On September 14, 2015 at 09:50:45 UTC the two detectors of the Laser Interferometer Gravitational-Wave Observatory simultaneously observed a transient gravitational-wave signal. The signal sweeps upwards in frequency from 35 to 250 Hz with a peak gravitational-wave strain of 1.0×10(-21). It matches the waveform predicted by general relativity for the inspiral and merger of a pair of black holes and the ringdown of the resulting single black hole. The signal was observed with a matched-filter signal-to-noise ratio of 24 and a false alarm rate estimated to be less than 1 event per 203,000 years, equivalent to a significance greater than 5.1σ. The source lies at a luminosity distance of 410(-180)(+160)  Mpc corresponding to a redshift z=0.09(-0.04)(+0.03). In the source frame, the initial black hole masses are 36(-4)(+5)M⊙ and 29(-4)(+4)M⊙, and the final black hole mass is 62(-4)(+4)M⊙, with 3.0(-0.5)(+0.5)M⊙c(2) radiated in gravitational waves. All uncertainties define 90% credible intervals. These observations demonstrate the existence of binary stellar-mass black hole systems. This is the first direct detection of gravitational waves and the first observation of a binary black hole merger.

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The Observation of Gravitational Waves from a Binary Black Hole Merger

A fast-Fourier-transform method of topography and interferometry is proposed. By computer processing of a noncontour type of fringe pattern, automatic discrimination is achieved between elevation and depression of the object or wave-front form, which has not been possible by the fringe-contour-generation techniques. The method has advantages over moire topography and conventional fringe-contour interferometry in both accuracy and sensitivity. Unlike fringe-scanning techniques, the method is easy to apply because it uses no moving components.

Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry

A self-scanned 1024 element photodiode array and minicomputer are used to measure the phase (wavefront) in the interference pattern of an interferometer to lambda/100. The photodiode array samples intensities over a 32 x 32 matrix in the interference pattern as the length of the reference arm is varied piezoelectrically. Using these data the minicomputer synchronously detects the phase at each of the 1024 points by a Fourier series method and displays the wavefront in contour and perspective plot on a storage oscilloscope in less than 1 min (Bruning et al. Paper WE16, OSA Annual Meeting, Oct. 1972). The array of intensities is sampled and averaged many times in a random fashion so that the effects of air turbulence, vibrations, and thermal drifts are minimized. Very significant is the fact that wavefront errors in the interferometer are easily determined and may be automatically subtracted from current or subsequent wavefrots. Various programs supporting the measurement system include software for determining the aperture boundary, sum and difference of wavefronts, removal or insertion of tilt and focus errors, and routines for spatial manipulation of wavefronts. FFT programs transform wavefront data into point spread function and modulus and phase of the optical transfer function of lenses. Display programs plot these functions in contour and perspective. The system has been designed to optimize the collection of data to give higher than usual accuracy in measuring the individual elements and final performance of assembled diffraction limited optical systems, and furthermore, the short loop time of a few minutes makes the system an attractive alternative to constraints imposed by test glasses in the optical shop.

Digital wavefront measuring interferometer for testing optical surfaces and lenses

Phase shifting algorithms for fringe projection profilometry: A review

Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques

In this paper, a novel accurate deformation distribution measurement technique by using sampling moire method is proposed. The basic principle and an experimental result of a steel beam in symmetric three-point bending are reported. In this method, the measurement area of a target is attached with an adhesive tape of a known pitch grating firstly. An ordinary CCD camera is installed on a fixed point to record the image during deformation. The captured image is analyzed by performing easy image processing, i.e., thinning-out and linear interpolation, to obtain the multiple phase-shifted moire patterns. Then, the phase distribution of the moire pattern can be calculated using phase-shifting method. Finally, the deformation distribution is calculated by the grating pitch times the phase difference of before deformation and after deformation. The experimental results in symmetric three-point bending test show that the displacement of the steel beam at loading point agree well with those obtained by an accurate displacement sensor. The average error of displacement measurement is less than 4 μm when 2 mm grating pitch is used, and it corresponds to 1/500 of the grating pitch accuracy. This indicates that noncontact deformation distribution measurement is possible by simple and easy procedure with high accuracy, high speed, and low cost for the structural evaluation of infrastructures.

