Focus Issue on Length Metrology

This Focus Issue of Metrologia was instigated by the Consultative Committee for Length's Working Group on Strategic Planning when it met online in 2020 during the COVID-19 pandemic. Submission of articles started closely thereafter and was closed in September 2022. Covering a wide range of topics, the issue shows that despite the Coronavirus pandemic disrupting laboratory work, length metrology researchers have continued to deliver cutting edge research. The revised definition of the metre and its Mise en Pratique, both published in 2019, have stimulated further research and opened additional opportunities for developing length metrology capabilities. Articles related to the new Mise en Pratique are included in the issue, together with a range of articles demonstrating the breadth and ingenuity of current leading edge research in length metrology. A final paper indicates how metrology, with length used as an example, may be enhanced by a transition to a digital framework for realising the SI.

The angular performance of a nano-angle generator (NANGO) developed at Diamond Light Source has been characterised using a dual-beam laser interferometer designed and built at the National Physical Laboratory (NPL). NANGO is a flexure-based, piezo actuated device which generates milli- to sub-nano-radian angles for the calibration of metrology instruments used to test the quality of synchrotron x-ray mirrors and angular nano-positioning stages at Diamond. The NPL interferometer provides traceability for small angle measurements made by NANGO. An uncertainty budget has been developed for measurements over a 50 nanoradian range. In closed-loop, using feedback from the NANGO's angle encoder, for the first time we show that 1 nanoradian steps made by NANGO are measurable by an external metrology device. The 200 kHz acquisition rate of the NPL angle interferometer also reveals new dynamic information about NANGO's angular motion. The NPL interferometer demonstrates that NANGO in open-loop can make: distinct steps of 500 picoradians; sinusoidal oscillations at 0.4 Hz with an amplitude of 125 picoradians; or 1 nanoradian oscillations at 40 Hz. Traceability to the SI though National Metrology Institute instrumentation for NANGO will provide enhanced accuracy for a wide range of angle metrology applications at Diamond, including inputs to deterministic polishing techniques for the creation of next-generation x-ray mirrors and dynamic characterisation of nano-positioning stages.

The National Metrology Institute of Japan (NMIJ) has developed a highly accurate two-point diameter measurement method of a sphere using a micro-coordinate measuring machine (µ-CMM) with a low-force two-dimensional touch probe. The measured sphere is used as a reference to calibrate the probe of other µ-CMM in industry. Our strategy for measuring sphere's diameter is that the probe radius of NMIJ's µ-CMM is calibrated using a reference gauge block, and the NMIJ's µ-CMM using the calibrated probe measures the mean two-point diameter of a sphere. To achieve the highly accurate calibration of the reference gauge block, we have adopted a double-sided interferometer (DSI) and an atomic force microscopy (AFM). The DSI measures the length of gauge block without wringing. The phase correction value of the gauge block is evaluated by measuring the surface roughness using the AFM. Using the DSI and AFM, the mechanical length of reference gauge block was obtained with the expanded uncertainty of 13 nm (k = 2). Then, the probe radius of NMIJ's µ-CMM was also calibrated using the reference gauge block calibrated by the DSI and the AFM. The µ-CMM's probe radius was obtained with the expanded uncertainty of 10 nm (k = 2). Finally, the µ-CMM measured the mean two-point diameter of a sphere with the expanded uncertainty 21 nm (k = 2) was demonstrated.

A comparison of 11 steel gauge blocks (8 short and 3 long) calibrated by absolute interferometry is presented. Thus, the results of the individual measurements of five designated signatories (three from INTI and two from CENAM) were analyzed in order to evaluate the impact of possible correlations and systematic effects in the reproducibility of the measurements. All measurements were carried out in the framework of a bilateral key comparison between CENAM and INTI and following their own calibration procedures. The measurement analysis was performed by using inclusive and exclusive statistics. The agreement between all measurements is satisfactory and no detectable systematic effects were observed. Average offsets were also calculated for each operator, verifying that systematic effects did not affect the measurements of the present exercise. Even though the approaches deployed to analyze the results are well-established, this work provides a detailed example of analysis in the field of gauge block metrology. This exercise has also allowed to validate the internal criteria considered by each institute for selecting final results.

A current focus of the international metrology community is the digitalisation of documents, certificates and services in response to initiatives underway throughout industry and to the requirement to follow the principles of data being Findable, Accessible, Interoperable, and Reusable. We propose the key elements of a digital framework for the SI metre, at the point of realisation, showing how it may be implemented in practice. We give examples of direct benefits of this approach, which may be extended to other SI units.

