Comparison of the Kirchhoff and the Rayleigh–Sommerfeld Theories of Diffraction at an Aperture

Wolf, Emil; Marchand, Erich W.

doi:10.1364/josa.54.000587

Cited by 81 publications

(27 citation statements)

References 5 publications

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“…It is well known that Sommerfeld's diffraction theory integral [e.g., Pearson et al, 1969) used in conjunction with some of Kirchhoff's approximations yields accurate results if the aperture is large in comparison with a wavelength and the diffracted field is observed far from the aperture [Silver, 1962;Wolf and Marchand, 1964;Marchand and Wolf. 1966).…”

Section: Slit Diffraction Method: Theoretical Background and Experimementioning

confidence: 99%

Dilatancy and precursory bulging along incipient fracture zones in uniaxially compressed westerly granite

Liu¹,

Livanos²

1976

J. Geophys. Res.

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The existence of precursory bulge along an incipient fracture zone in a uniaxially compressed Westerly granite sample has been investigated by two optical methods. The first method is the method of slit diffraction. The cross section of a cylindrical rock sample is monitored by bringing two straightedges next to the rock sample to form two slits (each slit being formed between one straightedge and one side of the rock sample) and by illuminating alternately the two slits with a collimated laser beam. The FraunhoFer diffraction pattern is recorded on film in the direction perpendicular to the straightedge and can be interpreted as the absolute value squared of the one-dimensional spatial Fourier transform of the slit under certain conditions, thereby providing a simple method of magnification of the rock surface geometry. The conditions under which the Fraunhofer diffraction pattern can be interpreted as the absolute value squared of a one-dimensional Fourier transform are related to the radius of the rock sample, width of the slit, position of the recording film plane, and nature of deformation of the rock surface and are presented in this paper. The films are digitized by a microdensitometer. The data are analyzed by digital filtering and interpolation techniques to give a strain resolution of 10-•. During a test with the slit diffraction method, strain inhomogeneities in terms of local bulges indicative of incipient failure zones were found to develop at -92% of the uniaxial compressive strength, and their propagation is traced at 2.66-s intervals until failure. Local strains in the incipient failure zones are of the order of 10-• before failure takes place. Because of the large amplitude of the strain inhomogeneity prior to failure recorded by the slit diffraction method, we then tried the faster method of recording without magnification by a motion picture camera. In the second test a precursory bulge in the middle of the sample first appeared at -3.75 s prior to failure at a load of >99.7% of the uniaxial compressive strength. The bulge developed rapidly in successive frames until eventually a failure plane passed through this sharp bulge. The results from both tests demonstrate the formation of a concentrated weak zone as a result of the interaction and coalescence among the microcracks in the final stage of the test, which then develop into fracture zones. The bulging is the result of accentuated deformation in the weak zone because of its reduced deformation moduli. It is considered that the local bulge and orientation of the fracture zones in the first test were controlled by the stress concentration at the sample-load block interface, whereas those in the second test were controlled mainly by the inhomogeneity in the material properties within the sample. The precursor times of both tests do not fit into the empirical relationship between precursor time and fault dimension as derived from earthquakes and mine rock bursts. The precursor times of these tests are too long by 3 orders of magnitude in compa...

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Section: Slit Diffraction Method: Theoretical Background and Experimementioning

confidence: 99%

Dilatancy and precursory bulging along incipient fracture zones in uniaxially compressed westerly granite

Liu¹,

Livanos²

1976

J. Geophys. Res.

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“…The Rayleigh-Sommerfield integral model has been used for these simulations, together with the piston-transducer approximation. [24] This allows the spatial distribution of the transmitter and detector fields to be determined at a particular frequency, and hence the sensitivity of the transducers to a certain location in 3D space to be determined. As an example, simulation results are shown in Figure 4 for operation at a depth of 5 mm into a polyurethane sample, where the combined sensitivity level of the two probes has been plotted at 300 kHz.…”

Section: Apparatus and Methodsmentioning

confidence: 99%

Ultrasonic NDE of thick polyurethane flexible riser stiffener material

Mohamed

Hutchins

Davis

et al. 2016

Nondestructive Testing and Evaluation

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a school of engineering, university of Warwick, coventry, uK; b dipartimento di ingegneria, università degli studi di Perugia, terni, italy ABSTRACT Ultrasonic signals at frequencies below 1 MHz have been used for the NDE of thick polymer samples. Coded signals such as chirps and bipolar Golay codes, together with pulse compression and signal processing, have been used to penetrate into thick sections of attenuating polypropylene riser stiffener material, using piezocomposite transducers to provide the required bandwidth. It is shown that this approach can be used to detect manufacturing defects such as air bubbles in flexible riser stiffeners, which might compromise their performance in the offshore oil and gas environments.

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“…1,2 The Rayleigh-Sommerfeld diffraction formula has been proved to yield exact evaluations of the light fields by methods of Fourier treatment [3][4][5] and gives correct results for both far-field and near-field diffraction. 6,7 Unfortunately, in most cases, these formulas cannot be solved analytically.…”

Section: Introductionmentioning

confidence: 99%

“…12 The DI method computes the diffraction integrals in the spatial domain by means of numerical integration, which can be treated as a linear convolution and can be effectively calculated by means of a FFT and an inverse FFT (IFFT). [13][14][15] Although numerical calculation of both the AS and the DI methods has been discussed extensively in the literature, 4,6,7,[12][13][14][15][16] there is still some confusion to be cleared up. First, the implementation of the diffraction integral by means of convolution and a FFT was not clearly presented.…”

Section: Introductionmentioning

confidence: 99%

Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula

2006

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The numerical calculation of the Rayleigh-Sommerfeld diffraction integral is investigated. The implementation of a fast-Fourier-transform (FFT) based direct integration (FFT-DI) method is presented, and Simpson's rule is used to improve the calculation accuracy. The sampling interval, the size of the computation window, and their influence on numerical accuracy and on computational complexity are discussed for the FFT-DI and the FFT-based angular spectrum (FFT-AS) methods. The performance of the FFT-DI method is verified by numerical simulation and compared with that of the FFT-AS method.

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Comparison of the Kirchhoff and the Rayleigh–Sommerfeld Theories of Diffraction at an Aperture

Cited by 81 publications

References 5 publications

Dilatancy and precursory bulging along incipient fracture zones in uniaxially compressed westerly granite

Dilatancy and precursory bulging along incipient fracture zones in uniaxially compressed westerly granite

Ultrasonic NDE of thick polyurethane flexible riser stiffener material

Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula

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