We investigate the bias due to the beam divergence of the collimator in a free-fall absolute gravimeter of type FG5X. First, we measure the beam parameters with a Shack-Hartmann sensor. Then, we use the parameters to simulate the relative gravitational acceleration error of an FG5X gravimeter, which employs an unbalanced Mach-Zehnder laser interferometer. This investigation we do with four different commercial collimators, providing different divergence angles. We compare the results to real gravity measurements using the same collimators. The larger the divergence angle, and the bigger the relative length error, the bigger is the bias in the gravity measurements. A good agreement between theory and experiment is found, resulting in a relative bias of - 2.77 ( 24 ) & sdot; 10 - 9 ( - 2.72 ( 24 ) mu Gal) for our standard collimator of type Thorlabs TC25APC, which is usually used for free-fall acceleration determinations. The outcome is also important for the realization of the SI unit kilogram via Kibble balance experiments that, on one side, employ laser interferometers for velocity measurements, and, on the other side, require accurate values of the gravitational acceleration. For example, if this divergence error is not corrected in the Kibble balance, then the mass determination would be biased by 2.77 ( 24 ) mu g kg-1 (numbers are valid only for our gravimeter with our collimator and fiber).

The feasibility of a new design and measurement concept of a combined optical and x-ray interferometer (COXI) is demonstrated here. Mechanical noise in the x-ray interferometer can be cancelled efficiently with fast and synchronous detection of the optical interferometer phase and x-ray counts. The scan-length requirement of the decadic step method of conventional COXI setups is reduced by more than one order of magnitude due to the correlation between single-photon interference events. The actual scan length of the shown setup is limited to 4 µm. It is estimated that a future update will enable us to measure the lattice parameter of 28Si, at least with the targeted relative statistical reproducibility of 3 × 10−9.

This paper describes how the angular spectrum of a laser beam is determined, in order to estimate the relevant correction in optical interferometers as applied to length measurements. Tests and numerical simulations carried out to assess the measurement procedure delivered clues of deviations from a free-space propagation due to scattering centers in the beam path. Based on the simulation results, additional measurements are ongoing to confirm this evidence.

This paper describes the measurement of the silicon lattice parameter in the SI-unit of length by combined optical and x-ray scanning interferometry. The detection of single x-ray photon quantum beats allows the continuous data acquisition of the x-ray interference fringes at increased scanning speeds compared to traditional experimental setups. MC-simulations show, that this scheme can reduce the scanning length by one order of magnitude while maintaining the same level of uncertainty. A high resolution optical interferometer with a continuous phase retrieval at a clock rate of minimum 10kHz is required for this scanning method.

Dimensional measurements by laser interferometry require a correction because of diffraction, which makes the fringe period different from the wavelength of a plane wave. The fractional correction—from parts in 10−7 to parts in 10−9, depending on the beam collimation—is half the central second moment of the angular power-spectrum of the beam, a generalization of the divergence concept. We report new insights into the second moment measurement and their consequences on the measurement of the silicon lattice parameter by combined x-ray and optical interferometry.

We report and correct errors in our recently published paper [J. Opt. Soc. Am. A32, 1403 (2015)10.1364/JOSAA.32.001403JOAOD61084-7529].

We report and correct errors in our recently published paper [J. Opt. Soc. Am. A 32, 1403 (2015)]. (C) 2016 Optical Society of America

The propagation of coherent laser light in optical systems is simulated by the vectorial ray-based diffraction integral (VRBDI) method which utilizes vectorial diffraction theory, ray aiming, differential ray tracing and matrix optics. On a global scale the method is not restricted to the paraxial approximation, whereas it is properly used for a local representation of the wavefront close to an aimed detection location. First, the field of a monochromatic continuous wave on an input plane is decomposed into spherical or plane wave components. Then, these components are represented by aimed ray tubes and traced through an optical system. Finally, the contributions are added coherently on an output plane whose position has to be chosen according to ray-aiming requirements. Provided that the apertures in the optical system are large with respect to the wavelength the results are fairly accurate.

Small angle deflectometry is commonly used to measure the topography of optical flats, mirrors or synchrotron optics with uncertainties down to the sub-nanometre range. Most of these deflectometric profilometers apply the slope measurement technique using commercially available autocollimators. These autocollimators can be calibrated, for example, at PTB and are then capable of measuring with uncertainties in the range of 0.01 arcsec. The lateral resolution of the measured topography is determined by the aperture of the autocollimator which is in the millimetre range for commercially available autocollimators.