Sampling Moiré Method for Accurate Small Deformation Distribution Measurement

Measuring accurate displacement distributions for large-scale structures is an important issue and a very challenging task. Recently, a simple and accurate phase measurement technique called sampling moire method [Exp Mech 50–4:501–508, (2010)] has been developed for small-displacement distribution measurements. In this method, the phase distribution of moire fringes can be analyzed from a single grating image by simultaneously performing down-sampling image processing and intensity-interpolation to generate multiple phase-shifted moire fringe images. In addition, the phase of the original grating can also be obtained from the phase of the moire fringe by adding the phase of the sampling grating. In this study, the measurement accuracy of the sampling moire method was analyzed through computer simulations and a displacement measurement experiment. Four factors of the sampling moire method were investigated, including the sampling pitch, the order of the intensity-interpolation, random noise, and the form of grating. The results show that determining the optimal sampling pitch is an important factor for obtaining better results but it is not critical. In addition, a practical application of the sampling moire method is presented that involves a deflection measurement on a 10-meter-long crane. The experimental results demonstrate that submillimeter deflections of the crane can be successfully detected.

Accuracy of the Sampling Moiré Method and its Application to Deflection Measurements of Large-Scale Structures

A camera based on the digital micromirror device (DMD) technology has been previously developed. In this optical system, the correspondence of each mirror of the DMD to each pixel of the CCD cannot readily be done since the pixel sizes of the DMD and the CCD are very small. An accurate pixel-to-pixel correspondence adjustment in the DMD camera by means of the phase-shifting moire method is proposed. To perform high accurate adjustment of the optical system, the phase distribution of a moire fringe pattern is analyzed when the CCD pixels and the DMD mirrors have a mismatch and/or misalignment with each other. This technique does not need a complicated setting or complex image processing to generate the moire fringe pattern, and it needs only one captured image. In the adjustment experiment, the proposed method provided very accurate adjustment whose error was less than 1/25 pixel. An experiment of phase analysis to demonstrate the usefulness was performed.

Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method.

Recently, a rapid and accurate single-shot phase measurement technique called the sampling moire method has been developed for small-displacement distribution measurements. In this study, the theoretical phase error of the sampling moire method caused by linear intensity interpolation in the case of a mismatch between the sampling pitch and the original grating pitch is analyzed. The periodic phase error is proportional to the square of the spatial angular frequency of the moire fringe. Moreover, an effective phase compensation methodology is developed to reduce the periodic phase error. Single-shot phase analysis can perform accurately even when the sampling pitch is not matched to the original grating pitch exactly. The primary simulation results demonstrate the effectiveness of the proposed phase compensation methodology.

Theoretical error analysis of the sampling moiré method and phase compensation methodology for single-shot phase analysis

Phase-measuring profilometry is an accurate and effective technique for performing three-dimensional (3D) shape and deformation measurements of diffuse objects by fringe projection. However, phase analysis cannot be performed in underexposed or overexposed areas of the detector when an object with wide reflectance is measured. A novel intensity range extension method using a digital micromirror device (DMD) camera is proposed. In the optics of the DMD camera, each pixel of the CCD corresponds exactly to each mirror of the DMD. The phase-shifted fringe patterns with high contrast can be easily captured by programming an inverse intensity pattern that depends on the reflectance of the object. Our method can provide a wider intensity range and higher accuracy for 3D shape measurement than other conventional methods in both underexposed and overexposed areas. The measurements of a replica of a metallic art object and a flat plane are analyzed experimentally to verify the effectiveness of our method. In the experiment, the percentage of invalid points due to underexposure and overexposure can be reduced from 20% to 1%.

Shien Ri

Papers

Sampling Moiré Method for Accurate Small Deformation Distribution Measurement

Accuracy of the Sampling Moiré Method and its Application to Deflection Measurements of Large-Scale Structures

Accurate pixel-to-pixel correspondence adjustment in a digital micromirror device camera by using the phase-shifting moiré method.

Theoretical error analysis of the sampling moiré method and phase compensation methodology for single-shot phase analysis

Intensity range extension method for three-dimensional shape measurement in phase-measuring profilometry using a digital micromirror device camera.