X-ray computed tomography (XCT) is increasingly used for dimensional metrology, where it can offer accurate measurements of internal features that are not accessible with other techniques. However, XCT scanning can be relatively slow, which often prevents routine uptake for many applications. This paper explores the feasibility of improving the speed of XCT measurements while maintaining the quality of the dimensional measurements derived from reconstructed volumes. In particular, we compare two approaches to fast XCT acquisition, the use of fewer XCT projections as well as the use of shortened x-ray exposure times for each projection. The study shows that the additional Poisson noise produced by reducing the exposure for each projection has significantly less impact on dimensional measurements compared to the artefacts associated with strategies that take fewer projection images, leading to about half the measurement error variability. Advanced reconstruction algorithms such as the conjugate gradient least squares method or total variation constrained approaches, are shown to allow further improvements in measurement speed, though this can come at the cost of increased measurement bias (e.g. 2.8% increase in relative error in one example) and variance (e.g. 25% in the same example).

Spectral interferometry is becoming a popular method of performing dimensional measurements, e.g. of surfaces, but such devices require pre-calibration of their spectrometers to achieve traceability. The use of a spectral lamp and imaging spectrometer to perform inherently traceable distance measurements using spectral interferometry, without the need for external calibration, is proposed and its experimental feasibility demonstrated. Initial experiments show measurements over a working range of 200 µm. The estimated standard uncertainty of the distance measurements is 0.6 nm, corresponding to an expanded uncertainty of 1.2 nm at k = 2.

Autocollimators are versatile devices for angle metrology used in a wide range of applications in engineering and manufacturing. A modern electronic autocollimator generally features two measuring axes and can thus fully determine the surface normal of an optical surface relative to it in space. Until recently, however, the calibration capabilities of the National Metrology Institutes were limited to plane angles. Although it was possible to calibrate both measuring axes independently of each other, it was not feasible to determine their crosstalk if angular deflections were present in both axes simultaneously. To expand autocollimator calibrations from plane angles to spatial angles, PTB and VTT MIKES have created dedicated calibration devices which are based on different measurement principles and accomplish the task of metrological traceability in different ways. Comparing calibrations of a transfer standard makes it possible to detect systematic measurement errors of the two devices and to evaluate the validity of their uncertainty budgets. The uncertainty levels of the devices are comparable to each other, with an expanded uncertainty U = 0.014 arcsecond (95.5% coverage probability) over a measuring range of ±1000 arcsecond in the case of PTB and U = 0.015 arcsecond over a range of ±500 arcsecond and U = 0.020 arcsecond over ±1000 arcsecond in the case of VTT MIKES. Over a range of ±1000 arcsecond with regards to both measuring axis of an Elcomat 3000 autocollimator as a transfer standard, no statistically significant differences between the calibrations were detected. The results thus confirm the calibration capabilities of PTB and VTT MIKES as stated in the calibration and measuring capability database of Bureau International des Poids et Mesures.

The realisation of the metre definition is still today mostly performed using iodine frequency-stabilized HeNe lasers. Unfortunately, they deliver little optical power. Instead of migrating to other expensive laser technologies, the use of an injection locked semiconductor laser is proposed in order to boost the optical output of the legacy HeNe laser primary reference. The characterisation of the system and its validation through a comparison measurement is reported whereby no impact on the frequency and the stability of the HeNe reference laser could be observed.

In this study, an optical system for simultaneous measurement of physical thickness, group refractive index, bow, and warp of a large silicon wafer is first proposed based on a reflection-type spectral-domain interferometer. Such key parameters are determined by combining four different optical path differences (OPDs) measured at each sampling point throughout two-axis sample scanning within area of 250 mm by 250 mm. To overcome the measurement limitations by the deflection of a free, unclamped large-sized wafer, two OPDs representing the surface profiles of both sides are utilized to facilitate the thickness and refractive index measurements insensitive to sample inclination. For verification of the proposed method, a 300 mm diameter silicon wafer with nominal thickness of 775 μm was used as a test sample. For measuring the bow and warp with gravity effect compensation, a silicon wafer was measured once again after turning over. Through theoretical analysis on the changes of OPDs with the wafer tilted based on the measured surface profiles, it was verified that the effect of wafer bending on thickness and refractive index measurements can be ignored. The measurement uncertainties (k = 1) of physical thickness, group refractive index, bow, and warp were evaluated to be approximately 0.692 μm, 0.003, 0.416 μm, and 0.589 μm, respectively.

The 2019 update to the Mise en Pratique for the metre adopted the lattice parameter of silicon as a secondary realisation of the metre for dimensional nanometrology. One route for this realisation is the use of amphitheatre like monoatomic steps of silicon. In response, in this paper we present new algorithms for one- and two-dimensional analysis of atomic force microscope images of these large area atomic terraces on the surface of silicon. These algorithms can be used to determine the spacing between the steps and identify errors in AFM scanning systems. Since the vertical separation of the steps is of the same order of magnitude as many errors associated with AFMs great care is needed in processing AFM measurements of the steps. However, using the algorithms presented in this paper, corrections may be made for AFM scanner bow and waviness as well as taking into account the edge effects on the silicon steps. Applicability of the data processing methods is demonstrated on data sets obtained from various instruments. Aspects of steps arrangement on surface and its impact on uncertainties are discussed as well.

