Consistent Formulation of Kirchhoff’s Diffraction Theory*

Marchand, Erich W.; Wolf, Emil

doi:10.1364/josa.56.001712

Cited by 55 publications

(22 citation statements)

References 11 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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“…At any rate, it isn't clear that the realist would be happy with turning to these two theories, since as a matter of experimental fact Kirchhoff's theory outperforms both Rayleigh-Sommerfeld theories in many circumstances. That is, in many circumstances the inconsistent theory is more accurate than the theories which are in a sense its nearest consistent alternatives, although the two mutually incompatible alternatives each hold sway over certain ranges of parameter values (Totzeck 1990 Marchand and Wolf (1966) show that Kirchhoff's diffraction formula (1) In the present paper we show that the inconsistency in Kirchhoff's diffraction theory is only apparent (Marchand andWolf 1966, 1713).…”

Section: The Inconsistency Puzzlementioning

confidence: 51%

“…The consistency of the Kirchhoff diffraction theory was first demonstrated by Marchand and Wolf (1966). (27) Prima facie, these remarks are rather puzzling: the derivation of Kirchhoff's formula makes essential use of boundary assumptions which are simply not recovered when the formula is evaluated at the boundary!…”

Section: The Inconsistency Puzzlementioning

confidence: 99%

Miraculous Success? Inconsistency and Untruth in Kirchhoff’s Diffraction Theory

Saatsi¹,

Vickers²

2011

The British Journal for the Philosophy of Science

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“…Thus, the Helmholtz-Kirchhoff diffraction integral [3,4] can be rewritten as: The Helmholtz-Kirchhoff diffraction integral expresses the disturbance at a given point in terms of its values and the values of its first derivative on the surrounding volumes A, B and C. In Equation (B.20), U is the disturbance to be determined and G ¼ (exp(jks))/s is the auxiliary Green's function.…”

Section: B12 Green's Theoremmentioning

confidence: 99%

“…If we further consider that the aperture is illuminated by a spherical diverging wave, the disturbance U, the auxiliary function G and their first derivatives are as follows: It has been experimentally proven [1,2] that Fresnel-Kirchhoff diffraction theory predicts the behavior of the diffracted fields with excellent accuracy [3,4]. Assumption #1: on surface B, we have U ¼ 0 and dU/dn ¼ 0 (i.e.…”

Section: B12 Green's Theoremmentioning

confidence: 99%