X-Ray-induced absorption changes in Cu-doped lithium niobate crystals (LiNbO3) are investigated. All induced photochromic effects are fully reversible, i.e., illumination with white light annihilates the absorption changes. With increasing doping level, a decrease in the saturation value of the absorption changes is found. Changes of the Li concentration of the crystal to a more stoichiometric composition via Li indiffusion further decreases the photochromic effect. For sufficient X-ray doses, the saturation value is independent of the radiation energy in the measured range from 50–150 kV. The spectral distribution of the X-ray-induced absorption changes and the slow dark decay indicate that the additional absorption results from an increase of the Cu+ concentration.

In dimensional measurements by laser interferometry, when the accuracy approaches 10(-9)lambda, wavefront aberrations cause systematic variations of the fringe period. This paper illustrates how these effects are modeled and experimentally studied in the measurements of the lattice parameter and the topographical survey of 1 kg Si spheres, which measurements are necessary to "count" atoms and to make it possible to realize the kilogram from the Planck constant value.

The PTB developed a new optical heterodyne interferometer in the context of the European joint research project ?Nanotrace?. A new optical concept using plane-parallel plates and spatially separated input beams to minimize the periodic nonlinearities was realized. Furthermore, the interferometer has the resolution of a double-path interferometer, compensates for possible angle variations between the mirrors and the interferometer optics and offers a minimal path difference between the reference and the measurement arm. Additionally, a new heterodyne phase evaluation based on an analogue to digital converter board with embedded field programmable gate arrays was developed, providing a high-resolving capability in the single-digit picometre range. The nonlinearities were characterized by a comparison with an x-ray interferometer, over a measurement range of 2.2 periods of the optical interferometer. Assuming an error-free x-ray interferometer, the nonlinearities are considered to be the deviation of the measured displacement from a best-fit line. For the proposed interferometer, nonlinearities smaller than ?10 pm were observed without any quadrature fringe correction.

The calculation of the phase correction and the associated uncertainty in the measurement of sphere diameters carried out by optical interferometers is of particular interest. With this view, a Gaussian-beam tracing method was utilized for application to the sphere-interferometer at the National Metrology Institute of Japan as it was used to measure the volume of the Si-28 spheres in the determination of the Avogadro constant. Monte Carlo simulations taking account also of possible interferometer misalignments reveal a correction and its uncertainty of 0.609(139) nm.

The Avogadro constant links the atomic and the macroscopic properties of matter. Since the molar Planck constant is well known via the measurement of the Rydberg constant, it is also closely related to the Planck constant. In addition, its accurate determination is of paramount importance for a definition of the kilogram in terms of a fundamental constant. We describe a new approach for its determination by counting the atoms in 1 kg single-crystal spheres, which are highly enriched with the 28Si isotope. It enabled isotope dilution mass spectroscopy to determine the molar mass of the silicon crystal with unprecedented accuracy. The value obtained, NA = 6.022,140,78(18) × 10(23) mol(-1), is the most accurate input datum for a new definition of the kilogram.

To determine the volume of solid density standards, manufactured as Si-crystal spheres, an optical interferometer is used to measure their diameter at the NMIJ. To support and complement these measurements, the effect of the Gouy phase has been studied analytically and numerically. In measurement, the sphere is placed between the end-mirrors of a Fizeau cavity and the distances between the cavity mirrors and the sphere are measured, as well as the cavity length. The present analysis outlines a model of the interferometer operation and quantifies the Gouy-phase correction in the diameter measurement.

This paper concerns an international research project aimed at determining the Avogadro constant by counting the atoms in an isotopically enriched silicon crystal. The counting procedure was based on the measurement of the molar volume and the volume of an atom in two 1 kg crystal spheres. The novelty was the use of isotope dilution mass spectrometry as a new and very accurate method for the determination of the molar mass of enriched silicon. Because of an unexpected metallic contamination of the sphere surfaces, the relative measurement uncertainty, 3 x 10(-8) N-A, is larger by a factor 1.5 than that targeted. The measured value of the Avogadro constant, N-A = 6.022 140 82(18) x 10(23) mol(-1), is the most accurate input datum for the kilogram redefinition and differs by 16 x 10(-8) N-A from the CODATA 2006 adjusted value. This value is midway between the NIST and NPL watt-balance values.