. Assuming that ${\hat{n}}_{1}$ is a real number n₁ (such as for air, glass or vacuum and other dielectric materials), this ratio results to: ${E}_{\mathrm{r}}/E=\left({n}_{\mathrm{1}}-\left({n}_{2}+\mathrm{i}\cdot {\kappa }_{2}\right)\right)/\left({n}_{\mathrm{1}}+\left({n}_{2}+\mathrm{i}\cdot {\kappa }_{2}\right)\right)$ and can be written as ${E}_{\mathrm{r}}/E=\left\vert {E}_{\mathrm{r}}/E\right\vert \cdot {\text{e}}^{\text{i}\cdot \delta {\varphi }_{\mathrm{P}}}$ where δφ_P describes the phase change on reflection ⁷ :

$$\begin{equation}\delta {\varphi }_{\mathrm{P}}=\mathrm{arg}\left[{E}_{\mathrm{r}}/E\right]=a\enspace \mathrm{t}\mathrm{a}\mathrm{n}\enspace 2\left[-2\cdot {n}_{1}\cdot {\kappa }_{2},{{n}_{1}}^{2}-{{n}_{2}}^{2}-{{\kappa }_{2}}^{2}\right].\end{equation} \tag{ 29 }$$

While for κ₂ = 0 and n₂ > n₁ equation (29) returns exactly π radians, which is the known phase change on reflection for dielectric materials, increased values of κ₂ lead to an increased phase change for non-dielectric materials, e.g. metals (see [69] and references therein). This effect can be interpreted as the virtual penetration of the light into the material and must be generally taken into consideration in the measurement of the (geometrical) length.

When measuring step heights in a wide field interferometer, e.g. the measurement of a gauge block central length when wrung to a platen, any difference in the complex refractive index between the upper and lower surfaces will cause an erroneous difference in the phase of the reflected wavefronts leading to a fringe shift and a length measurement error. Measurements of n₂ and κ₂ for a number of steel gauge blocks using ellipsometry [70] indicated a 20% variation in κ₂, resulting in a 5° variation of δφ_P, corresponding to a variation in measured length of ∼5 nm, for λ ≅ 633 nm [71]. Typical values include ∼20 nm difference between e.g. steel and glass.

Semi-transparent surface layers on the material cause multiple reflections that lead to an additional phase change. This must be taken into account in particular if the desired measurement uncertainty is in the sub-nm range. A prominent example is the volume determination of highly enriched ²⁸Si spheres [72], where it is important to collect information about the composition and thickness of several surface layers and to calculate the influence of these layers on the measurement. Another example is the correction for the phase change occurring on dielectric mirrors of a Fabry–Perot cavity refractometer [73].