We revisit the problem of choosing combinations of sources for multi-wavelength interferometry, using the risk of assigning the wrong integer interference order to a measured phase as a quantitative criterion. We model this risk, explicitly accounting for the experimenter's ability to reject implausible measurement results. We present a general reliability limit for the method of exact fractions, which can be used as a benchmark when evaluating wavelength combinations, as well as performance estimates for several promising combinations.

The revised International System of Units (SI) came into force on May 20, 2019. Simultaneously, updated versions of supporting documents for the practical realization of the SI base units (mises en pratique) were published. This review paper provides an overview of the updated mise en pratique for the SI base unit of length, the metre, that now gives practical guidance on realisation of traceable length metrology spanning 24 orders of magnitude. The review begins by showing how the metre may be primarily realized through time of flight and interferometric techniques using a variety of types of interferometer. Examples of techniques for measuring the interferometric phase and coping when the integer interference order is unknown are then described, together with examples of typical uncertainty contributions that may be encountered. The requirements for traceable nanoscale metrology and the need for an alternative secondary metre as identified by the Consultative Committee for Length's Working Group on Nanometrology are outlined. These led to the inclusion in the mise en pratique of secondary realisations of the length unit at the nanometre and sub nanometre scale, based on the lattice spacing of silicon. Three measurement techniques using this secondary realisation are then described in detail. The paper concludes by emphasising that measurements made today over 24 order of magnitude are still compatible with measurements made using the metre as adopted over 200 years ago.

Mutual calibration by a combination of a zero-offset method and a length-unit traceable method has been suggested as a promising approach to determine the traceable thickness of ultra-thin oxide films. However, the measurement uncertainty is somewhat complicated to calculate because the standard uncertainties from the two measurement methods and the linear regression process should be combined. In this study, the mutual calibration method to evaluate the film thickness and uncertainty of ultra-thin oxide films was investigated. The algorithm of the linear regression equation in the mutual calibration method was studied and the uncertainty calculation program for the thickness measurement by mutual calibration was developed. The result will provide an effective and useful guideline to certify the thickness of the ultra-thin oxide film on Si(100) substrate which is used as the gate oxide in the semiconductor devices. The magnitude of the relative expanded uncertainty in the thickness measurement by mutual calibration is in the range from 8.6% to 9.3%.

The measurement of the angle between an interferometer's front mirror and the diffracting planes is a critical aspect of the measurement of Si lattice parameters by combined x-ray and optical interferometry. In addition to being measured offline by x-ray diffraction, it was checked online by moving the analyser crystal transversely and observing the phase shift of the interference fringe. We describe the measurement procedure and give the miscut angle of the ²⁸Si crystal, whose lattice parameter was an essential input datum for the determination of the Avogadro constant in the past and which is now used in the definition of the kilogram, based on counting atoms. These data will benefit others that might wish to repeat the measurement of the lattice parameters of this unique crystal.

The Length Standards Group at the National Metrology Institute of Japan has developed a double-sided interferometer (DSI) for non-contact thickness measurement, that is traceable to SI units. The measurement uncertainty of DSI was evaluated. The results of gauge block measurements were compared with those of a conventional gauge block interferometer and their equivalency within the claimed uncertainty was confirmed. The advantage of applying a DSI in measuring the thickness of silicon wafers is that it has absolute measurement traceable to SI units with high precision and independence from the refractive index of the sample material. The expanded uncertainty of silicon wafer thickness measurement was evaluated as 20 nm (the coverage factor k = 2). A bilateral comparison on silicon wafers' thicknesses was conducted with Korea Research Institute of Standards and Science, using a different measurement technique from the spectral-domain interferometer. Results of 100 μm, 300 μm, 600 μm, and 950 μm-thick silicon wafers agreed well with absolute values of E_n-numbers of 0.15 or less.

High-accuracy length measurements of prismatic bodies (e.g. gauge blocks) are usually performed by means of single-ended interferometers. To perform these measurements, the gauge block must be wrung to a reference plate. The quality of contact affects the measured length and also the wringing process wears down or damages the measuring faces. Furthermore, it limits the use of such interferometers to bodies that are suitable for wringing. PTB's double-ended interferometer allows high-accuracy length measurements that are traceable to the International System of Units to be performed without a reference plate. However, because the setup of this interferometer is complex and additional optical components are required the alignment process is challenging. Compliance with the defined gauge-block length in ISO 3650 is also challenging, especially for non-perfect shaped gauge blocks. In this work, we develop a precise alignment method for the double-ended interferometer and systematically study the contributions of misalignments to the uncertainty of the measured length. In order to explore the accuracy of the developed procedure and to estimate the size of the uncertainty caused by deviations from perfect gauge block shapes, virtual experiments are carried out using the PTB library SimOptDevice. The virtual experiment is validated by a comparison to experimental data. In addition, theoretical relations are confirmed. Finally, Monte Carlo runs of the virtual experiment are performed to quantitatively explore the size of different sources of uncertainty on the developed alignment method. The results suggest that the developed alignment method is highly accurate and is expected to yield an uncertainty contribution to the final length measurement in the sub-nanometer range.