J. Carlos López-Vázquez

Propagation of the measurement uncertainty in Fourier transform digital holographic interferometry

Abstract. The derivation of expressions to evaluate the local standard uncertainty of the complex amplitude of the numerically reconstructed field as well as of the phase-change measurements resulting from Fourier- and quasi-Fourier transform digital holographic interferometry is presented. Applying the law of propagation of uncertainty, as defined in the “Guide to the expression of uncertainty in measurement,” to the digital reconstruction of holograms by Fourier transformation and to the subsequent calculation of the phase change between two such reconstructions results in a set of expressions, which allow the evaluation of the uncertainties of the complex amplitude and of the phase change at every pixel of the reconstruction in terms of the measured values and their standard uncertainty in the pixels of the original digital holograms. These expressions are increasingly simplified by first assuming a linear dependence between the squared uncertainty and the local value of the original holograms, and then considering that the object beam is a speckle pattern. We assess the behavior of the method by comparing the predicted standard uncertainty with the sample variance obtained from experiments conducted under repeatability conditions, and find a good agreement between both quantities.

Transient elastic wave propagation and scattering in plates: comparison between pulsed TV-holography measurements and finite element method predictions

Abstract. Pulsed TV-holography (PTVH) can be used for obtaining two-dimensional maps of instantaneous out-of-plane displacements in plates. In particular, our group has demonstrated that scattering patterns generated by the interaction of elastic waves with defects can be measured with PTVH and employed for the characterization of damage in nondestructive inspection of plate structures. Recently, we have succeeded in obtaining a quantitative description of experimental scattering patterns of quasi-Rayleigh (qR) waves produced by holes in harmonic regime using a finite element method (FEM) combined with a two-dimensional scalar wave equation, avoiding the standard and more complex vector approaches based on the rigorous linear elasticity theory. This scheme has been extended here for characterizing equivalent scattering phenomena in transient regime. Simulated scattering patterns, obtained with the scalar FEM, and the corresponding experimental patterns associated to the interaction of qR waves with holes, measured with our specifically developed PTVH system, have been compared. Results have shown that, except for the evaluation of the backscattering coefficient, a reasonable agreement between theory and experiment is obtained in both amplitude and phase, which confirms the feasibility and potential of the proposed scalar approximation for the characterization of experimental transient scattering patterns measured with our PTVH technique.

Numerical modeling and measurement by pulsed television holography of ultrasonic displacement maps in plates with through-thickness defects

We present a novel numerical modeling of ultrasonic Lamb and Rayleigh wave propagation and scattering by through-thickness defects like holes and slots in homogeneous plates, and its experimental verification in both near and far field by a self-developed pulsed TV holography system. In contrast to rigorous vectorial formulation of elasticity theory, our model is based on the 2-D scalar wave equation over the plate surface, with specific boundary conditions in the defects and plate edges. The experimental data include complex amplitude maps of the out-of-plane displacements of the plate surface, obtained by a two-step spatiotemporal Fourier transform method. We find a fair match between the numerical and experimental results, which allows for quantitative characterization of the defects.

Characterization of the Sound Field Generated by an Ultrasonic Transducer in a Solid Medium by Rayleigh-Sommerfeld Back-Propagation of Bulk Acoustic Waves Measured with Double-Pulsed TV Holography

The established approach for the characterization of sound beams is the acquisition of experimental data by measuring the acoustic field pressure distribution in a fluid medium, which is usually done with point detectors (hydrophones, microphones) or arrays of detectors. From these point measurements, important characteristics of the sound field emitted by the transducer (such as the axial and transversal beam profiles, focal length or beam spread) can be derived. When the propagation medium is a solid, these characteristics are normally obtained from the measurement of pulse-echo signals that arise from the interaction of the sound beam with targets placed in the material, such as metal balls embedded in plastics, or flat-bottom or side holes drilled in metallic blocks [1]. These measurements are more difficult to perform and provide a sparser set of data compared to the immersion techniques.

Characterization of defects in plates by two-dimensional ultrasonic displacement maps: comparison between pulsed TV-holography measurements and finite element method predictions

Pulsed TV-holography (PTVH) can be used for obtaining two-dimensional maps of instantaneous out-of-plane displacements in plates. In particular, scattering patterns generated by the interaction of elastic waves with defects can be measured with PTVH and employed for non-destructive inspection and damage detection in plate structures. For quantitative characterization of damage (position, dimensions, orientation, etc.) on this basis, modeling of elastic wave scattering is usually performed in terms of full-vector three-dimensional formulations based on elasticity theory. In this work, a finite element method (FEM) applied to a two-dimensional scalar model based on Helmholtz equation is employed for obtaining a quantitative description of the scattering patterns, avoiding the aforementioned more complex and rigorous standard approach. Simulated scattering patterns are obtained with the scalar FEM assuming harmonic regime and free-stress boundary conditions. The corresponding experimental interaction of narrowband Rayleigh-Lamb waves with artificial defects in plates are measured using our specifically developed PTVH system. In our case, the raw optical phase-difference values are processed by employing a specially developed procedure, based on a two step spatial Fourier transform method, to derive a high quality two-dimensional acoustic field map from which an important part of the noise component has been filtered out. A comparison between filtered experimental maps and FEM simulated maps is developed, considering defects with different sizes in relation to the acoustic wavelength.

