Diffracted field by an arbitrary aperture
01 Jan 1999-American Journal of Physics (American Association of Physics Teachers)-Vol. 67, Iss: 1, pp 87-92
TL;DR: In this paper, the 2D integration is reduced to a 1D parametric integration over the perimeter of the aperture, similar to the Rubinowicz representation of the Kirchhoff diffraction integral.
Abstract: Diffraction patterns of apertures on screens uniformly illuminated are standard calculations in undergraduate optics and acoustics courses. These calculations imply two-dimensional (2-D) integrations which are often performed in the far zone, at moderate angles of diffraction, i.e., using Fresnel and Fraunhofer approximations. In this note, the 2-D integration is reduced to a 1-D parametric integration over the perimeter of the aperture—resembling the Rubinowicz’s representation of the Kirchhoff diffraction integral—which allows numerical evaluation with few computational resources. The proposed formula allows mathematically exact calculation of the near-field, in the context of scalar wave theory. Explicit calculations for circular and elliptical apertures are shown.
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TL;DR: The implementation of a fast-Fourier-transform based direct integration method is presented, and Simpson's rule is used to improve the calculation accuracy, and the Rayleigh-Sommerfeld diffraction integral integral is investigated.
Abstract: 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.
263 citations
TL;DR: The diffraction theory of Kirchhoff is reinterpreted and a new form of a surface diffraction integral is developed by using the axioms of the modified theory of physical optics, which leads to the exact scattered fields by conducting bodies.
Abstract: The diffraction theory of Kirchhoff is reinterpreted and a new form of a surface diffraction integral is developed by using the axioms of the modified theory of physical optics, which leads to the exact scattered fields by conducting bodies. The new integral is arranged according to the interpretation of Young, and the diffracted waves are expressed in terms of a line integral. The method is applied to the diffraction problem by a semi-infinite edge contour.
42 citations
TL;DR: In this paper, the numerical determination of diffraction patterns using plane wave spectrum decomposition (PWS) is investigated and a simple formula for sampling selection for error-free numerical computation is proposed and its applicability is discussed.
38 citations
TL;DR: In this article, an analytic approach to the analysis of these apodization functions is presented, which is used to develop a tolerance analysis suitable for use in designing practical starshades.
Abstract: External occulters, otherwise known as starshades, have been proposed as a solution to one of the highest priority yet technically vexing problems facing astrophysics—the direct imaging and characterization of terrestrial planets around other stars. New apodization functions, developed over the past few years, now enable starshades of just a few tens of meters diameter to occult central stars so efficiently that the orbiting exoplanets can be revealed and other high-contrast imaging challenges addressed. In this paper, an analytic approach to the analysis of these apodization functions is presented. It is used to develop a tolerance analysis suitable for use in designing practical starshades. The results provide a mathematical basis for understanding starshades and a quantitative approach to setting tolerances.
35 citations
TL;DR: In this paper, an analytic approach to analysis of these apodization functions is presented, which is used to develop a tolerance analysis suitable for use in designing practical starshades.
Abstract: External occulters, otherwise known as starshades, have been proposed as a solution to one of the highest priority yet technically vexing problems facing astrophysics - the direct imaging and characterization of terrestrial planets around other stars. New apodization functions, developed over the past few years, now enable starshades of just a few tens of meters diameter to occult central stars so efficiently that the orbiting exoplanets can be revealed and other high contrast imaging challenges addressed. In this paper an analytic approach to analysis of these apodization functions is presented. It is used to develop a tolerance analysis suitable for use in designing practical starshades. The results provide a mathematical basis for understanding starshades and a quantitative approach to setting tolerances.
34 citations
Cites methods from "Diffracted field by an arbitrary ap..."
...Dubra and Ferrari (1999) published a paper entitled “Diffracted field by an arbitrary aperture” in which they integrated the Kirchoff formulation of diffraction theory by means of a Green’s function approach and converted the two-dimensional integral to a onedimensional parametric integral....
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TL;DR: In this article, a comparison of the predictions relating to the field diffracted at an aperture, based on Kirchhoff's theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR), is made.
Abstract: With a view to elucidating the effect of a well-known mathematical inconsistency in Kirchhoff’s diffraction theory, a comparison is made of the predictions relating to the field diffracted at an aperture, based on Kirchhoff’s theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR). It is shown that, when the incident wave is plane or spherical, the difference δ = UK−UR represents a boundary wave, i.e., a wave which may be thought of as originating at each point of the edge of the aperture. It is shown further that, when the linear dimensions of the aperture are large compared with the wavelength, the boundary values of δ in the plane of the aperture change very rapidly and almost periodically from point to point, with the mean period close to the wavelength of the incident radiation. This result is shown to imply that if the linear dimensions of the aperture are large compared with the wavelength, the two theories predict essentially the same behavior for the diffracted field in the far zone, at moderate angles of diffraction.
79 citations
TL;DR: In this paper, it is shown that the axial focal point of the converged spherical wave falls inside, at, or outside the geometrical focal point according to the angular semi-aperture θm of the lens.
Abstract: Rayleigh’s diffraction integral is solved in closed form as regards all axial points when a divergent or a convergent spherical wave is specified as the electromagnetic disturbance incident upon a circular aperture or obstacle. Diffraction of divergent waves is treated briefly. The method is applied more fully to the diffraction of convergent waves by circular apertures. It is shown that the axial “focal point” of the converged spherical wave falls inside, at, or outside the geometrical focal point according as the angular semiaperture θm of the lens is less than, equal to, or greater than a particular angle that falls near 70.5°. The magnitudes of the departures of the focal point from the geometrical focal point are illustrated by examples for both the radar and optical regions.
67 citations
TL;DR: In this article, the physical reality of the edge-diffracted wave in the boundary wave theory of diffraction is described, and the basic philosophy is simple and the experimental evidence is strong.
Abstract: An experiment that deals with the physical reality of the edge‐diffracted wave in the boundary wave theory of diffraction is described in this article. The basic philosophy is simple and the experimental evidence is strong.
30 citations