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Numerical evaluation of the Fresnel diffraction integrals

Numerical evaluation of the Fresnel diffraction integrals,Christine Adelle,L. Ricoand Mark,Nolan P. Confesor,A. Bonifacio

Numerical evaluation of the Fresnel diffraction integrals  
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The simplest and most commonly described examples of diffraction and interference from two- dimensional apertures are those for which the light incident on the apertures and the light passage through the apertures can be described as plane waves. In this limit the diffraction is described as Fraunhofer or far-field diffraction. If one of the aperture dimensions is very small compared to the other - an example would be a slit with a width small compared to its length - the intensity of the light transmitted through the aperture and observed some distance away will vary in a direction perpen- dicualr to the slit width and the light beam, but will be constant in a direction along the slit. Thus the variation in the pattern can be described by only a single dimension and it is called one-dimensional. Fraunhofer diffraction has a particularly simple mathematical description. The amplitude of the diffracted wave can be described as the Fourier transform of the aperture function (which is for a suitable aperture function is one that is a constant equal to 1 over the aperture and 0 elsewhere). It is, of course, the intensity that is observed. Because it is most convenient to treat the amplitude as a complex quantity, the intensity or irradiance is proportional to the amplitude times its complex conjugate. For many examples of diffraction, the light source and the point of observation are sufficiently far from the diffracting aperture that both the incident and diffracted light can be treated as plane waves. If these conditions are met, the diffraction is described as Fraunhofer or far-field diffraction. If the condition that the light source and point of observation are far from the diffracting aperture is not met, so that one cannot employ the approximation of plane waves, then the curvature of the wavefront must be considered in deriving the diffraction pattern. This diffraction is described as Fresnel or near-field diffraction. The mathematics involved in Fresnel diffraction is not as simple as the Fourier transforms of the far-field diffraction. However, a description has been developed in terms of what are called Fresnel zones, that will yield understandable, qualitative results. If more quantitaive answers are needed, special integrals called Fresnel diffraction integrals must be evaluated. In this paper, the Fresnel diffraction integrals for apertures bounded by infinite straight line such as slits will be numerically evaluated. A simple numerical integration method for diffraction integrals to determine the scalar wave function as well as the irradiance, which is based on elementary geometrical considerations in which different portions of the incident wavefront contributes to the diffracted field, will also be presented.
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