Modelling for characterizing defects in plates using two-dimensional maps of instantaneous ultrasonic out-of-plane displacement obtained by pulsed TV-holography

It has been demonstrated that non-destructive inspection of plates can be performed by using two-dimensional maps of instantaneous out-of-plane displacements obtained with a self-developed pulsed TV-holography system. Specifically, the interaction of guided elastic waves with defects produces scattering patterns that contain information about the defects (position, dimensions, orientation, etc.). For quantitative characterization on this basis, modeling of the wave propagation and interaction with the defects is necessary. In fact, the development of models for scattering of waves in plates is yet an active research field in which the most reliable approach is usually based on the rigorous formulation of elasticity theory. By contrast, in this work the capability of a simple two-dimensional scalar model for obtaining a quantitative description of the output two-dimensional maps associated to artificial defects in plates is studied. Some experiments recording the interaction of narrowband Rayleigh waves with artificial defects in aluminum plates are presented, in which the acoustic field is obtained from the TV-holography optical phase-change maps by means of a specially developed two-step spatio-temporal Fourier transform method. For the modeling, harmonic regime and free-stress boundary conditions are assumed. Comparisons between experimental and simulated maps are included for defects with different shapes.

Quantitative analysis of the agreement between scalar finite element simulation and pulsed TV-holography detection of the scattering of Rayleigh-Lamb waves in plates

The instantaneous out-of-plane displacement two-dimensional (2-D) maps associated to the scattering generated by the interaction of Rayleigh-Lamb waves with defects in plate structures can be measured using pulsed TV-holography (PTVH) and employed to characterize damage in non-destructive inspection applications. On the basis of visual comparisons we have shown previously that, except for the amplitude in the backscattering zone, a reasonable description of the measured experimental scattering patterns produced by holes both in harmonic and transient regimes can be obtained using the finite element method (FEM) combined with a 2-D model based on the scalar wave equation. In this work a systematic quantitative analysis of the agreement between FEM simulated maps and filtered experimental PTVH maps is developed considering both the spatial distribution of the local (pixel-wise) error in amplitude and phase and the corresponding global (averaged) errors over different areas in the 2-D image of the acoustic field. Changes produced in the experimental values by the speckle noise and variations introduced in the numerical values by the uncertainty in the characterization of the incident acoustic wave and the shape and position of the hole are characterized in order to obtain the net value of the error between theory and experiment.

Wide-Field, Low-Cost Mapping of Power Ultrasound Fields in Water by Time-Average Moiré Deflectometry

Characterization of acoustic fields in the power ultrasound range in water is a common problem in diverse application areas like sonochemistry, biomedicine, or industrial cleaning. Different approaches exist for the visualization and mapping of such acoustic fields, being a classical solution the mechanical scanning with pressure sensors (typically, hydrophones) over a grid of points [1]. For high intensity ultrasound, the analysis of bubbles trajectory has also been employed [2]. Alternative optical techniques are the scanning of a pointwise sensor (PIV, LDV) [3, 4], and also full field techniques like deflectometry or schlieren [5], smooth wavefront interferometry [6], holographic interferometry [7], ESPI and similar interferometric speckle techniques [4] or light diffraction tomography [8].

Characterization of Transducer Performance and Narrowband Transient Ultrasonic Fields in Metals by Rayleigh-Sommerfeld Backpropagation of Compression Acoustic Waves Measured with Double-Pulsed Tv Holography

This article presents a method aimed at the characterization of the narrowband transient acoustic field radiated by an ultrasonic plane transducer into a homogeneous, isotropic and optically opaque prismatic solid, and the assessment of the performance of the acoustic source. The method relies on a previous technique based on the full-field optical measurement of an acoustic wavepacket at the surface of a solid and its subsequent numerical backpropagation within the material. The experimental results show that quantitative transversal and axial profiles of the complex amplitude of the beam can be obtained at any plane between the measurement and excitation surfaces. The reconstruction of the acoustic field at the transducer face, carried out on a defective transducer model, shows that the method could also be suitable for the nondestructive testing of the performance of ultrasonic sources. In all cases, the measurements were performed with the transducer working under realistic loading conditions.

Determination of the frequency spectrum of Lamb waves from a sequence of maps of the instantaneous acoustic displacement obtained with TV holography

A novel approach to an established method to calculate the frequency spectrum of Lamb waves is introduced. Lamb wavetrains are generated with the wedge method in aluminium plates, and a sequence of instantaneous acoustic out-of-plane displacement fields at the plate surface is measured with a self-developed double-pulsed TV holography system. This is achieved by emitting two laser pulses synchronized with the piezoelectric transducer that generates the waves and conveniently delayed. As a result, a 2D optical phase-change map, proportional to the aforementioned acoustic displacement field, is obtained for the instant of emission of the second laser pulse. Then, a series of maps is acquired under repeatability conditions by successively delaying the second laser pulse, so that the resulting sequence of maps records successive instants of the propagation of the wavetrain. The frequency spectrum of the wavetrain is obtained from a 3D spatio-temporal Fourier transform of the whole sequence of optical phase-change maps, as the relation between the temporal frequency and the spatial frequency along the principal propagation direction of the wavetrain. The use of a 3D Fourier transform permits to calculate the frequency spectrum regardless of the propagation direction of the wavetrain, with non-perfectly plane wavefronts and also increases the signal to noise ratio with respect to the 2D spatio-temporal Fourier transform approach. Experiments show that the resulting branches for the Lamb modes existing in the wavetrain are in agreement with the theoretical frequency spectrum of Lamb waves in aluminium.